diff --git a/FILE-CONVENTIONS.md b/FILE-CONVENTIONS.md index 0467bccfbc..7dba35bc5a 100644 --- a/FILE-CONVENTIONS.md +++ b/FILE-CONVENTIONS.md @@ -35,27 +35,42 @@ Every file should begin with a header in the following format: # The title of the file ``` -and immediately after this, any option pragmas, then the module declaration then -public imports, if any, should be declared. E.g. +Directly after the header, include an Agda code block containing + +- any option pragmas, +- the main module declaration, +- any public import statements + +in this order. E.g. ````md ```agda -{-# OPTIONS --safe #-} +{-# OPTIONS ... #-} -module foundation.dependent-pair-types where +module ... where -open import foundation-core.dependent-pair-types public +open import ... public ``` ```` ### Imports block -After the module declaration, include an Agda code block of all module imports -starting with `
Imports` and ending with -`
`. This Agda block should only contain module imports. Do not import -further modules later in the file. On the documentation pages, this Agda imports -block will be hidden by default, but it can be revealed by clicking on the -_Imports_ link. +After the module declaration, include an Agda code block of all non-public +module imports starting with `
Imports` and ending +with `
`. This block should only contain module imports and there +should have no further import statements after it. In the rendered markdown, the +contents of this block will be hidden by default, but can be revealed by +clicking on _Imports_. + +````md +
Imports + +```agda +open import ... +``` + +
+```` ### Sections and headings @@ -65,10 +80,10 @@ subsubsections. Use `##` headings for the main sections of the file and reserve `## Definitions`, and `## Properties`. Occasionally, you might include a section like `## Examples` or `## Theorem`, based on the purpose of the file. -Ideally, the first section of a file explains the idea, the second section -proceeds to give the main definition that is the focus of the file, then the -third section proceeds possibly with examples or by deriving basic properties of -the defined concept. +Ideally, depending on the purpose of the file, the first section explains the +main idea, the second section proceeds to give the main definition that is the +focus of the file, then the third section proceeds possibly with examples or by +deriving basic properties of the defined concept. #### Subsections @@ -105,6 +120,6 @@ contents of the file. - An instructive example of a file with the expected structure is [`foundation.cantor-schroder-bernstein-escardo`](https://raw.githubusercontent.com/UniMath/agda-unimath/master/src/foundation/cantor-schroder-bernstein-escardo.lagda.md). -Please note that some conventions above are enforced by our `pre-commit` hooks. -You can read more about them in our +Please note that some of the conventions above are enforced by our `pre-commit` +hooks. You can read more about them in our [installation guide](HOWTO-INSTALL.md#pre-commit-hooks). diff --git a/src/elementary-number-theory/powers-of-two.lagda.md b/src/elementary-number-theory/powers-of-two.lagda.md index c3dde3d93a..ca73b53c07 100644 --- a/src/elementary-number-theory/powers-of-two.lagda.md +++ b/src/elementary-number-theory/powers-of-two.lagda.md @@ -16,16 +16,16 @@ open import elementary-number-theory.natural-numbers open import elementary-number-theory.parity-natural-numbers open import elementary-number-theory.strong-induction-natural-numbers +open import foundation.dependent-pair-types +open import foundation.equality-cartesian-product-types +open import foundation.functions open import foundation.split-surjective-maps open import foundation.universe-levels open import foundation-core.cartesian-product-types open import foundation-core.coproduct-types -open import foundation-core.dependent-pair-types open import foundation-core.empty-types -open import foundation-core.equality-cartesian-product-types open import foundation-core.equivalences -open import foundation-core.functions open import foundation-core.identity-types open import foundation-core.injective-maps ``` diff --git a/src/elementary-number-theory/type-arithmetic-natural-numbers.lagda.md b/src/elementary-number-theory/type-arithmetic-natural-numbers.lagda.md index 820686c868..b0812df41d 100644 --- a/src/elementary-number-theory/type-arithmetic-natural-numbers.lagda.md +++ b/src/elementary-number-theory/type-arithmetic-natural-numbers.lagda.md @@ -15,6 +15,8 @@ open import elementary-number-theory.natural-numbers open import elementary-number-theory.parity-natural-numbers open import elementary-number-theory.powers-of-two +open import foundation.dependent-pair-types +open import foundation.functions open import foundation.functoriality-cartesian-product-types open import foundation.functoriality-coproduct-types open import foundation.iterating-functions @@ -26,10 +28,8 @@ open import foundation.unit-type open import foundation-core.cartesian-product-types open import foundation-core.coproduct-types -open import foundation-core.dependent-pair-types open import foundation-core.empty-types open import foundation-core.equivalences -open import foundation-core.functions open import foundation-core.identity-types open import foundation-core.injective-maps open import foundation-core.negation diff --git a/src/foundation-core.lagda.md b/src/foundation-core.lagda.md index 06b0da59a8..64f0b316a2 100644 --- a/src/foundation-core.lagda.md +++ b/src/foundation-core.lagda.md @@ -5,52 +5,36 @@ ```agda module foundation-core where -open import foundation-core.0-maps public open import foundation-core.1-types public -open import foundation-core.automorphisms public open import foundation-core.cartesian-product-types public open import foundation-core.coherently-invertible-maps public -open import foundation-core.commuting-3-simplices-of-homotopies public -open import foundation-core.commuting-3-simplices-of-maps public open import foundation-core.commuting-cubes-of-maps public open import foundation-core.commuting-squares-of-maps public -open import foundation-core.commuting-triangles-of-homotopies public open import foundation-core.commuting-triangles-of-maps public -open import foundation-core.cones-over-cospans public open import foundation-core.constant-maps public open import foundation-core.contractible-maps public open import foundation-core.contractible-types public open import foundation-core.coproduct-types public -open import foundation-core.cospans public open import foundation-core.decidable-propositions public -open import foundation-core.dependent-pair-types public open import foundation-core.diagonal-maps-of-types public open import foundation-core.discrete-types public open import foundation-core.embeddings public open import foundation-core.empty-types public open import foundation-core.endomorphisms public -open import foundation-core.equality-cartesian-product-types public open import foundation-core.equality-dependent-pair-types public -open import foundation-core.equality-fibers-of-maps public open import foundation-core.equivalence-induction public open import foundation-core.equivalence-relations public open import foundation-core.equivalences public -open import foundation-core.faithful-maps public open import foundation-core.fibers-of-maps public open import foundation-core.function-extensionality public -open import foundation-core.functions public open import foundation-core.functoriality-dependent-function-types public open import foundation-core.functoriality-dependent-pair-types public -open import foundation-core.functoriality-fibers-of-maps public open import foundation-core.functoriality-function-types public -open import foundation-core.fundamental-theorem-of-identity-types public open import foundation-core.homotopies public -open import foundation-core.identity-systems public open import foundation-core.identity-types public open import foundation-core.injective-maps public open import foundation-core.involutions public open import foundation-core.logical-equivalences public -open import foundation-core.morphisms-cospans public open import foundation-core.negation public open import foundation-core.path-split-maps public open import foundation-core.propositional-maps public @@ -61,15 +45,11 @@ open import foundation-core.sections public open import foundation-core.sets public open import foundation-core.singleton-induction public open import foundation-core.small-types public -open import foundation-core.subtype-identity-principle public open import foundation-core.subtypes public open import foundation-core.truncated-maps public open import foundation-core.truncated-types public open import foundation-core.truncation-levels public -open import foundation-core.type-arithmetic-cartesian-product-types public -open import foundation-core.type-arithmetic-dependent-pair-types public open import foundation-core.univalence public open import foundation-core.universal-property-pullbacks public open import foundation-core.universal-property-truncation public -open import foundation-core.universe-levels public ``` diff --git a/src/foundation-core/0-maps.lagda.md b/src/foundation-core/0-maps.lagda.md deleted file mode 100644 index 46e5e5e53c..0000000000 --- a/src/foundation-core/0-maps.lagda.md +++ /dev/null @@ -1,156 +0,0 @@ -# 0-Maps - -```agda -module foundation-core.0-maps where -``` - -
Imports - -```agda -open import foundation-core.dependent-pair-types -open import foundation-core.fibers-of-maps -open import foundation-core.functions -open import foundation-core.functoriality-dependent-pair-types -open import foundation-core.homotopies -open import foundation-core.sets -open import foundation-core.truncated-maps -open import foundation-core.truncation-levels -open import foundation-core.universe-levels -``` - -
- -## Definition - -Maps `f : A → B` of which the fibers are sets, i.e., 0-truncated types, are -called 0-maps. We will show in `foundation-core.faithful-maps` that a map f is a -0-map if and only if f is faithful, i.e., f induces embeddings on identity -types. - -```agda -module _ - {l1 l2 : Level} - where - - is-0-map : {A : UU l1} {B : UU l2} → (A → B) → UU (l1 ⊔ l2) - is-0-map {A} {B} f = (y : B) → is-set (fib f y) - - 0-map : (A : UU l1) (B : UU l2) → UU (l1 ⊔ l2) - 0-map A B = Σ (A → B) is-0-map - - map-0-map : {A : UU l1} {B : UU l2} → 0-map A B → A → B - map-0-map = pr1 - - is-0-map-map-0-map : - {A : UU l1} {B : UU l2} (f : 0-map A B) → is-0-map (map-0-map f) - is-0-map-map-0-map = pr2 -``` - -## Properties - -### Projections of families of sets are `0`-maps - -```agda -module _ - {l1 l2 : Level} {A : UU l1} - where - - abstract - is-0-map-pr1 : - {B : A → UU l2} → ((x : A) → is-set (B x)) → is-0-map (pr1 {B = B}) - is-0-map-pr1 {B} H x = - is-set-equiv (B x) (equiv-fib-pr1 B x) (H x) - - pr1-0-map : - (B : A → Set l2) → 0-map (Σ A (λ x → type-Set (B x))) A - pr1 (pr1-0-map B) = pr1 - pr2 (pr1-0-map B) = is-0-map-pr1 (λ x → is-set-type-Set (B x)) -``` - -### `0`-maps are closed under homotopies - -```agda -module _ - {l1 l2 : Level} {A : UU l1} {B : UU l2} {f g : A → B} (H : f ~ g) - where - - is-0-map-htpy : is-0-map g → is-0-map f - is-0-map-htpy = is-trunc-map-htpy zero-𝕋 H -``` - -### `0`-maps are closed under composition - -```agda -module _ - {l1 l2 l3 : Level} {A : UU l1} {B : UU l2} {X : UU l3} - where - - is-0-map-comp : - (g : B → X) (h : A → B) → - is-0-map g → is-0-map h → is-0-map (g ∘ h) - is-0-map-comp = is-trunc-map-comp zero-𝕋 - - is-0-map-comp-htpy : - (f : A → X) (g : B → X) (h : A → B) (H : f ~ (g ∘ h)) → - is-0-map g → is-0-map h → is-0-map f - is-0-map-comp-htpy = is-trunc-map-comp-htpy zero-𝕋 -``` - -### If a composite is a 0-map, then so is its right factor - -```agda -module _ - {l1 l2 l3 : Level} {A : UU l1} {B : UU l2} {X : UU l3} - where - - is-0-map-right-factor : - (g : B → X) (h : A → B) → - is-0-map g → is-0-map (g ∘ h) → is-0-map h - is-0-map-right-factor = is-trunc-map-right-factor zero-𝕋 - - is-0-map-right-factor-htpy : - (f : A → X) (g : B → X) (h : A → B) (H : f ~ (g ∘ h)) → - is-0-map g → is-0-map f → is-0-map h - is-0-map-right-factor-htpy = is-trunc-map-right-factor-htpy zero-𝕋 -``` - -### A family of `0`-maps induces a `0`-map on total spaces - -```agda -module _ - {l1 l2 l3 : Level} {A : UU l1} {B : A → UU l2} {C : A → UU l3} - {f : (x : A) → B x → C x} - where - - abstract - is-0-map-tot : ((x : A) → is-0-map (f x)) → is-0-map (tot f) - is-0-map-tot = is-trunc-map-tot zero-𝕋 -``` - -### For any type family over the codomain, a `0`-map induces a `0`-map on total spaces - -In other words, `0`-maps are stable under pullbacks. We will come to this point -when we introduce homotopy pullbacks. - -```agda -module _ - {l1 l2 l3 : Level} {A : UU l1} {B : UU l2} {f : A → B} (C : B → UU l3) - where - - abstract - is-0-map-map-Σ-map-base : is-0-map f → is-0-map (map-Σ-map-base f C) - is-0-map-map-Σ-map-base = is-trunc-map-map-Σ-map-base zero-𝕋 C -``` - -### The functorial action of `Σ` preserves `0`-maps - -```agda -module _ - {l1 l2 l3 l4 : Level} {A : UU l1} {B : UU l2} {C : A → UU l3} - (D : B → UU l4) {f : A → B} {g : (x : A) → C x → D (f x)} - where - - is-0-map-map-Σ : - is-0-map f → ((x : A) → is-0-map (g x)) → is-0-map (map-Σ D f g) - is-0-map-map-Σ = is-trunc-map-map-Σ zero-𝕋 D -``` diff --git a/src/foundation-core/1-types.lagda.md b/src/foundation-core/1-types.lagda.md index 92c1414cda..ddc5b7a9ab 100644 --- a/src/foundation-core/1-types.lagda.md +++ b/src/foundation-core/1-types.lagda.md @@ -7,13 +7,14 @@ module foundation-core.1-types where
Imports ```agda -open import foundation-core.dependent-pair-types +open import foundation.dependent-pair-types +open import foundation.universe-levels + open import foundation-core.equivalences open import foundation-core.identity-types open import foundation-core.sets open import foundation-core.truncated-types open import foundation-core.truncation-levels -open import foundation-core.universe-levels ```
diff --git a/src/foundation-core/automorphisms.lagda.md b/src/foundation-core/automorphisms.lagda.md deleted file mode 100644 index c34c6027a7..0000000000 --- a/src/foundation-core/automorphisms.lagda.md +++ /dev/null @@ -1,44 +0,0 @@ -# Automorphisms - -```agda -module foundation-core.automorphisms where -``` - -
Imports - -```agda -open import foundation.sets - -open import foundation-core.dependent-pair-types -open import foundation-core.equivalences -open import foundation-core.universe-levels - -open import structured-types.pointed-types -``` - -
- -## Idea - -An automorphism on a type `A` is an equivalence `A ≃ A`. We will just reuse the -infrastructure of equivalences for automorphisms. - -## Definitions - -### The type of automorphisms on a type - -```agda -Aut : {l : Level} → UU l → UU l -Aut Y = Y ≃ Y - -is-set-Aut : {l : Level} {A : UU l} → is-set A → is-set (Aut A) -is-set-Aut H = is-set-equiv-is-set H H - -Aut-Set : {l : Level} → Set l → Set l -pr1 (Aut-Set A) = Aut (type-Set A) -pr2 (Aut-Set A) = is-set-Aut (is-set-type-Set A) - -Aut-Pointed-Type : {l : Level} → UU l → Pointed-Type l -pr1 (Aut-Pointed-Type A) = Aut A -pr2 (Aut-Pointed-Type A) = id-equiv -``` diff --git a/src/foundation-core/cartesian-product-types.lagda.md b/src/foundation-core/cartesian-product-types.lagda.md index 0ada89c06d..86f2147537 100644 --- a/src/foundation-core/cartesian-product-types.lagda.md +++ b/src/foundation-core/cartesian-product-types.lagda.md @@ -9,8 +9,8 @@ module foundation-core.cartesian-product-types where
Imports ```agda -open import foundation-core.dependent-pair-types -open import foundation-core.universe-levels +open import foundation.dependent-pair-types +open import foundation.universe-levels ```
diff --git a/src/foundation-core/coherently-invertible-maps.lagda.md b/src/foundation-core/coherently-invertible-maps.lagda.md index 107d51cd1e..bd6ebddcd7 100644 --- a/src/foundation-core/coherently-invertible-maps.lagda.md +++ b/src/foundation-core/coherently-invertible-maps.lagda.md @@ -10,13 +10,13 @@ module foundation-core.coherently-invertible-maps where ```agda open import foundation.commuting-squares-of-identifications +open import foundation.dependent-pair-types +open import foundation.functions +open import foundation.universe-levels open import foundation-core.cartesian-product-types -open import foundation-core.dependent-pair-types -open import foundation-core.functions open import foundation-core.homotopies open import foundation-core.identity-types -open import foundation-core.universe-levels ``` diff --git a/src/foundation-core/commuting-3-simplices-of-homotopies.lagda.md b/src/foundation-core/commuting-3-simplices-of-homotopies.lagda.md deleted file mode 100644 index a17dacd09b..0000000000 --- a/src/foundation-core/commuting-3-simplices-of-homotopies.lagda.md +++ /dev/null @@ -1,76 +0,0 @@ -# Commuting 3-simplices of homotopies - -```agda -{-# OPTIONS --safe #-} - -module foundation-core.commuting-3-simplices-of-homotopies where -``` - -
Imports - -```agda -open import foundation-core.commuting-triangles-of-homotopies -open import foundation-core.homotopies -open import foundation-core.universe-levels -``` - -
- -## Idea - -A commuting 3-simplex of homotopies is a commuting diagram of the form - -```text - f ----------> g - | \ ^ | - | \ / | - | / | - | / \ | - V / v V - h ----------> i. -``` - -where f, g, h, and i are functions. - -## Definition - -```agda -module _ - {l1 l2 : Level} {A : UU l1} {B : A → UU l2} - {f g h i : (x : A) → B x} - (top : f ~ g) (left : f ~ h) (right : g ~ i) (bottom : h ~ i) - (diagonal-up : h ~ g) (diagonal-down : f ~ i) - (upper-left : coherence-triangle-homotopies top diagonal-up left) - (lower-right : coherence-triangle-homotopies bottom right diagonal-up) - (upper-right : coherence-triangle-homotopies diagonal-down right top) - (lower-left : coherence-triangle-homotopies diagonal-down bottom left) - where - - coherence-3-simplex-homotopies : UU (l1 ⊔ l2) - coherence-3-simplex-homotopies = - ( upper-right ∙h - left-whisk-htpy-coherence-triangle-homotopies - ( diagonal-up) - ( right) - ( upper-left)) ~ - ( ( lower-left ∙h - right-whisk-htpy-coherence-triangle-homotopies - ( right) - ( lower-right) - ( left)) ∙h - assoc-htpy left diagonal-up right) - - coherence-3-simplex-homotopies' : UU (l1 ⊔ l2) - coherence-3-simplex-homotopies' = - ( ( lower-left ∙h - right-whisk-htpy-coherence-triangle-homotopies - ( right) - ( lower-right) - ( left)) ∙h - assoc-htpy left diagonal-up right) ~ - ( upper-right ∙h - left-whisk-htpy-coherence-triangle-homotopies - ( diagonal-up) - ( right) - ( upper-left)) -``` diff --git a/src/foundation-core/commuting-3-simplices-of-maps.lagda.md b/src/foundation-core/commuting-3-simplices-of-maps.lagda.md deleted file mode 100644 index 7949669a58..0000000000 --- a/src/foundation-core/commuting-3-simplices-of-maps.lagda.md +++ /dev/null @@ -1,55 +0,0 @@ -# Commuting 3-simplices of maps - -```agda -{-# OPTIONS --safe #-} - -module foundation-core.commuting-3-simplices-of-maps where -``` - -
Imports - -```agda -open import foundation-core.commuting-triangles-of-maps -open import foundation-core.homotopies -open import foundation-core.universe-levels -``` - -
- -## Idea - -A commuting 3-simplex is a commuting diagram of the form - -```text - A ----------> B - | \ ^ | - | \ / | - | / | - | / \ | - V / v V - X ----------> Y. -``` - -## Definition - -```agda -module _ - {l1 l2 l3 l4 : Level} {A : UU l1} {B : UU l2} {X : UU l3} {Y : UU l4} - (top : A → B) (left : A → X) (right : B → Y) (bottom : X → Y) - (diagonal-up : X → B) (diagonal-down : A → Y) - (upper-left : coherence-triangle-maps top diagonal-up left) - (lower-right : coherence-triangle-maps bottom right diagonal-up) - (upper-right : coherence-triangle-maps diagonal-down right top) - (lower-left : coherence-triangle-maps diagonal-down bottom left) - where - - coherence-3-simplex-maps : UU (l1 ⊔ l4) - coherence-3-simplex-maps = - ( upper-right ∙h (right ·l upper-left)) ~ - ( lower-left ∙h (lower-right ·r left)) - - coherence-3-simplex-maps' : UU (l1 ⊔ l4) - coherence-3-simplex-maps' = - ( lower-left ∙h (lower-right ·r left)) ~ - ( upper-right ∙h (right ·l upper-left)) -``` diff --git a/src/foundation-core/commuting-cubes-of-maps.lagda.md b/src/foundation-core/commuting-cubes-of-maps.lagda.md index f3f6ef8662..6dad78d70e 100644 --- a/src/foundation-core/commuting-cubes-of-maps.lagda.md +++ b/src/foundation-core/commuting-cubes-of-maps.lagda.md @@ -7,14 +7,14 @@ module foundation-core.commuting-cubes-of-maps where
Imports ```agda +open import foundation.cones-over-cospans +open import foundation.dependent-pair-types +open import foundation.functions open import foundation.hexagons-of-identifications +open import foundation.universe-levels -open import foundation-core.cones-over-cospans -open import foundation-core.dependent-pair-types -open import foundation-core.functions open import foundation-core.homotopies open import foundation-core.identity-types -open import foundation-core.universe-levels ```
diff --git a/src/foundation-core/commuting-squares-of-maps.lagda.md b/src/foundation-core/commuting-squares-of-maps.lagda.md index 5d9344c51b..0ff971f15c 100644 --- a/src/foundation-core/commuting-squares-of-maps.lagda.md +++ b/src/foundation-core/commuting-squares-of-maps.lagda.md @@ -9,10 +9,11 @@ module foundation-core.commuting-squares-of-maps where
Imports ```agda +open import foundation.functions +open import foundation.universe-levels + open import foundation-core.commuting-triangles-of-maps -open import foundation-core.functions open import foundation-core.homotopies -open import foundation-core.universe-levels ```
diff --git a/src/foundation-core/commuting-triangles-of-homotopies.lagda.md b/src/foundation-core/commuting-triangles-of-homotopies.lagda.md deleted file mode 100644 index 72bbe5f655..0000000000 --- a/src/foundation-core/commuting-triangles-of-homotopies.lagda.md +++ /dev/null @@ -1,132 +0,0 @@ -# Commuting triangles of homotopies - -```agda -{-# OPTIONS --safe #-} - -module foundation-core.commuting-triangles-of-homotopies where -``` - -
Imports - -```agda -open import foundation-core.functions -open import foundation-core.homotopies -open import foundation-core.identity-types -open import foundation-core.universe-levels -``` - -
- -## Idea - -A triangle of homotopies of maps - -```text - f ----> g - \ / - \ / - V V - h -``` - -is said to commute if there is a homotopy between the homotopy on the left -`f ~ h` and the composite homotopy `f ~ g ~ h`. - -## Definition - -```agda -module _ - {l1 l2 : Level} {A : UU l1} {B : A → UU l2} - {f g h : (x : A) → B x} - where - - coherence-triangle-homotopies : - (left : f ~ h) (right : g ~ h) (top : f ~ g) → UU (l1 ⊔ l2) - coherence-triangle-homotopies left right top = left ~ (top ∙h right) - - coherence-triangle-homotopies' : - (left : f ~ h) (right : g ~ h) (top : f ~ g) → UU (l1 ⊔ l2) - coherence-triangle-homotopies' left right top = (top ∙h right) ~ left -``` - -## Properties - -### Distributive law for left whiskering - -```agda -module _ - {l1 l2 : Level} {A : UU l1} {B : UU l2} - {f g h : A → B} - {l3 : Level} {X : UU l3} (i : B → X) - (left : f ~ h) (right : g ~ h) (top : f ~ g) - where - - distributivity-left-whisk : - coherence-triangle-homotopies left right top → - (i ·l left) ~ ((i ·l top) ∙h (i ·l right)) - distributivity-left-whisk T x = - ap-concat-eq i (top x) (right x) (left x) (T x) -``` - -### Left whiskering triangles of homotopies - -```agda -module _ - {l1 l2 : Level} {A : UU l1} {B : A → UU l2} - {f g h : (x : A) → B x} - {left : f ~ h} (right : g ~ h) {top : f ~ g} - where - - left-whisk-htpy-coherence-triangle-homotopies : - {i : (x : A) → B x} - (H : h ~ i) (T : coherence-triangle-homotopies left right top) → - coherence-triangle-homotopies {h = i} (left ∙h H) (right ∙h H) top - left-whisk-htpy-coherence-triangle-homotopies H T = - (λ x → ap (_∙ H x) (T x)) ∙h assoc-htpy top right H - -module _ - {l1 l2 : Level} {A : UU l1} {B : UU l2} - {f g h : A → B} - {left : f ~ h} (right : g ~ h) {top : f ~ g} - where - - left-whisk-coherence-triangle-homotopies : - {l3 : Level} {X : UU l3} (i : B → X) - (T : coherence-triangle-homotopies left right top) → - coherence-triangle-homotopies - {f = i ∘ f} {i ∘ g} {i ∘ h} - (i ·l left) (i ·l right) (i ·l top) - left-whisk-coherence-triangle-homotopies i = - distributivity-left-whisk i left right top -``` - -### Right whiskering triangles of homotopies - -```agda -module _ - {l1 l2 : Level} {A : UU l1} {B : A → UU l2} - {f g h : (x : A) → B x} - {left : f ~ h} (right : g ~ h) {top : f ~ g} - where - - right-whisk-htpy-coherence-triangle-homotopies : - {i : (x : A) → B x} - (T : coherence-triangle-homotopies left right top) (H : i ~ f) → - coherence-triangle-homotopies {f = i} (H ∙h left) right (H ∙h top) - right-whisk-htpy-coherence-triangle-homotopies T H = - (λ x → ap (H x ∙_) (T x)) ∙h (inv-htpy-assoc-htpy H top right) - -module _ - {l1 l2 : Level} {A : UU l1} {B : UU l2} - {f g h : A → B} - {left : f ~ h} (right : g ~ h) {top : f ~ g} - where - - right-whisk-coherence-triangle-homotopies : - {l3 : Level} {X : UU l3} - (T : coherence-triangle-homotopies left right top) (i : X → A) → - coherence-triangle-homotopies - {f = f ∘ i} {g ∘ i} {h ∘ i} - (left ·r i) (right ·r i) (top ·r i) - right-whisk-coherence-triangle-homotopies T i = T ∘ i -``` diff --git a/src/foundation-core/commuting-triangles-of-maps.lagda.md b/src/foundation-core/commuting-triangles-of-maps.lagda.md index 2097067e0a..22bc8f6928 100644 --- a/src/foundation-core/commuting-triangles-of-maps.lagda.md +++ b/src/foundation-core/commuting-triangles-of-maps.lagda.md @@ -9,9 +9,10 @@ module foundation-core.commuting-triangles-of-maps where
Imports ```agda -open import foundation-core.functions +open import foundation.functions +open import foundation.universe-levels + open import foundation-core.homotopies -open import foundation-core.universe-levels ```
diff --git a/src/foundation-core/cones-over-cospans.lagda.md b/src/foundation-core/cones-over-cospans.lagda.md deleted file mode 100644 index 617ff9b952..0000000000 --- a/src/foundation-core/cones-over-cospans.lagda.md +++ /dev/null @@ -1,270 +0,0 @@ -# Cones on pullback diagrams - -```agda -module foundation-core.cones-over-cospans where -``` - -
Imports - -```agda -open import foundation.homotopies -open import foundation.structure-identity-principle - -open import foundation-core.commuting-squares-of-maps -open import foundation-core.contractible-types -open import foundation-core.dependent-pair-types -open import foundation-core.equivalences -open import foundation-core.functions -open import foundation-core.fundamental-theorem-of-identity-types -open import foundation-core.identity-types -open import foundation-core.universe-levels -``` - -
- -## Idea - -A cone on a cospan `A --f--> X <--g-- B` with vertex `C` is a triple `(p,q,H)` -consisting of a map `p : C → A`, a map `q : C → B`, and a homotopy `H` -witnessing that the square - -```text - q - C -----> B - | | - p| |g - V V - A -----> X - f -``` - -commutes. - -## Definitions - -### Cones on cospans - -A cone on a cospan with a vertex C is a pair of functions from C into the -domains of the maps in the cospan, equipped with a homotopy witnessing that the -resulting square commutes. - -```agda -module _ - {l1 l2 l3 : Level} {A : UU l1} {B : UU l2} {X : UU l3} - (f : A → X) (g : B → X) - where - - cone : {l4 : Level} → UU l4 → UU (l1 ⊔ l2 ⊔ l3 ⊔ l4) - cone C = Σ (C → A) (λ p → Σ (C → B) (λ q → coherence-square-maps q p g f)) - -module _ - {l1 l2 l3 l4 : Level} {A : UU l1} {B : UU l2} {X : UU l3} {C : UU l4} - (f : A → X) (g : B → X) (c : cone f g C) - where - - vertical-map-cone : C → A - vertical-map-cone = pr1 c - - horizontal-map-cone : C → B - horizontal-map-cone = pr1 (pr2 c) - - coherence-square-cone : - coherence-square-maps horizontal-map-cone vertical-map-cone g f - coherence-square-cone = pr2 (pr2 c) -``` - -### Dependent cones - -```agda -cone-family : - {l1 l2 l3 l4 l5 l6 l7 l8 : Level} - {X : UU l1} {A : UU l2} {B : UU l3} {C : UU l4} - (PX : X → UU l5) {PA : A → UU l6} {PB : B → UU l7} - {f : A → X} {g : B → X} → - (f' : (a : A) → PA a → PX (f a)) (g' : (b : B) → PB b → PX (g b)) → - cone f g C → (C → UU l8) → UU (l4 ⊔ l5 ⊔ l6 ⊔ l7 ⊔ l8) -cone-family {C = C} PX {f = f} {g} f' g' c PC = - (x : C) → - cone - ( ( tr PX (coherence-square-cone f g c x)) ∘ - ( f' (vertical-map-cone f g c x))) - ( g' (horizontal-map-cone f g c x)) - ( PC x) -``` - -### Identifications of cones - -Next we characterize the identity type of the type of cones with a given vertex -C. Note that in the definition of htpy-cone we do not use pattern matching on -the cones c and c'. This is to ensure that the type htpy-cone f g c c' is a -Σ-type for any c and c', not just for c and c' of the form (pair p (pair q H)) -and (pair p' (pair q' H')) respectively. - -```agda -module _ - {l1 l2 l3 l4 : Level} {A : UU l1} {B : UU l2} {X : UU l3} - (f : A → X) (g : B → X) {C : UU l4} - where - - coherence-htpy-cone : - (c c' : cone f g C) (K : vertical-map-cone f g c ~ vertical-map-cone f g c') - (L : horizontal-map-cone f g c ~ horizontal-map-cone f g c') → UU (l4 ⊔ l3) - coherence-htpy-cone c c' K L = - ( coherence-square-cone f g c ∙h (g ·l L)) ~ - ( (f ·l K) ∙h coherence-square-cone f g c') - - htpy-cone : cone f g C → cone f g C → UU (l1 ⊔ l2 ⊔ l3 ⊔ l4) - htpy-cone c c' = - Σ ( vertical-map-cone f g c ~ vertical-map-cone f g c') - ( λ K → - Σ ( horizontal-map-cone f g c ~ horizontal-map-cone f g c') - ( coherence-htpy-cone c c' K)) - - refl-htpy-cone : (c : cone f g C) → htpy-cone c c - pr1 (refl-htpy-cone c) = refl-htpy - pr1 (pr2 (refl-htpy-cone c)) = refl-htpy - pr2 (pr2 (refl-htpy-cone c)) = right-unit-htpy - - htpy-eq-cone : (c c' : cone f g C) → c ＝ c' → htpy-cone c c' - htpy-eq-cone c .c refl = refl-htpy-cone c - - is-contr-total-htpy-cone : - (c : cone f g C) → is-contr (Σ (cone f g C) (htpy-cone c)) - is-contr-total-htpy-cone c = - is-contr-total-Eq-structure - ( λ p qH K → - Σ ( horizontal-map-cone f g c ~ pr1 qH) - ( coherence-htpy-cone c (pair p qH) K)) - ( is-contr-total-htpy (vertical-map-cone f g c)) - ( pair (vertical-map-cone f g c) refl-htpy) - ( is-contr-total-Eq-structure - ( λ q H → - coherence-htpy-cone c - ( pair (vertical-map-cone f g c) (pair q H)) - ( refl-htpy)) - ( is-contr-total-htpy (horizontal-map-cone f g c)) - ( pair (horizontal-map-cone f g c) refl-htpy) - ( is-contr-total-htpy (coherence-square-cone f g c ∙h refl-htpy))) - - is-equiv-htpy-eq-cone : (c c' : cone f g C) → is-equiv (htpy-eq-cone c c') - is-equiv-htpy-eq-cone c = - fundamental-theorem-id (is-contr-total-htpy-cone c) (htpy-eq-cone c) - - extensionality-cone : (c c' : cone f g C) → (c ＝ c') ≃ htpy-cone c c' - pr1 (extensionality-cone c c') = htpy-eq-cone c c' - pr2 (extensionality-cone c c') = is-equiv-htpy-eq-cone c c' - - eq-htpy-cone : (c c' : cone f g C) → htpy-cone c c' → (c ＝ c') - eq-htpy-cone c c' = map-inv-equiv (extensionality-cone c c') -``` - -### Precomposing cones - -```agda -module _ - {l1 l2 l3 : Level} {A : UU l1} {B : UU l2} {X : UU l3} - (f : A → X) (g : B → X) - where - - cone-map : - {l4 l5 : Level} {C : UU l4} {C' : UU l5} → - cone f g C → (C' → C) → cone f g C' - pr1 (cone-map c h) = vertical-map-cone f g c ∘ h - pr1 (pr2 (cone-map c h)) = horizontal-map-cone f g c ∘ h - pr2 (pr2 (cone-map c h)) = coherence-square-cone f g c ·r h -``` - -### Pasting cones horizontally - -```agda -module _ - {l1 l2 l3 l4 l5 l6 : Level} - {A : UU l1} {B : UU l2} {C : UU l3} {X : UU l4} {Y : UU l5} {Z : UU l6} - (i : X → Y) (j : Y → Z) (h : C → Z) - where - - pasting-horizontal-cone : - (c : cone j h B) → cone i (vertical-map-cone j h c) A → cone (j ∘ i) h A - pr1 (pasting-horizontal-cone c (pair f (pair p H))) = f - pr1 (pr2 (pasting-horizontal-cone c (pair f (pair p H)))) = - (horizontal-map-cone j h c) ∘ p - pr2 (pr2 (pasting-horizontal-cone c (pair f (pair p H)))) = - pasting-horizontal-coherence-square-maps p - ( horizontal-map-cone j h c) - ( f) - ( vertical-map-cone j h c) - ( h) - ( i) - ( j) - ( H) - ( coherence-square-cone j h c) -``` - -### Vertical composition of cones - -```agda -module _ - {l1 l2 l3 l4 l5 l6 : Level} - {A : UU l1} {B : UU l2} {C : UU l3} {X : UU l4} {Y : UU l5} {Z : UU l6} - (f : C → Z) (g : Y → Z) (h : X → Y) - where - - pasting-vertical-cone : - (c : cone f g B) → cone (horizontal-map-cone f g c) h A → cone f (g ∘ h) A - pr1 (pasting-vertical-cone c (pair p' (pair q' H'))) = - ( vertical-map-cone f g c) ∘ p' - pr1 (pr2 (pasting-vertical-cone c (pair p' (pair q' H')))) = q' - pr2 (pr2 (pasting-vertical-cone c (pair p' (pair q' H')))) = - pasting-vertical-coherence-square-maps q' p' h - ( horizontal-map-cone f g c) - ( vertical-map-cone f g c) - ( g) - ( f) - ( H') - ( coherence-square-cone f g c) -``` - -### The swapping function on cones - -```agda -swap-cone : - {l1 l2 l3 l4 : Level} {A : UU l1} {B : UU l2} {X : UU l3} {C : UU l4} - (f : A → X) (g : B → X) → cone f g C → cone g f C -pr1 (swap-cone f g c) = horizontal-map-cone f g c -pr1 (pr2 (swap-cone f g c)) = vertical-map-cone f g c -pr2 (pr2 (swap-cone f g c)) = inv-htpy (coherence-square-cone f g c) -``` - -### Parallel cones - -```agda -module _ - {l1 l2 l3 : Level} {A : UU l1} {B : UU l2} {X : UU l3} - {f f' : A → X} (Hf : f ~ f') {g g' : B → X} (Hg : g ~ g') - where - - coherence-htpy-parallel-cone : - {l4 : Level} {C : UU l4} (c : cone f g C) (c' : cone f' g' C) - (Hp : vertical-map-cone f g c ~ vertical-map-cone f' g' c') - (Hq : horizontal-map-cone f g c ~ horizontal-map-cone f' g' c') → - UU (l3 ⊔ l4) - coherence-htpy-parallel-cone c c' Hp Hq = - ( ( coherence-square-cone f g c) ∙h - ( (g ·l Hq) ∙h (Hg ·r horizontal-map-cone f' g' c'))) ~ - ( ( (f ·l Hp) ∙h (Hf ·r (vertical-map-cone f' g' c'))) ∙h - ( coherence-square-cone f' g' c')) - - fam-htpy-parallel-cone : - {l4 : Level} {C : UU l4} (c : cone f g C) → (c' : cone f' g' C) → - (vertical-map-cone f g c ~ vertical-map-cone f' g' c') → UU (l2 ⊔ l3 ⊔ l4) - fam-htpy-parallel-cone c c' Hp = - Σ ( horizontal-map-cone f g c ~ horizontal-map-cone f' g' c') - ( coherence-htpy-parallel-cone c c' Hp) - - htpy-parallel-cone : - {l4 : Level} {C : UU l4} → - cone f g C → cone f' g' C → UU (l1 ⊔ l2 ⊔ l3 ⊔ l4) - htpy-parallel-cone c c' = - Σ ( vertical-map-cone f g c ~ vertical-map-cone f' g' c') - ( fam-htpy-parallel-cone c c') -``` diff --git a/src/foundation-core/constant-maps.lagda.md b/src/foundation-core/constant-maps.lagda.md index 7dc391726e..6064a63519 100644 --- a/src/foundation-core/constant-maps.lagda.md +++ b/src/foundation-core/constant-maps.lagda.md @@ -9,7 +9,7 @@ module foundation-core.constant-maps where
Imports ```agda -open import foundation-core.universe-levels +open import foundation.universe-levels ```
diff --git a/src/foundation-core/contractible-maps.lagda.md b/src/foundation-core/contractible-maps.lagda.md index 7c85b3b8c1..88ad0fb7b5 100644 --- a/src/foundation-core/contractible-maps.lagda.md +++ b/src/foundation-core/contractible-maps.lagda.md @@ -7,15 +7,16 @@ module foundation-core.contractible-maps where
Imports ```agda +open import foundation.dependent-pair-types +open import foundation.functions +open import foundation.universe-levels + open import foundation-core.coherently-invertible-maps open import foundation-core.contractible-types -open import foundation-core.dependent-pair-types open import foundation-core.equivalences open import foundation-core.fibers-of-maps -open import foundation-core.functions open import foundation-core.homotopies open import foundation-core.identity-types -open import foundation-core.universe-levels ```
diff --git a/src/foundation-core/contractible-types.lagda.md b/src/foundation-core/contractible-types.lagda.md index ecb4d32bea..dbaf3b5341 100644 --- a/src/foundation-core/contractible-types.lagda.md +++ b/src/foundation-core/contractible-types.lagda.md @@ -7,16 +7,16 @@ module foundation-core.contractible-types where
Imports ```agda +open import foundation.dependent-pair-types +open import foundation.equality-cartesian-product-types open import foundation.function-extensionality +open import foundation.universe-levels open import foundation-core.cartesian-product-types -open import foundation-core.dependent-pair-types -open import foundation-core.equality-cartesian-product-types open import foundation-core.equality-dependent-pair-types open import foundation-core.equivalences open import foundation-core.identity-types open import foundation-core.retractions -open import foundation-core.universe-levels ```
diff --git a/src/foundation-core/coproduct-types.lagda.md b/src/foundation-core/coproduct-types.lagda.md index 79239a1a9a..a13e3d222c 100644 --- a/src/foundation-core/coproduct-types.lagda.md +++ b/src/foundation-core/coproduct-types.lagda.md @@ -9,7 +9,7 @@ module foundation-core.coproduct-types where
Imports ```agda -open import foundation-core.universe-levels +open import foundation.universe-levels ```
diff --git a/src/foundation-core/cospans.lagda.md b/src/foundation-core/cospans.lagda.md deleted file mode 100644 index b76928faf0..0000000000 --- a/src/foundation-core/cospans.lagda.md +++ /dev/null @@ -1,31 +0,0 @@ -# Cospans - -```agda -{-# OPTIONS --safe #-} - -module foundation-core.cospans where -``` - -
Imports - -```agda -open import foundation-core.cartesian-product-types -open import foundation-core.dependent-pair-types -open import foundation-core.universe-levels -``` - -
- -## Definition - -### Cospans - -A cospan is a pair of functions with a common codomain - -```agda -cospan : - {l1 l2 : Level} (l : Level) (A : UU l1) (B : UU l2) → - UU (l1 ⊔ l2 ⊔ lsuc l) -cospan l A B = - Σ (UU l) (λ X → (A → X) × (B → X)) -``` diff --git a/src/foundation-core/decidable-propositions.lagda.md b/src/foundation-core/decidable-propositions.lagda.md index 07ce742e58..6353e975dc 100644 --- a/src/foundation-core/decidable-propositions.lagda.md +++ b/src/foundation-core/decidable-propositions.lagda.md @@ -9,19 +9,19 @@ module foundation-core.decidable-propositions where ```agda open import foundation.coproduct-types open import foundation.decidable-types +open import foundation.dependent-pair-types open import foundation.double-negation +open import foundation.functions open import foundation.negation open import foundation.propositional-truncations open import foundation.unit-type +open import foundation.universe-levels open import foundation-core.cartesian-product-types -open import foundation-core.dependent-pair-types open import foundation-core.empty-types -open import foundation-core.functions open import foundation-core.functoriality-dependent-pair-types open import foundation-core.propositions open import foundation-core.subtypes -open import foundation-core.universe-levels ``` diff --git a/src/foundation-core/dependent-pair-types.lagda.md b/src/foundation-core/dependent-pair-types.lagda.md deleted file mode 100644 index c70191de41..0000000000 --- a/src/foundation-core/dependent-pair-types.lagda.md +++ /dev/null @@ -1,78 +0,0 @@ -# Dependent pair types - -```agda -{-# OPTIONS --safe #-} - -module foundation-core.dependent-pair-types where -``` - -
Imports - -```agda -open import foundation-core.universe-levels -``` - -
- -## Idea - -When `B` is a family of types over `A`, then we can form the type of pairs -`pair a b` consisting of an element `a : A` and an element `b : B a`. Such pairs -are called dependent pairs, since the type of the second component depends on -the first component. - -## Definition - -```agda -record Σ {l1 l2 : Level} (A : UU l1) (B : A → UU l2) : UU (l1 ⊔ l2) where - constructor pair - field - pr1 : A - pr2 : B pr1 - -open Σ public - -{-# BUILTIN SIGMA Σ #-} - -infixr 10 _,_ -pattern _,_ a b = pair a b -``` - -## Constructions - -```agda -ind-Σ : - {l1 l2 l3 : Level} {A : UU l1} {B : A → UU l2} {C : Σ A B → UU l3} → - ((x : A) (y : B x) → C (pair x y)) → ((t : Σ A B) → C t) -ind-Σ f (pair x y) = f x y - -ev-pair : - {l1 l2 l3 : Level} {A : UU l1} {B : A → UU l2} {C : Σ A B → UU l3} → - ((t : Σ A B) → C t) → (x : A) (y : B x) → C (pair x y) -ev-pair f x y = f (pair x y) - -triple : - {l1 l2 l3 : Level} {A : UU l1} {B : A → UU l2} {C : (x : A) → B x → UU l3} → - (a : A) (b : B a) → C a b → Σ A (λ x → Σ (B x) (C x)) -pr1 (triple a b c) = a -pr1 (pr2 (triple a b c)) = b -pr2 (pr2 (triple a b c)) = c - -triple' : - {l1 l2 l3 : Level} {A : UU l1} {B : A → UU l2} {C : Σ A B → UU l3} → - (a : A) (b : B a) → C (pair a b) → Σ (Σ A B) C -pr1 (pr1 (triple' a b c)) = a -pr2 (pr1 (triple' a b c)) = b -pr2 (triple' a b c) = c -``` - -### Families on dependent pair types - -```agda -module _ - {l1 l2 l3 : Level} {A : UU l1} {B : A → UU l2} - where - - fam-Σ : ((x : A) → B x → UU l3) → Σ A B → UU l3 - fam-Σ C (pair x y) = C x y -``` diff --git a/src/foundation-core/diagonal-maps-of-types.lagda.md b/src/foundation-core/diagonal-maps-of-types.lagda.md index 09d2962f2f..5f336318b6 100644 --- a/src/foundation-core/diagonal-maps-of-types.lagda.md +++ b/src/foundation-core/diagonal-maps-of-types.lagda.md @@ -7,16 +7,17 @@ module foundation-core.diagonal-maps-of-types where
Imports ```agda +open import foundation.dependent-pair-types +open import foundation.equality-cartesian-product-types +open import foundation.functions +open import foundation.universe-levels + open import foundation-core.cartesian-product-types -open import foundation-core.dependent-pair-types -open import foundation-core.equality-cartesian-product-types open import foundation-core.equivalences open import foundation-core.fibers-of-maps -open import foundation-core.functions open import foundation-core.homotopies open import foundation-core.identity-types open import foundation-core.propositions -open import foundation-core.universe-levels ```
diff --git a/src/foundation-core/discrete-types.lagda.md b/src/foundation-core/discrete-types.lagda.md index b1826189ee..bd922b360f 100644 --- a/src/foundation-core/discrete-types.lagda.md +++ b/src/foundation-core/discrete-types.lagda.md @@ -8,10 +8,10 @@ module foundation-core.discrete-types where ```agda open import foundation.decidable-equality +open import foundation.dependent-pair-types +open import foundation.universe-levels -open import foundation-core.dependent-pair-types open import foundation-core.sets -open import foundation-core.universe-levels ``` diff --git a/src/foundation-core/embeddings.lagda.md b/src/foundation-core/embeddings.lagda.md index be4a3b17ed..ea6a233ec3 100644 --- a/src/foundation-core/embeddings.lagda.md +++ b/src/foundation-core/embeddings.lagda.md @@ -9,11 +9,12 @@ module foundation-core.embeddings where
Imports ```agda -open import foundation-core.dependent-pair-types +open import foundation.dependent-pair-types +open import foundation.functions +open import foundation.universe-levels + open import foundation-core.equivalences -open import foundation-core.functions open import foundation-core.identity-types -open import foundation-core.universe-levels ```
diff --git a/src/foundation-core/empty-types.lagda.md b/src/foundation-core/empty-types.lagda.md index ca74a7cb45..5a2c12bd60 100644 --- a/src/foundation-core/empty-types.lagda.md +++ b/src/foundation-core/empty-types.lagda.md @@ -7,16 +7,16 @@ module foundation-core.empty-types where
Imports ```agda +open import foundation.dependent-pair-types +open import foundation.functions open import foundation.propositions +open import foundation.universe-levels -open import foundation-core.dependent-pair-types open import foundation-core.embeddings open import foundation-core.equivalences -open import foundation-core.functions open import foundation-core.sets open import foundation-core.truncated-types open import foundation-core.truncation-levels -open import foundation-core.universe-levels ```
diff --git a/src/foundation-core/endomorphisms.lagda.md b/src/foundation-core/endomorphisms.lagda.md index 73e94109fb..c7cdc965b5 100644 --- a/src/foundation-core/endomorphisms.lagda.md +++ b/src/foundation-core/endomorphisms.lagda.md @@ -7,13 +7,13 @@ module foundation-core.endomorphisms where
Imports ```agda +open import foundation.dependent-pair-types +open import foundation.functions open import foundation.sets +open import foundation.universe-levels -open import foundation-core.dependent-pair-types -open import foundation-core.functions open import foundation-core.truncated-types open import foundation-core.truncation-levels -open import foundation-core.universe-levels open import structured-types.pointed-types ``` diff --git a/src/foundation-core/equality-cartesian-product-types.lagda.md b/src/foundation-core/equality-cartesian-product-types.lagda.md deleted file mode 100644 index c6f6fafe1c..0000000000 --- a/src/foundation-core/equality-cartesian-product-types.lagda.md +++ /dev/null @@ -1,147 +0,0 @@ -# Equality of cartesian product types - -```agda -{-# OPTIONS --safe #-} - -module foundation-core.equality-cartesian-product-types where -``` - -
Imports - -```agda -open import foundation-core.cartesian-product-types -open import foundation-core.dependent-pair-types -open import foundation-core.equivalences -open import foundation-core.functions -open import foundation-core.homotopies -open import foundation-core.identity-types -open import foundation-core.universe-levels -``` - -
- -## Idea - -Identifications `Id (pair x y) (pair x' y')` in a cartesian product are -equivalently described as pairs of identifications `Id x x'` and `Id y y'`. This -provides us with a characterization of the identity type of cartesian product -types. - -## Definition - -```agda -module _ - {l1 l2 : Level} {A : UU l1} {B : UU l2} - where - - Eq-prod : (s t : A × B) → UU (l1 ⊔ l2) - Eq-prod s t = ((pr1 s) ＝ (pr1 t)) × ((pr2 s) ＝ (pr2 t)) -``` - -## Properties - -### The type `Eq-prod s t` is equivalent to `Id s t` - -```agda -module _ - {l1 l2 : Level} {A : UU l1} {B : UU l2} - where - - eq-pair' : {s t : A × B} → Eq-prod s t → s ＝ t - eq-pair' {pair x y} {pair .x .y} (pair refl refl) = refl - - eq-pair : - {s t : A × B} → (pr1 s) ＝ (pr1 t) → (pr2 s) ＝ (pr2 t) → s ＝ t - eq-pair p q = eq-pair' (pair p q) - - pair-eq : {s t : A × B} → s ＝ t → Eq-prod s t - pr1 (pair-eq α) = ap pr1 α - pr2 (pair-eq α) = ap pr2 α - - isretr-pair-eq : - {s t : A × B} → ((pair-eq {s} {t}) ∘ (eq-pair' {s} {t})) ~ id - isretr-pair-eq {pair x y} {pair .x .y} (pair refl refl) = refl - - issec-pair-eq : - {s t : A × B} → ((eq-pair' {s} {t}) ∘ (pair-eq {s} {t})) ~ id - issec-pair-eq {pair x y} {pair .x .y} refl = refl - - abstract - is-equiv-eq-pair : - (s t : A × B) → is-equiv (eq-pair' {s} {t}) - is-equiv-eq-pair s t = - is-equiv-has-inverse pair-eq issec-pair-eq isretr-pair-eq - - equiv-eq-pair : - (s t : A × B) → Eq-prod s t ≃ (s ＝ t) - pr1 (equiv-eq-pair s t) = eq-pair' - pr2 (equiv-eq-pair s t) = is-equiv-eq-pair s t - - abstract - is-equiv-pair-eq : - (s t : A × B) → is-equiv (pair-eq {s} {t}) - is-equiv-pair-eq s t = - is-equiv-has-inverse eq-pair' isretr-pair-eq issec-pair-eq - - equiv-pair-eq : - (s t : A × B) → (s ＝ t) ≃ Eq-prod s t - pr1 (equiv-pair-eq s t) = pair-eq - pr2 (equiv-pair-eq s t) = is-equiv-pair-eq s t -``` - -## Properties - -### Commuting triangles for `eq-pair` - -```agda -module _ - {l1 l2 : Level} {A : UU l1} {B : UU l2} - where - - triangle-eq-pair : - {a0 a1 : A} {b0 b1 : B} (p : a0 ＝ a1) (q : b0 ＝ b1) → - eq-pair p q ＝ ((eq-pair p refl) ∙ (eq-pair refl q)) - triangle-eq-pair refl refl = refl - - triangle-eq-pair' : - {a0 a1 : A} {b0 b1 : B} (p : a0 ＝ a1) (q : b0 ＝ b1) → - eq-pair p q ＝ ((eq-pair refl q) ∙ (eq-pair p refl)) - triangle-eq-pair' refl refl = refl -``` - -### `eq-pair` preserves concatenation - -```agda -eq-pair-concat : - {l1 l2 : Level} {A : UU l1} {B : UU l2} {x x' x'' : A} {y y' y'' : B} - (p : x ＝ x') (p' : x' ＝ x'') (q : y ＝ y') (q' : y' ＝ y'') → - ( eq-pair {s = pair x y} {t = pair x'' y''} (p ∙ p') (q ∙ q')) ＝ - ( ( eq-pair {s = pair x y} {t = pair x' y'} p q) ∙ - ( eq-pair p' q')) -eq-pair-concat refl p' refl q' = refl -``` - -### `eq-pair` computes in the expected way when the action on paths of the projections is applies - -```agda -ap-pr1-eq-pair : - {l1 l2 : Level} {A : UU l1} {B : UU l2} - {x x' : A} (p : x ＝ x') {y y' : B} (q : y ＝ y') → - ap pr1 (eq-pair {s = pair x y} {pair x' y'} p q) ＝ p -ap-pr1-eq-pair refl refl = refl - -ap-pr2-eq-pair : - {l1 l2 : Level} {A : UU l1} {B : UU l2} - {x x' : A} (p : x ＝ x') {y y' : B} (q : y ＝ y') → - ap pr2 (eq-pair {s = pair x y} {pair x' y'} p q) ＝ q -ap-pr2-eq-pair refl refl = refl -``` - -## See also - -- Equality proofs in dependent pair types are characterized in - [`foundation.equality-dependent-pair-types`](foundation.equality-dependent-pair-types.md). -- Equality proofs in dependent product types are characterized in - [`foundation.equality-dependent-function-types`](foundation.equality-dependent-function-types.md). -- Equality proofs in coproduct types are characterized in - [`foundation.equality-coproduct-types`](foundation.equality-coproduct-types.md). diff --git a/src/foundation-core/equality-dependent-pair-types.lagda.md b/src/foundation-core/equality-dependent-pair-types.lagda.md index 59e8fcdf27..9b9559087e 100644 --- a/src/foundation-core/equality-dependent-pair-types.lagda.md +++ b/src/foundation-core/equality-dependent-pair-types.lagda.md @@ -9,12 +9,13 @@ module foundation-core.equality-dependent-pair-types where
Imports ```agda -open import foundation-core.dependent-pair-types +open import foundation.dependent-pair-types +open import foundation.functions +open import foundation.universe-levels + open import foundation-core.equivalences -open import foundation-core.functions open import foundation-core.homotopies open import foundation-core.identity-types -open import foundation-core.universe-levels ```
diff --git a/src/foundation-core/equality-fibers-of-maps.lagda.md b/src/foundation-core/equality-fibers-of-maps.lagda.md deleted file mode 100644 index 8c564a0889..0000000000 --- a/src/foundation-core/equality-fibers-of-maps.lagda.md +++ /dev/null @@ -1,120 +0,0 @@ -# Equality in the fibers of a map - -```agda -module foundation-core.equality-fibers-of-maps where -``` - -
Imports - -```agda -open import foundation.identity-types - -open import foundation-core.dependent-pair-types -open import foundation-core.equality-dependent-pair-types -open import foundation-core.equivalences -open import foundation-core.fibers-of-maps -open import foundation-core.functions -open import foundation-core.functoriality-dependent-pair-types -open import foundation-core.homotopies -open import foundation-core.universe-levels -``` - -
- -## Idea - -In the file -[`foundation-core.fibers-of-maps`](foundation-core.fibers-of-maps.md) we already -gave one characterization of the identity type of `fib f b`, for an arbitrary -map `f : A → B`. Here we give a second characterization, using the fibers of the -action on identifications of `f`. - -## Theorem - -For any map `f : A → B` any `b : B` and any `x y : fib f b`, there is an -equivalence - -```text -(x ＝ y) ≃ fib (ap f) ((pr2 x) ∙ (inv (pr2 y))) -``` - -### Proof - -```agda -module _ - {l1 l2 : Level} {A : UU l1} {B : UU l2} (f : A → B) {b : B} - where - - fib-ap-eq-fib-fiberwise : - (s t : fib f b) (p : (pr1 s) ＝ (pr1 t)) → - ((tr (λ (a : A) → (f a) ＝ b) p (pr2 s)) ＝ (pr2 t)) → - (ap f p ＝ ((pr2 s) ∙ (inv (pr2 t)))) - fib-ap-eq-fib-fiberwise (pair .x' p) (pair x' refl) refl = - inv ∘ (concat right-unit refl) - - abstract - is-fiberwise-equiv-fib-ap-eq-fib-fiberwise : - (s t : fib f b) → is-fiberwise-equiv (fib-ap-eq-fib-fiberwise s t) - is-fiberwise-equiv-fib-ap-eq-fib-fiberwise (pair x y) (pair .x refl) refl = - is-equiv-comp - ( inv) - ( concat right-unit refl) - ( is-equiv-concat right-unit refl) - ( is-equiv-inv (y ∙ refl) refl) - - fib-ap-eq-fib : - (s t : fib f b) → s ＝ t → - fib (ap f {x = pr1 s} {y = pr1 t}) ((pr2 s) ∙ (inv (pr2 t))) - pr1 (fib-ap-eq-fib s .s refl) = refl - pr2 (fib-ap-eq-fib s .s refl) = inv (right-inv (pr2 s)) - - triangle-fib-ap-eq-fib : - (s t : fib f b) → - ( fib-ap-eq-fib s t) ~ - ( (tot (fib-ap-eq-fib-fiberwise s t)) ∘ (pair-eq-Σ {s = s} {t})) - triangle-fib-ap-eq-fib (pair x refl) .(pair x refl) refl = refl - - abstract - is-equiv-fib-ap-eq-fib : (s t : fib f b) → is-equiv (fib-ap-eq-fib s t) - is-equiv-fib-ap-eq-fib s t = - is-equiv-comp-htpy - ( fib-ap-eq-fib s t) - ( tot (fib-ap-eq-fib-fiberwise s t)) - ( pair-eq-Σ {s = s} {t}) - ( triangle-fib-ap-eq-fib s t) - ( is-equiv-pair-eq-Σ s t) - ( is-equiv-tot-is-fiberwise-equiv - ( is-fiberwise-equiv-fib-ap-eq-fib-fiberwise s t)) - - equiv-fib-ap-eq-fib : - (s t : fib f b) → - (s ＝ t) ≃ fib (ap f {x = pr1 s} {y = pr1 t}) ((pr2 s) ∙ (inv (pr2 t))) - pr1 (equiv-fib-ap-eq-fib s t) = fib-ap-eq-fib s t - pr2 (equiv-fib-ap-eq-fib s t) = is-equiv-fib-ap-eq-fib s t - -module _ - {l1 l2 : Level} {A : UU l1} {B : UU l2} (f : A → B) (x y : A) - where - - eq-fib-fib-ap : - (q : f x ＝ f y) → (pair x q) ＝ (pair y refl) → fib (ap f {x} {y}) q - eq-fib-fib-ap q = - (tr (fib (ap f)) right-unit) ∘ (fib-ap-eq-fib f (pair x q) (pair y refl)) - - abstract - is-equiv-eq-fib-fib-ap : - (q : (f x) ＝ (f y)) → is-equiv (eq-fib-fib-ap q) - is-equiv-eq-fib-fib-ap q = - is-equiv-comp - ( tr (fib (ap f)) right-unit) - ( fib-ap-eq-fib f (pair x q) (pair y refl)) - ( is-equiv-fib-ap-eq-fib f (pair x q) (pair y refl)) - ( is-equiv-tr (fib (ap f)) right-unit) -``` - -## See also - -- Equality proofs in dependent pair types are characterized in - [`foundation.equality-dependent-pair-types`](foundation.equality-dependent-pair-types.md). -- Equality proofs in dependent function types are characterized in - [`foundation.equality-dependent-function-types`](foundation.equality-dependent-function-types.md). diff --git a/src/foundation-core/equivalence-induction.lagda.md b/src/foundation-core/equivalence-induction.lagda.md index 1275f83308..62ed07fccd 100644 --- a/src/foundation-core/equivalence-induction.lagda.md +++ b/src/foundation-core/equivalence-induction.lagda.md @@ -7,15 +7,16 @@ module foundation-core.equivalence-induction where
Imports ```agda +open import foundation.dependent-pair-types +open import foundation.functions +open import foundation.universe-levels + open import foundation-core.contractible-types -open import foundation-core.dependent-pair-types open import foundation-core.equivalences -open import foundation-core.functions open import foundation-core.homotopies open import foundation-core.identity-types open import foundation-core.sections open import foundation-core.singleton-induction -open import foundation-core.universe-levels ```
diff --git a/src/foundation-core/equivalence-relations.lagda.md b/src/foundation-core/equivalence-relations.lagda.md index 26eaae930f..86509c34bf 100644 --- a/src/foundation-core/equivalence-relations.lagda.md +++ b/src/foundation-core/equivalence-relations.lagda.md @@ -8,16 +8,16 @@ module foundation-core.equivalence-relations where ```agda open import foundation.binary-relations +open import foundation.dependent-pair-types open import foundation.inhabited-subtypes open import foundation.propositional-truncations open import foundation.unit-type +open import foundation.universe-levels open import foundation-core.cartesian-product-types -open import foundation-core.dependent-pair-types open import foundation-core.equivalences open import foundation-core.logical-equivalences open import foundation-core.propositions -open import foundation-core.universe-levels ```
diff --git a/src/foundation-core/equivalences.lagda.md b/src/foundation-core/equivalences.lagda.md index f804227979..523f00b928 100644 --- a/src/foundation-core/equivalences.lagda.md +++ b/src/foundation-core/equivalences.lagda.md @@ -9,15 +9,16 @@ module foundation-core.equivalences where
Imports ```agda +open import foundation.dependent-pair-types +open import foundation.functions +open import foundation.universe-levels + open import foundation-core.cartesian-product-types open import foundation-core.coherently-invertible-maps -open import foundation-core.dependent-pair-types -open import foundation-core.functions open import foundation-core.homotopies open import foundation-core.identity-types open import foundation-core.retractions open import foundation-core.sections -open import foundation-core.universe-levels ```
diff --git a/src/foundation-core/faithful-maps.lagda.md b/src/foundation-core/faithful-maps.lagda.md deleted file mode 100644 index a4df7cbb1b..0000000000 --- a/src/foundation-core/faithful-maps.lagda.md +++ /dev/null @@ -1,260 +0,0 @@ -# Faithful maps - -```agda -module foundation-core.faithful-maps where -``` - -
Imports - -```agda -open import foundation-core.0-maps -open import foundation-core.dependent-pair-types -open import foundation-core.embeddings -open import foundation-core.equivalences -open import foundation-core.functions -open import foundation-core.functoriality-dependent-pair-types -open import foundation-core.homotopies -open import foundation-core.identity-types -open import foundation-core.propositional-maps -open import foundation-core.sets -open import foundation-core.truncated-maps -open import foundation-core.truncation-levels -open import foundation-core.universe-levels -``` - -
- -## Idea - -Since we sometimes think of types as ∞-groupoids, with the groupoid structure -provided implicitly by the identity type and its induction principle, we can -think of maps as functors of ∞-groupoids. We borrow some terminology of -functors, and call a map faithful if it induces embeddings on identity types. - -## Definition - -```agda -module _ - {l1 l2 : Level} {A : UU l1} {B : UU l2} - where - - is-faithful : (A → B) → UU (l1 ⊔ l2) - is-faithful f = (x y : A) → is-emb (ap f {x} {y}) - -faithful-map : {l1 l2 : Level} → UU l1 → UU l2 → UU (l1 ⊔ l2) -faithful-map A B = Σ (A → B) is-faithful - -module _ - {l1 l2 : Level} {A : UU l1} {B : UU l2} - where - - map-faithful-map : faithful-map A B → A → B - map-faithful-map = pr1 - - is-faithful-map-faithful-map : - (f : faithful-map A B) → is-faithful (map-faithful-map f) - is-faithful-map-faithful-map = pr2 - - emb-ap-faithful-map : - (f : faithful-map A B) {x y : A} → - (x ＝ y) ↪ (map-faithful-map f x ＝ map-faithful-map f y) - pr1 (emb-ap-faithful-map f {x} {y}) = ap (map-faithful-map f) - pr2 (emb-ap-faithful-map f {x} {y}) = is-faithful-map-faithful-map f x y - - is-faithful-is-emb : {f : A → B} → is-emb f → is-faithful f - is-faithful-is-emb {f} H x y = is-emb-is-equiv (H x y) - - faithful-map-emb : (A ↪ B) → faithful-map A B - pr1 (faithful-map-emb f) = map-emb f - pr2 (faithful-map-emb f) = is-faithful-is-emb (is-emb-map-emb f) - - is-faithful-is-equiv : {f : A → B} → is-equiv f → is-faithful f - is-faithful-is-equiv H = is-faithful-is-emb (is-emb-is-equiv H) - - faithful-map-equiv : (A ≃ B) → faithful-map A B - pr1 (faithful-map-equiv e) = map-equiv e - pr2 (faithful-map-equiv e) = is-faithful-is-equiv (is-equiv-map-equiv e) - - emb-ap : (f : A ↪ B) (x y : A) → (x ＝ y) ↪ (map-emb f x ＝ map-emb f y) - pr1 (emb-ap f x y) = ap (map-emb f) {x} {y} - pr2 (emb-ap f x y) = is-faithful-is-emb (is-emb-map-emb f) x y -``` - -## Examples - -### The identity map is faithful - -```agda -module _ - {l : Level} {A : UU l} - where - - id-faithful-map : faithful-map A A - id-faithful-map = faithful-map-emb id-emb - - is-faithful-id-faithful-map : is-faithful (id {A = A}) - is-faithful-id-faithful-map = is-faithful-map-faithful-map id-faithful-map -``` - -### Any `0`-map is faithful - -```agda -module _ - {l1 l2 : Level} {A : UU l1} {B : UU l2} {f : A → B} - where - - is-0-map-is-faithful : is-faithful f → is-0-map f - is-0-map-is-faithful H = - is-trunc-map-is-trunc-map-ap neg-one-𝕋 f - ( λ x y → is-prop-map-is-emb (H x y)) - - is-faithful-is-0-map : is-0-map f → is-faithful f - is-faithful-is-0-map H x y = - is-emb-is-prop-map (is-trunc-map-ap-is-trunc-map neg-one-𝕋 f H x y) -``` - -## Properties - -### The projection map of a family of sets is faithful - -```agda -module _ - {l1 l2 : Level} {A : UU l1} - where - - abstract - is-faithful-pr1 : - {B : A → UU l2} → ((x : A) → is-set (B x)) → is-faithful (pr1 {B = B}) - is-faithful-pr1 H = is-faithful-is-0-map (is-0-map-pr1 H) - - pr1-faithful-map : - (B : A → Set l2) → faithful-map (Σ A (λ x → type-Set (B x))) A - pr1 (pr1-faithful-map B) = pr1 - pr2 (pr1-faithful-map B) = is-faithful-pr1 (λ x → is-set-type-Set (B x)) -``` - -### Faithful maps are closed under homotopies - -```agda -module _ - {l1 l2 : Level} {A : UU l1} {B : UU l2} {f g : A → B} (H : f ~ g) - where - - abstract - is-faithful-htpy : is-faithful g → is-faithful f - is-faithful-htpy K = - is-faithful-is-0-map (is-0-map-htpy H (is-0-map-is-faithful K)) -``` - -### Faithful maps are closed under composition - -```agda -module _ - {l1 l2 l3 : Level} {A : UU l1} {B : UU l2} {X : UU l3} - where - - abstract - is-faithful-comp : - (g : B → X) (h : A → B) → - is-faithful g → is-faithful h → is-faithful (g ∘ h) - is-faithful-comp g h is-faithful-g is-faithful-h = - is-faithful-is-0-map - ( is-0-map-comp g h - ( is-0-map-is-faithful is-faithful-g) - ( is-0-map-is-faithful is-faithful-h)) - - abstract - is-faithful-comp-htpy : - (f : A → X) (g : B → X) (h : A → B) (H : f ~ (g ∘ h)) → - is-faithful g → is-faithful h → is-faithful f - is-faithful-comp-htpy f g h H is-faithful-g is-faithful-h = - is-faithful-is-0-map - ( is-0-map-comp-htpy f g h H - ( is-0-map-is-faithful is-faithful-g) - ( is-0-map-is-faithful is-faithful-h)) -``` - -### If a composite is faithful, then its right factor is faithful - -```agda -module _ - {l1 l2 l3 : Level} {A : UU l1} {B : UU l2} {X : UU l3} - where - - is-faithful-right-factor : - (g : B → X) (h : A → B) → - is-faithful g → is-faithful (g ∘ h) → is-faithful h - is-faithful-right-factor g h is-faithful-g is-faithful-gh = - is-faithful-is-0-map - ( is-0-map-right-factor g h - ( is-0-map-is-faithful is-faithful-g) - ( is-0-map-is-faithful is-faithful-gh)) - - is-faithful-right-factor-htpy : - (f : A → X) (g : B → X) (h : A → B) (H : f ~ (g ∘ h)) → - is-faithful g → is-faithful f → is-faithful h - is-faithful-right-factor-htpy f g h H is-faithful-g is-faithful-f = - is-faithful-is-0-map - ( is-0-map-right-factor-htpy f g h H - ( is-0-map-is-faithful is-faithful-g) - ( is-0-map-is-faithful is-faithful-f)) -``` - -### The map on total spaces induced by a family of truncated maps is truncated - -```agda -module _ - {l1 l2 l3 : Level} {A : UU l1} {B : A → UU l2} {C : A → UU l3} - {f : (x : A) → B x → C x} - where - - is-faithful-tot : ((x : A) → is-faithful (f x)) → is-faithful (tot f) - is-faithful-tot H = - is-faithful-is-0-map (is-0-map-tot (λ x → is-0-map-is-faithful (H x))) - -module _ - {l1 l2 l3 : Level} {A : UU l1} {B : A → UU l2} {C : A → UU l3} - where - - tot-faithful-map : - ((x : A) → faithful-map (B x) (C x)) → faithful-map (Σ A B) (Σ A C) - pr1 (tot-faithful-map f) = tot (λ x → map-faithful-map (f x)) - pr2 (tot-faithful-map f) = - is-faithful-tot (λ x → is-faithful-map-faithful-map (f x)) - -module _ - {l1 l2 l3 : Level} {A : UU l1} {B : UU l2} - where - - module _ - {f : A → B} (C : B → UU l3) - where - - abstract - is-faithful-map-Σ-map-base : - is-faithful f → is-faithful (map-Σ-map-base f C) - is-faithful-map-Σ-map-base H = - is-faithful-is-0-map - ( is-0-map-map-Σ-map-base C (is-0-map-is-faithful H)) - - faithful-map-Σ-faithful-map-base : - (f : faithful-map A B) (C : B → UU l3) → - faithful-map (Σ A (λ a → C (map-faithful-map f a))) (Σ B C) - pr1 (faithful-map-Σ-faithful-map-base f C) = - map-Σ-map-base (map-faithful-map f) C - pr2 (faithful-map-Σ-faithful-map-base f C) = - is-faithful-map-Σ-map-base C (is-faithful-map-faithful-map f) - -module _ - {l1 l2 l3 l4 : Level} {A : UU l1} {B : UU l2} {C : A → UU l3} - (D : B → UU l4) {f : A → B} {g : (x : A) → C x → D (f x)} - where - - is-faithful-map-Σ : - is-faithful f → ((x : A) → is-faithful (g x)) → is-faithful (map-Σ D f g) - is-faithful-map-Σ H K = - is-faithful-is-0-map - ( is-0-map-map-Σ D - ( is-0-map-is-faithful H) - ( λ x → is-0-map-is-faithful (K x))) -``` diff --git a/src/foundation-core/fibers-of-maps.lagda.md b/src/foundation-core/fibers-of-maps.lagda.md index 485674a254..5022b6a2b0 100644 --- a/src/foundation-core/fibers-of-maps.lagda.md +++ b/src/foundation-core/fibers-of-maps.lagda.md @@ -7,14 +7,14 @@ module foundation-core.fibers-of-maps where
Imports ```agda +open import foundation.dependent-pair-types open import foundation.function-extensionality +open import foundation.functions +open import foundation.universe-levels -open import foundation-core.dependent-pair-types open import foundation-core.equivalences -open import foundation-core.functions open import foundation-core.homotopies open import foundation-core.identity-types -open import foundation-core.universe-levels ```
diff --git a/src/foundation-core/function-extensionality.lagda.md b/src/foundation-core/function-extensionality.lagda.md index 7f19112c2b..c8a73a4d36 100644 --- a/src/foundation-core/function-extensionality.lagda.md +++ b/src/foundation-core/function-extensionality.lagda.md @@ -9,11 +9,12 @@ module foundation-core.function-extensionality where
Imports ```agda +open import foundation.functions +open import foundation.universe-levels + open import foundation-core.equivalences -open import foundation-core.functions open import foundation-core.homotopies open import foundation-core.identity-types -open import foundation-core.universe-levels ```
diff --git a/src/foundation-core/functions.lagda.md b/src/foundation-core/functions.lagda.md deleted file mode 100644 index b2cf903e9d..0000000000 --- a/src/foundation-core/functions.lagda.md +++ /dev/null @@ -1,87 +0,0 @@ -# Functions - -```agda -{-# OPTIONS --safe #-} - -module foundation-core.functions where -``` - -
Imports - -```agda -open import foundation-core.universe-levels -``` - -
- -## Idea - -Functions are primitive in Agda. Here we construct some basic functions - -## Examples - -### The identity function - -```agda -id : {l : Level} {A : UU l} → A → A -id a = a - -idω : {A : UUω} → A → A -idω a = a -``` - -### Dependent composition of functions - -```agda -_∘_ : - {l1 l2 l3 : Level} {A : UU l1} {B : A → UU l2} {C : (a : A) → B a → UU l3} → - ({a : A} → (b : B a) → C a b) → (f : (a : A) → B a) → (a : A) → C a (f a) -(g ∘ f) a = g (f a) -``` - -### Evaluation at a point - -```agda -ev-point : - {l1 l2 : Level} {A : UU l1} (a : A) {P : A → UU l2} → ((x : A) → P x) → P a -ev-point a f = f a - -ev-point' : - {l1 l2 : Level} {A : UU l1} (a : A) {X : UU l2} → (A → X) → X -ev-point' a f = f a -``` - -### Precomposition functions - -```agda -precomp-Π : - {l1 l2 l3 : Level} {A : UU l1} {B : UU l2} (f : A → B) (C : B → UU l3) → - ((b : B) → C b) → ((a : A) → C (f a)) -precomp-Π f C h a = h (f a) - -precomp : - {l1 l2 l3 : Level} {A : UU l1} {B : UU l2} (f : A → B) (C : UU l3) → - (B → C) → (A → C) -precomp f C = precomp-Π f (λ b → C) -``` - -### Postcomposition functions - -```agda -postcomp : - {l1 l2 l3 : Level} {X : UU l1} {Y : UU l2} (A : UU l3) → - (X → Y) → (A → X) → (A → Y) -postcomp A f h = f ∘ h - -map-Π : - {l1 l2 l3 : Level} {I : UU l1} {A : I → UU l2} {B : I → UU l3} - (f : (i : I) → A i → B i) → - ((i : I) → A i) → ((i : I) → B i) -map-Π f h i = f i (h i) - -map-Π' : - {l1 l2 l3 l4 : Level} {I : UU l1} {A : I → UU l2} {B : I → UU l3} - {J : UU l4} (α : J → I) → - ((i : I) → A i → B i) → ((j : J) → A (α j)) → ((j : J) → B (α j)) -map-Π' α f = map-Π (λ j → f (α j)) -``` diff --git a/src/foundation-core/functoriality-dependent-function-types.lagda.md b/src/foundation-core/functoriality-dependent-function-types.lagda.md index 22bc3a9970..a3646b3b63 100644 --- a/src/foundation-core/functoriality-dependent-function-types.lagda.md +++ b/src/foundation-core/functoriality-dependent-function-types.lagda.md @@ -7,22 +7,22 @@ module foundation-core.functoriality-dependent-function-types where
Imports ```agda +open import foundation.dependent-pair-types open import foundation.function-extensionality +open import foundation.functions open import foundation.type-theoretic-principle-of-choice +open import foundation.universe-levels open import foundation-core.coherently-invertible-maps open import foundation-core.constant-maps open import foundation-core.contractible-maps open import foundation-core.contractible-types -open import foundation-core.dependent-pair-types open import foundation-core.equivalences open import foundation-core.fibers-of-maps -open import foundation-core.functions open import foundation-core.functoriality-dependent-pair-types open import foundation-core.homotopies open import foundation-core.identity-types open import foundation-core.path-split-maps -open import foundation-core.universe-levels ```
diff --git a/src/foundation-core/functoriality-dependent-pair-types.lagda.md b/src/foundation-core/functoriality-dependent-pair-types.lagda.md index ada4ee85c4..0b6007d626 100644 --- a/src/foundation-core/functoriality-dependent-pair-types.lagda.md +++ b/src/foundation-core/functoriality-dependent-pair-types.lagda.md @@ -7,16 +7,17 @@ module foundation-core.functoriality-dependent-pair-types where
Imports ```agda +open import foundation.dependent-pair-types +open import foundation.functions +open import foundation.universe-levels + open import foundation-core.contractible-maps open import foundation-core.contractible-types -open import foundation-core.dependent-pair-types open import foundation-core.equality-dependent-pair-types open import foundation-core.equivalences open import foundation-core.fibers-of-maps -open import foundation-core.functions open import foundation-core.homotopies open import foundation-core.identity-types -open import foundation-core.universe-levels ```
diff --git a/src/foundation-core/functoriality-fibers-of-maps.lagda.md b/src/foundation-core/functoriality-fibers-of-maps.lagda.md deleted file mode 100644 index 04993cd4a4..0000000000 --- a/src/foundation-core/functoriality-fibers-of-maps.lagda.md +++ /dev/null @@ -1,120 +0,0 @@ -# The functoriality of `fib` - -```agda -module foundation-core.functoriality-fibers-of-maps where -``` - -
Imports - -```agda -open import foundation-core.cones-over-cospans -open import foundation-core.dependent-pair-types -open import foundation-core.equality-dependent-pair-types -open import foundation-core.fibers-of-maps -open import foundation-core.functions -open import foundation-core.functoriality-dependent-pair-types -open import foundation-core.homotopies -open import foundation-core.identity-types -open import foundation-core.universe-levels -``` - -
- -## Idea - -Any commuting square - -induces a map between the fibers of the vertical maps - -## Definitions - -### Any cone induces a family of maps between the fibers of the vertical maps - -```agda -module _ - {l1 l2 l3 l4 : Level} {A : UU l1} {B : UU l2} {C : UU l3} {X : UU l4} - (f : A → X) (g : B → X) (c : cone f g C) - where - - map-fib-cone : (x : A) → fib (pr1 c) x → fib g (f x) - pr1 (map-fib-cone x t) = pr1 (pr2 c) (pr1 t) - pr2 (map-fib-cone x t) = (inv (pr2 (pr2 c) (pr1 t))) ∙ (ap f (pr2 t)) - -map-fib-cone-id : - {l1 l2 : Level} {B : UU l1} {X : UU l2} (g : B → X) (x : X) → - map-fib-cone id g (triple g id refl-htpy) x ~ id -map-fib-cone-id g .(g b) (pair b refl) = - refl -``` - -## Properties - -### Computing `map-fib-cone` of a horizontal pasting of cones - -```agda -module _ - {l1 l2 l3 l4 l5 l6 : Level} - {A : UU l1} {B : UU l2} {C : UU l3} {X : UU l4} {Y : UU l5} {Z : UU l6} - (i : X → Y) (j : Y → Z) (h : C → Z) - where - - map-fib-pasting-horizontal-cone : - (c : cone j h B) (d : cone i (pr1 c) A) → (x : X) → - ( map-fib-cone (j ∘ i) h (pasting-horizontal-cone i j h c d) x) ~ - ( (map-fib-cone j h c (i x)) ∘ (map-fib-cone i (pr1 c) d x)) - map-fib-pasting-horizontal-cone - (pair g (pair q K)) (pair f (pair p H)) .(f a) (pair a refl) = - eq-pair-Σ - ( refl) - ( ( ap - ( concat' (h (q (p a))) refl) - ( distributive-inv-concat (ap j (H a)) (K (p a)))) ∙ - ( ( assoc (inv (K (p a))) (inv (ap j (H a))) refl) ∙ - ( ap - ( concat (inv (K (p a))) (j (i (f a)))) - ( ( ap (concat' (j (g (p a))) refl) (inv (ap-inv j (H a)))) ∙ - ( inv (ap-concat j (inv (H a)) refl)))))) -``` - -### Computing `map-fib-cone` of a horizontal pasting of cones - -```agda -module _ - {l1 l2 l3 l4 l5 l6 : Level} - {A : UU l1} {B : UU l2} {C : UU l3} {X : UU l4} {Y : UU l5} {Z : UU l6} - (f : C → Z) (g : Y → Z) (h : X → Y) - where - - map-fib-pasting-vertical-cone : - (c : cone f g B) (d : cone (pr1 (pr2 c)) h A) (x : C) → - ( ( map-fib-cone f (g ∘ h) (pasting-vertical-cone f g h c d) x) ∘ - ( inv-map-compute-fib-comp (pr1 c) (pr1 d) x)) ~ - ( ( inv-map-compute-fib-comp g h (f x)) ∘ - ( map-Σ - ( λ t → fib h (pr1 t)) - ( map-fib-cone f g c x) - ( λ t → map-fib-cone (pr1 (pr2 c)) h d (pr1 t)))) - map-fib-pasting-vertical-cone - (pair p (pair q H)) (pair p' (pair q' H')) .(p (p' a)) - (pair (pair .(p' a) refl) (pair a refl)) = - eq-pair-Σ refl - ( ( right-unit) ∙ - ( ( distributive-inv-concat (H (p' a)) (ap g (H' a))) ∙ - ( ( ap - ( concat (inv (ap g (H' a))) (f (p (p' a)))) - ( inv right-unit)) ∙ - ( ap - ( concat' (g (h (q' a))) - ( pr2 - ( map-fib-cone f g - ( triple p q H) - ( p (p' a)) - ( pair (p' a) refl)))) - ( ( inv (ap-inv g (H' a))) ∙ - ( ap (ap g) (inv right-unit))))))) -``` - -## See also - -- Equality proofs in the fiber of a map are characterized in - [`foundation.equality-fibers-of-maps`](foundation.equality-fibers-of-maps.md). diff --git a/src/foundation-core/functoriality-function-types.lagda.md b/src/foundation-core/functoriality-function-types.lagda.md index 97a94f7507..9b14fe042c 100644 --- a/src/foundation-core/functoriality-function-types.lagda.md +++ b/src/foundation-core/functoriality-function-types.lagda.md @@ -7,16 +7,16 @@ module foundation-core.functoriality-function-types where
Imports ```agda +open import foundation.dependent-pair-types open import foundation.function-extensionality +open import foundation.functions +open import foundation.universe-levels open import foundation-core.contractible-maps open import foundation-core.contractible-types -open import foundation-core.dependent-pair-types open import foundation-core.equivalences -open import foundation-core.functions open import foundation-core.homotopies open import foundation-core.identity-types -open import foundation-core.universe-levels ```
diff --git a/src/foundation-core/fundamental-theorem-of-identity-types.lagda.md b/src/foundation-core/fundamental-theorem-of-identity-types.lagda.md deleted file mode 100644 index f9b03ef4be..0000000000 --- a/src/foundation-core/fundamental-theorem-of-identity-types.lagda.md +++ /dev/null @@ -1,131 +0,0 @@ -# The fundamental theorem of identity types - -```agda -module foundation-core.fundamental-theorem-of-identity-types where -``` - -
Imports - -```agda -open import foundation-core.contractible-types -open import foundation-core.dependent-pair-types -open import foundation-core.equivalences -open import foundation-core.functoriality-dependent-pair-types -open import foundation-core.homotopies -open import foundation-core.identity-types -open import foundation-core.retractions -open import foundation-core.sections -open import foundation-core.universe-levels -``` - -
- -## Idea - -The fundamental theorem of identity type provides a way to characterize identity -types. It uses the fact that a family of maps `f : (x : A) → a ＝ x → B x` is a -family of equivalences if and only if it induces an equivalence -`Σ A (Id a) → Σ A B` on total spaces. Note that the total space `Σ A (Id a)` is -contractible. Therefore, any map `Σ A (Id a) → Σ A B` is an equivalence if and -only if `Σ A B` is contractible. - -## Theorem - -For any family of maps `f : (x : A) → a ＝ x → B x`, the following are -equivalent: - -1. Each `f x` is an equivalence -2. The total space `Σ A B` is contractible. - -```agda -module _ - {l1 l2 : Level} {A : UU l1} {B : A → UU l2} {a : A} - where - - abstract - fundamental-theorem-id : - is-contr (Σ A B) → (f : (x : A) → a ＝ x → B x) → is-fiberwise-equiv f - fundamental-theorem-id is-contr-AB f = - is-fiberwise-equiv-is-equiv-tot - ( is-equiv-is-contr (tot f) (is-contr-total-path a) is-contr-AB) - - abstract - fundamental-theorem-id' : - (f : (x : A) → a ＝ x → B x) → is-fiberwise-equiv f → is-contr (Σ A B) - fundamental-theorem-id' f is-fiberwise-equiv-f = - is-contr-is-equiv' - ( Σ A (Id a)) - ( tot f) - ( is-equiv-tot-is-fiberwise-equiv is-fiberwise-equiv-f) - ( is-contr-total-path a) -``` - -## Corollaries - -```agda -module _ - {l1 l2 : Level} {A : UU l1} {B : A → UU l2} (a : A) (b : B a) - where - - abstract - fundamental-theorem-id-J : - is-contr (Σ A B) → is-fiberwise-equiv (ind-Id a (λ x p → B x) b) - fundamental-theorem-id-J is-contr-AB = - fundamental-theorem-id is-contr-AB (ind-Id a (λ x p → B x) b) - - abstract - fundamental-theorem-id-J' : - (is-fiberwise-equiv (ind-Id a (λ x p → B x) b)) → is-contr (Σ A B) - fundamental-theorem-id-J' H = - is-contr-is-equiv' - ( Σ A (Id a)) - ( tot (ind-Id a (λ x p → B x) b)) - ( is-equiv-tot-is-fiberwise-equiv H) - ( is-contr-total-path a) -``` - -### Retracts of the identity type are equivalent to the identity type - -```agda -module _ - {l1 l2 : Level} {A : UU l1} {B : A → UU l2} (a : A) - where - - abstract - fundamental-theorem-id-retr : - (i : (x : A) → B x → a ＝ x) → (R : (x : A) → retr (i x)) → - is-fiberwise-equiv i - fundamental-theorem-id-retr i R = - is-fiberwise-equiv-is-equiv-tot - ( is-equiv-is-contr (tot i) - ( is-contr-retract-of (Σ _ (λ y → a ＝ y)) - ( pair (tot i) - ( pair (tot λ x → pr1 (R x)) - ( ( inv-htpy (preserves-comp-tot i (λ x → pr1 (R x)))) ∙h - ( ( tot-htpy λ x → pr2 (R x)) ∙h (tot-id B))))) - ( is-contr-total-path a)) - ( is-contr-total-path a)) -``` - -### The fundamental theorem of identity types, using sections - -```agda -module _ - {l1 l2 : Level} {A : UU l1} {B : A → UU l2} (a : A) - where - - abstract - fundamental-theorem-id-sec : - (f : (x : A) → a ＝ x → B x) → ((x : A) → sec (f x)) → - is-fiberwise-equiv f - fundamental-theorem-id-sec f sec-f x = - is-equiv-sec-is-equiv (f x) (sec-f x) (is-fiberwise-equiv-i x) - where - i : (x : A) → B x → a ＝ x - i = λ x → pr1 (sec-f x) - retr-i : (x : A) → retr (i x) - pr1 (retr-i x) = f x - pr2 (retr-i x) = pr2 (sec-f x) - is-fiberwise-equiv-i : is-fiberwise-equiv i - is-fiberwise-equiv-i = fundamental-theorem-id-retr a i retr-i -``` diff --git a/src/foundation-core/homotopies.lagda.md b/src/foundation-core/homotopies.lagda.md index 41d884d426..fd7a4fe7ff 100644 --- a/src/foundation-core/homotopies.lagda.md +++ b/src/foundation-core/homotopies.lagda.md @@ -9,9 +9,10 @@ module foundation-core.homotopies where
Imports ```agda -open import foundation-core.functions +open import foundation.functions +open import foundation.universe-levels + open import foundation-core.identity-types -open import foundation-core.universe-levels ```
diff --git a/src/foundation-core/identity-systems.lagda.md b/src/foundation-core/identity-systems.lagda.md deleted file mode 100644 index 06ccd2e5b8..0000000000 --- a/src/foundation-core/identity-systems.lagda.md +++ /dev/null @@ -1,81 +0,0 @@ -# Identity systems - -```agda -module foundation-core.identity-systems where -``` - -
Imports - -```agda -open import foundation-core.contractible-types -open import foundation-core.dependent-pair-types -open import foundation-core.equivalences -open import foundation-core.fundamental-theorem-of-identity-types -open import foundation-core.identity-types -open import foundation-core.sections -open import foundation-core.universe-levels -``` - -
- -## Idea - -A unary identity system on a type `A` equipped with a point `a : A` consists of -a type family `B` over `A` equipped with a point `b : B a` that satisfies an -induction principle analogous to the induction principle of the identity type at -`a`. - -```agda -module _ - {l1 l2 : Level} (l : Level) {A : UU l1} (B : A → UU l2) (a : A) (b : B a) - where - - IND-identity-system : UU (l1 ⊔ l2 ⊔ lsuc l) - IND-identity-system = - ( P : (x : A) (y : B x) → UU l) → - sec (λ (h : (x : A) (y : B x) → P x y) → h a b) -``` - -## Properties - -### A type family over `A` is an identity system if and only if it is equivalent to the identity type - -```agda -module _ - {l1 l2 : Level} {A : UU l1} {B : A → UU l2} (a : A) (b : B a) - where - - abstract - Ind-identity-system : - (is-contr-AB : is-contr (Σ A B)) → - {l : Level} → IND-identity-system l B a b - pr1 (Ind-identity-system is-contr-AB P) p x y = - tr - ( fam-Σ P) - ( eq-is-contr is-contr-AB) - ( p) - pr2 (Ind-identity-system is-contr-AB P) p = - ap - ( λ t → tr (fam-Σ P) t p) - ( eq-is-contr' - ( is-prop-is-contr is-contr-AB (pair a b) (pair a b)) - ( eq-is-contr is-contr-AB) - ( refl)) - - abstract - is-contr-total-space-IND-identity-system : - ({l : Level} → IND-identity-system l B a b) → is-contr (Σ A B) - pr1 (pr1 (is-contr-total-space-IND-identity-system ind)) = a - pr2 (pr1 (is-contr-total-space-IND-identity-system ind)) = b - pr2 (is-contr-total-space-IND-identity-system ind) (pair x y) = - pr1 (ind (λ x' y' → (pair a b) ＝ (pair x' y'))) refl x y - - abstract - fundamental-theorem-id-IND-identity-system : - ({l : Level} → IND-identity-system l B a b) → - (f : (x : A) → a ＝ x → B x) → (x : A) → is-equiv (f x) - fundamental-theorem-id-IND-identity-system ind f = - fundamental-theorem-id - ( is-contr-total-space-IND-identity-system ind) - ( f) -``` diff --git a/src/foundation-core/identity-types.lagda.md b/src/foundation-core/identity-types.lagda.md index c83f7bd483..06d0ba0c06 100644 --- a/src/foundation-core/identity-types.lagda.md +++ b/src/foundation-core/identity-types.lagda.md @@ -9,9 +9,10 @@ module foundation-core.identity-types where
Imports ```agda +open import foundation.functions +open import foundation.universe-levels + open import foundation-core.constant-maps -open import foundation-core.functions -open import foundation-core.universe-levels ```
diff --git a/src/foundation-core/injective-maps.lagda.md b/src/foundation-core/injective-maps.lagda.md index 5a5bfe7346..3d5827272a 100644 --- a/src/foundation-core/injective-maps.lagda.md +++ b/src/foundation-core/injective-maps.lagda.md @@ -7,18 +7,19 @@ module foundation-core.injective-maps where
Imports ```agda +open import foundation.dependent-pair-types +open import foundation.functions +open import foundation.universe-levels + open import foundation-core.contractible-types -open import foundation-core.dependent-pair-types open import foundation-core.embeddings open import foundation-core.equivalences -open import foundation-core.functions open import foundation-core.homotopies open import foundation-core.identity-types open import foundation-core.propositional-maps open import foundation-core.propositions open import foundation-core.sections open import foundation-core.sets -open import foundation-core.universe-levels ```
diff --git a/src/foundation-core/involutions.lagda.md b/src/foundation-core/involutions.lagda.md index bff728f691..7b78a45da1 100644 --- a/src/foundation-core/involutions.lagda.md +++ b/src/foundation-core/involutions.lagda.md @@ -7,14 +7,15 @@ module foundation-core.involutions where
Imports ```agda -open import foundation-core.automorphisms -open import foundation-core.dependent-pair-types +open import foundation.automorphisms +open import foundation.dependent-pair-types +open import foundation.functions +open import foundation.universe-levels + open import foundation-core.equivalences -open import foundation-core.functions open import foundation-core.homotopies open import foundation-core.truncated-types open import foundation-core.truncation-levels -open import foundation-core.universe-levels open import structured-types.pointed-types ``` diff --git a/src/foundation-core/logical-equivalences.lagda.md b/src/foundation-core/logical-equivalences.lagda.md index ac0c74494b..2cbd73d4b4 100644 --- a/src/foundation-core/logical-equivalences.lagda.md +++ b/src/foundation-core/logical-equivalences.lagda.md @@ -7,12 +7,13 @@ module foundation-core.logical-equivalences where
Imports ```agda +open import foundation.dependent-pair-types +open import foundation.functions +open import foundation.universe-levels + open import foundation-core.cartesian-product-types -open import foundation-core.dependent-pair-types open import foundation-core.equivalences -open import foundation-core.functions open import foundation-core.propositions -open import foundation-core.universe-levels ```
diff --git a/src/foundation-core/morphisms-cospans.lagda.md b/src/foundation-core/morphisms-cospans.lagda.md deleted file mode 100644 index f7729d33de..0000000000 --- a/src/foundation-core/morphisms-cospans.lagda.md +++ /dev/null @@ -1,143 +0,0 @@ -# Morphisms of cospans - -```agda -{-# OPTIONS --safe #-} - -module foundation-core.morphisms-cospans where -``` - -
Imports - -```agda -open import foundation-core.cartesian-product-types -open import foundation-core.dependent-pair-types -open import foundation-core.functions -open import foundation-core.homotopies -open import foundation-core.universe-levels -``` - -
- -## Idea - -A morphism of cospans is a commuting diagram of the form - -```text - A -----> X <----- B - | | | - | | | - V V V - A' ----> X' <---- B' -``` - -## Definitions - -### Morphisms of cospans - -```agda -hom-cospan : - {l1 l2 l3 l1' l2' l3' : Level} - {A : UU l1} {B : UU l2} {X : UU l3} (f : A → X) (g : B → X) - {A' : UU l1'} {B' : UU l2'} {X' : UU l3'} (f' : A' → X') (g' : B' → X') → - UU (l1 ⊔ l2 ⊔ l3 ⊔ l1' ⊔ l2' ⊔ l3') -hom-cospan {A = A} {B} {X} f g {A'} {B'} {X'} f' g' = - Σ (A → A') (λ hA → - Σ (B → B') (λ hB → - Σ (X → X') (λ hX → - ((f' ∘ hA) ~ (hX ∘ f)) × ((g' ∘ hB) ~ (hX ∘ g))))) -``` - -### Identity morphisms of cospans - -```agda -id-hom-cospan : - {l1 l2 l3 l1' l2' l3' : Level} - {A : UU l1} {B : UU l2} {X : UU l3} (f : A → X) (g : B → X) → - hom-cospan f g f g -pr1 (id-hom-cospan f g) = id -pr1 (pr2 (id-hom-cospan f g)) = id -pr1 (pr2 (pr2 (id-hom-cospan f g))) = id -pr1 (pr2 (pr2 (pr2 (id-hom-cospan f g)))) = refl-htpy -pr2 (pr2 (pr2 (pr2 (id-hom-cospan f g)))) = refl-htpy -``` - -### Rotating cospans of cospans - -```agda -cospan-hom-cospan-rotate : - {l1 l2 l3 l1' l2' l3' l1'' l2'' l3'' : Level} - {A : UU l1} {B : UU l2} {X : UU l3} (f : A → X) (g : B → X) - {A' : UU l1'} {B' : UU l2'} {X' : UU l3'} (f' : A' → X') (g' : B' → X') - {A'' : UU l1''} {B'' : UU l2''} {X'' : UU l3''} - (f'' : A'' → X'') (g'' : B'' → X'') - (h : hom-cospan f' g' f g) (h' : hom-cospan f'' g'' f g) → - hom-cospan (pr1 h) (pr1 h') (pr1 (pr2 (pr2 h))) (pr1 (pr2 (pr2 h'))) -pr1 - ( cospan-hom-cospan-rotate f g f' g' f'' g'' - ( pair hA (pair hB (pair hX (pair HA HB)))) - ( pair hA' (pair hB' (pair hX' (pair HA' HB'))))) = f' -pr1 - ( pr2 - ( cospan-hom-cospan-rotate f g f' g' f'' g'' - ( pair hA (pair hB (pair hX (pair HA HB)))) - ( pair hA' (pair hB' (pair hX' (pair HA' HB')))))) = f'' -pr1 - ( pr2 - ( pr2 - ( cospan-hom-cospan-rotate f g f' g' f'' g'' - ( pair hA (pair hB (pair hX (pair HA HB)))) - ( pair hA' (pair hB' (pair hX' (pair HA' HB'))))))) = f -pr1 - ( pr2 - ( pr2 - ( pr2 - ( cospan-hom-cospan-rotate f g f' g' f'' g'' - ( pair hA (pair hB (pair hX (pair HA HB)))) - ( pair hA' (pair hB' (pair hX' (pair HA' HB')))))))) = inv-htpy HA -pr2 - ( pr2 - ( pr2 - ( pr2 - ( cospan-hom-cospan-rotate f g f' g' f'' g'' - ( pair hA (pair hB (pair hX (pair HA HB)))) - ( pair hA' (pair hB' (pair hX' (pair HA' HB')))))))) = inv-htpy HA' - -cospan-hom-cospan-rotate' : - {l1 l2 l3 l1' l2' l3' l1'' l2'' l3'' : Level} - {A : UU l1} {B : UU l2} {X : UU l3} (f : A → X) (g : B → X) - {A' : UU l1'} {B' : UU l2'} {X' : UU l3'} (f' : A' → X') (g' : B' → X') - {A'' : UU l1''} {B'' : UU l2''} {X'' : UU l3''} - (f'' : A'' → X'') (g'' : B'' → X'') - (h : hom-cospan f' g' f g) (h' : hom-cospan f'' g'' f g) → - hom-cospan - (pr1 (pr2 h)) (pr1 (pr2 h')) (pr1 (pr2 (pr2 h))) (pr1 (pr2 (pr2 h'))) -pr1 - ( cospan-hom-cospan-rotate' f g f' g' f'' g'' - ( pair hA (pair hB (pair hX (pair HA HB)))) - ( pair hA' (pair hB' (pair hX' (pair HA' HB'))))) = g' -pr1 - ( pr2 - ( cospan-hom-cospan-rotate' f g f' g' f'' g'' - ( pair hA (pair hB (pair hX (pair HA HB)))) - ( pair hA' (pair hB' (pair hX' (pair HA' HB')))))) = g'' -pr1 - ( pr2 - ( pr2 - ( cospan-hom-cospan-rotate' f g f' g' f'' g'' - ( pair hA (pair hB (pair hX (pair HA HB)))) - ( pair hA' (pair hB' (pair hX' (pair HA' HB'))))))) = g -pr1 - ( pr2 - ( pr2 - ( pr2 - ( cospan-hom-cospan-rotate' f g f' g' f'' g'' - ( pair hA (pair hB (pair hX (pair HA HB)))) - ( pair hA' (pair hB' (pair hX' (pair HA' HB')))))))) = inv-htpy HB -pr2 - ( pr2 - ( pr2 - ( pr2 - ( cospan-hom-cospan-rotate' f g f' g' f'' g'' - ( pair hA (pair hB (pair hX (pair HA HB)))) - ( pair hA' (pair hB' (pair hX' (pair HA' HB')))))))) = inv-htpy HB' -``` diff --git a/src/foundation-core/negation.lagda.md b/src/foundation-core/negation.lagda.md index 709110b47b..e565a46dfc 100644 --- a/src/foundation-core/negation.lagda.md +++ b/src/foundation-core/negation.lagda.md @@ -7,8 +7,9 @@ module foundation-core.negation where
Imports ```agda +open import foundation.universe-levels + open import foundation-core.empty-types -open import foundation-core.universe-levels ```
diff --git a/src/foundation-core/path-split-maps.lagda.md b/src/foundation-core/path-split-maps.lagda.md index f8f526be14..56a34e9152 100644 --- a/src/foundation-core/path-split-maps.lagda.md +++ b/src/foundation-core/path-split-maps.lagda.md @@ -7,14 +7,15 @@ module foundation-core.path-split-maps where
Imports ```agda +open import foundation.dependent-pair-types +open import foundation.functions +open import foundation.universe-levels + open import foundation-core.cartesian-product-types open import foundation-core.coherently-invertible-maps -open import foundation-core.dependent-pair-types open import foundation-core.equivalences -open import foundation-core.functions open import foundation-core.identity-types open import foundation-core.sections -open import foundation-core.universe-levels ```
diff --git a/src/foundation-core/propositional-maps.lagda.md b/src/foundation-core/propositional-maps.lagda.md index 90a0f66f75..8eec0ccb64 100644 --- a/src/foundation-core/propositional-maps.lagda.md +++ b/src/foundation-core/propositional-maps.lagda.md @@ -7,14 +7,15 @@ module foundation-core.propositional-maps where
Imports ```agda +open import foundation.dependent-pair-types +open import foundation.fundamental-theorem-of-identity-types +open import foundation.universe-levels + open import foundation-core.contractible-types -open import foundation-core.dependent-pair-types open import foundation-core.embeddings open import foundation-core.fibers-of-maps -open import foundation-core.fundamental-theorem-of-identity-types open import foundation-core.identity-types open import foundation-core.propositions -open import foundation-core.universe-levels ```
diff --git a/src/foundation-core/propositions.lagda.md b/src/foundation-core/propositions.lagda.md index ab2e572b99..159ddfde16 100644 --- a/src/foundation-core/propositions.lagda.md +++ b/src/foundation-core/propositions.lagda.md @@ -7,16 +7,17 @@ module foundation-core.propositions where
Imports ```agda +open import foundation.dependent-pair-types +open import foundation.functions +open import foundation.universe-levels + open import foundation-core.cartesian-product-types open import foundation-core.contractible-types -open import foundation-core.dependent-pair-types open import foundation-core.equality-dependent-pair-types open import foundation-core.equivalences open import foundation-core.function-extensionality -open import foundation-core.functions open import foundation-core.homotopies open import foundation-core.identity-types -open import foundation-core.universe-levels ```
diff --git a/src/foundation-core/pullbacks.lagda.md b/src/foundation-core/pullbacks.lagda.md index 9faaa6b966..64d0f2a7ef 100644 --- a/src/foundation-core/pullbacks.lagda.md +++ b/src/foundation-core/pullbacks.lagda.md @@ -7,27 +7,27 @@ module foundation-core.pullbacks where
Imports ```agda +open import foundation.cones-over-cospans +open import foundation.dependent-pair-types +open import foundation.equality-cartesian-product-types +open import foundation.functions open import foundation.functoriality-cartesian-product-types +open import foundation.functoriality-fibers-of-maps open import foundation.identity-types +open import foundation.morphisms-cospans open import foundation.structure-identity-principle +open import foundation.type-arithmetic-dependent-pair-types open import foundation.type-theoretic-principle-of-choice +open import foundation.universe-levels open import foundation-core.cartesian-product-types -open import foundation-core.cones-over-cospans -open import foundation-core.dependent-pair-types open import foundation-core.diagonal-maps-of-types -open import foundation-core.equality-cartesian-product-types open import foundation-core.equality-dependent-pair-types open import foundation-core.equivalences open import foundation-core.fibers-of-maps -open import foundation-core.functions open import foundation-core.functoriality-dependent-pair-types -open import foundation-core.functoriality-fibers-of-maps open import foundation-core.homotopies -open import foundation-core.morphisms-cospans -open import foundation-core.type-arithmetic-dependent-pair-types open import foundation-core.universal-property-pullbacks -open import foundation-core.universe-levels ```
diff --git a/src/foundation-core/retractions.lagda.md b/src/foundation-core/retractions.lagda.md index 714a793513..df4622cadc 100644 --- a/src/foundation-core/retractions.lagda.md +++ b/src/foundation-core/retractions.lagda.md @@ -9,11 +9,12 @@ module foundation-core.retractions where
Imports ```agda -open import foundation-core.dependent-pair-types -open import foundation-core.functions +open import foundation.dependent-pair-types +open import foundation.functions +open import foundation.universe-levels + open import foundation-core.homotopies open import foundation-core.identity-types -open import foundation-core.universe-levels ```
diff --git a/src/foundation-core/sections.lagda.md b/src/foundation-core/sections.lagda.md index 75a4aefb23..751e185ca7 100644 --- a/src/foundation-core/sections.lagda.md +++ b/src/foundation-core/sections.lagda.md @@ -9,10 +9,11 @@ module foundation-core.sections where
Imports ```agda -open import foundation-core.dependent-pair-types -open import foundation-core.functions +open import foundation.dependent-pair-types +open import foundation.functions +open import foundation.universe-levels + open import foundation-core.homotopies -open import foundation-core.universe-levels ```
diff --git a/src/foundation-core/sets.lagda.md b/src/foundation-core/sets.lagda.md index 32eab931a6..9a8d083dc6 100644 --- a/src/foundation-core/sets.lagda.md +++ b/src/foundation-core/sets.lagda.md @@ -7,15 +7,16 @@ module foundation-core.sets where
Imports ```agda +open import foundation.dependent-pair-types +open import foundation.fundamental-theorem-of-identity-types +open import foundation.universe-levels + open import foundation-core.contractible-types -open import foundation-core.dependent-pair-types open import foundation-core.equivalences -open import foundation-core.fundamental-theorem-of-identity-types open import foundation-core.identity-types open import foundation-core.propositions open import foundation-core.truncated-types open import foundation-core.truncation-levels -open import foundation-core.universe-levels ```
diff --git a/src/foundation-core/singleton-induction.lagda.md b/src/foundation-core/singleton-induction.lagda.md index 5ce2b68ec5..51bae44392 100644 --- a/src/foundation-core/singleton-induction.lagda.md +++ b/src/foundation-core/singleton-induction.lagda.md @@ -7,13 +7,14 @@ module foundation-core.singleton-induction where
Imports ```agda +open import foundation.dependent-pair-types +open import foundation.functions +open import foundation.universe-levels + open import foundation-core.contractible-types -open import foundation-core.dependent-pair-types -open import foundation-core.functions open import foundation-core.homotopies open import foundation-core.identity-types open import foundation-core.sections -open import foundation-core.universe-levels ```
diff --git a/src/foundation-core/small-types.lagda.md b/src/foundation-core/small-types.lagda.md index 7f87d2b7a2..e1935fd2aa 100644 --- a/src/foundation-core/small-types.lagda.md +++ b/src/foundation-core/small-types.lagda.md @@ -7,6 +7,7 @@ module foundation-core.small-types where
Imports ```agda +open import foundation.dependent-pair-types open import foundation.equivalences open import foundation.functoriality-coproduct-types open import foundation.functoriality-dependent-function-types @@ -14,18 +15,17 @@ open import foundation.identity-types open import foundation.mere-equivalences open import foundation.propositional-truncations open import foundation.raising-universe-levels +open import foundation.type-arithmetic-dependent-pair-types open import foundation.unit-type open import foundation.univalence +open import foundation.universe-levels open import foundation-core.cartesian-product-types open import foundation-core.contractible-types open import foundation-core.coproduct-types -open import foundation-core.dependent-pair-types open import foundation-core.functoriality-dependent-pair-types open import foundation-core.logical-equivalences open import foundation-core.propositions -open import foundation-core.type-arithmetic-dependent-pair-types -open import foundation-core.universe-levels ```
diff --git a/src/foundation-core/subtype-identity-principle.lagda.md b/src/foundation-core/subtype-identity-principle.lagda.md deleted file mode 100644 index f7ed2d53c1..0000000000 --- a/src/foundation-core/subtype-identity-principle.lagda.md +++ /dev/null @@ -1,108 +0,0 @@ -# The subtype identity principle - -```agda -module foundation-core.subtype-identity-principle where -``` - -
Imports - -```agda -open import foundation-core.contractible-types -open import foundation-core.dependent-pair-types -open import foundation-core.equivalences -open import foundation-core.functions -open import foundation-core.fundamental-theorem-of-identity-types -open import foundation-core.identity-types -open import foundation-core.propositions -open import foundation-core.type-arithmetic-dependent-pair-types -open import foundation-core.universe-levels -``` - -
- -## Idea - -The subtype identity principle allows us to efficiently characterize the -identity type of a subtype, using a characterization of the identity type of the -base type. - -## Lemma - -The following is a general construction that will help us show that the identity -type of a subtype agrees with the identity type of the original type. We already -know that the first projection of a family of propositions is an embedding, but -the following lemma still has its uses. - -```agda -module _ - {l1 l2 : Level} {A : UU l1} {B : A → UU l2} - where - - abstract - is-contr-total-Eq-subtype : - {l3 : Level} {P : A → UU l3} → - is-contr (Σ A B) → ((x : A) → is-prop (P x)) → - (a : A) (b : B a) (p : P a) → - is-contr (Σ (Σ A P) (B ∘ pr1)) - is-contr-total-Eq-subtype {l3} {P} - is-contr-AB is-subtype-P a b p = - is-contr-equiv - ( Σ (Σ A B) (P ∘ pr1)) - ( equiv-right-swap-Σ) - ( is-contr-equiv - ( P a) - ( left-unit-law-Σ-is-contr - ( is-contr-AB) - ( pair a b)) - ( is-proof-irrelevant-is-prop (is-subtype-P a) p)) -``` - -## Theorem - -### The subtype identity principle - -```agda -module _ - {l1 l2 l3 : Level} {A : UU l1} {P : A → UU l2} - (is-prop-P : (x : A) → is-prop (P x)) {Eq-A : A → UU l3} - {a : A} (p : P a) (refl-A : Eq-A a) - where - - abstract - subtype-identity-principle : - {f : (x : A) → a ＝ x → Eq-A x} - (h : (z : (Σ A P)) → (pair a p) ＝ z → Eq-A (pr1 z)) → - ((x : A) → is-equiv (f x)) → (z : Σ A P) → is-equiv (h z) - subtype-identity-principle {f} h H = - fundamental-theorem-id - ( is-contr-total-Eq-subtype - ( fundamental-theorem-id' f H) - ( is-prop-P) - ( a) - ( refl-A) - ( p)) - ( h) - -module _ - {l1 l2 l3 : Level} {A : UU l1} (P : A → Prop l2) {Eq-A : A → UU l3} - {a : A} (p : type-Prop (P a)) (refl-A : Eq-A a) - where - - map-extensionality-type-subtype : - (f : (x : A) → (a ＝ x) ≃ Eq-A x) → - (z : Σ A (type-Prop ∘ P)) → (pair a p) ＝ z → Eq-A (pr1 z) - map-extensionality-type-subtype f .(pair a p) refl = refl-A - - extensionality-type-subtype : - (f : (x : A) → (a ＝ x) ≃ Eq-A x) → - (z : Σ A (type-Prop ∘ P)) → (pair a p ＝ z) ≃ Eq-A (pr1 z) - pr1 (extensionality-type-subtype f z) = map-extensionality-type-subtype f z - pr2 (extensionality-type-subtype f z) = - subtype-identity-principle - ( is-prop-type-Prop ∘ P) - ( p) - ( refl-A) - ( map-extensionality-type-subtype f) - ( is-equiv-map-equiv ∘ f) - ( z) -``` diff --git a/src/foundation-core/subtypes.lagda.md b/src/foundation-core/subtypes.lagda.md index 20577bd95d..012bcab7c1 100644 --- a/src/foundation-core/subtypes.lagda.md +++ b/src/foundation-core/subtypes.lagda.md @@ -7,21 +7,22 @@ module foundation-core.subtypes where
Imports ```agda -open import foundation-core.dependent-pair-types +open import foundation.dependent-pair-types +open import foundation.functions +open import foundation.subtype-identity-principle +open import foundation.universe-levels + open import foundation-core.embeddings open import foundation-core.equivalences open import foundation-core.fibers-of-maps -open import foundation-core.functions open import foundation-core.functoriality-dependent-pair-types open import foundation-core.identity-types open import foundation-core.logical-equivalences open import foundation-core.propositional-maps open import foundation-core.propositions open import foundation-core.sets -open import foundation-core.subtype-identity-principle open import foundation-core.truncated-types open import foundation-core.truncation-levels -open import foundation-core.universe-levels ```
diff --git a/src/foundation-core/truncated-maps.lagda.md b/src/foundation-core/truncated-maps.lagda.md index 126f4b2c63..273bc4cd14 100644 --- a/src/foundation-core/truncated-maps.lagda.md +++ b/src/foundation-core/truncated-maps.lagda.md @@ -7,20 +7,21 @@ module foundation-core.truncated-maps where
Imports ```agda +open import foundation.dependent-pair-types +open import foundation.equality-fibers-of-maps +open import foundation.functions +open import foundation.universe-levels + open import foundation-core.commuting-squares-of-maps open import foundation-core.contractible-maps -open import foundation-core.dependent-pair-types -open import foundation-core.equality-fibers-of-maps open import foundation-core.equivalences open import foundation-core.fibers-of-maps -open import foundation-core.functions open import foundation-core.functoriality-dependent-pair-types open import foundation-core.homotopies open import foundation-core.identity-types open import foundation-core.propositional-maps open import foundation-core.truncated-types open import foundation-core.truncation-levels -open import foundation-core.universe-levels ```
diff --git a/src/foundation-core/truncated-types.lagda.md b/src/foundation-core/truncated-types.lagda.md index 9cc0efd413..13bd9f8eaf 100644 --- a/src/foundation-core/truncated-types.lagda.md +++ b/src/foundation-core/truncated-types.lagda.md @@ -7,13 +7,14 @@ module foundation-core.truncated-types where
Imports ```agda +open import foundation.dependent-pair-types +open import foundation.equality-cartesian-product-types open import foundation.function-extensionality +open import foundation.universe-levels open import foundation-core.cartesian-product-types open import foundation-core.contractible-types -open import foundation-core.dependent-pair-types open import foundation-core.embeddings -open import foundation-core.equality-cartesian-product-types open import foundation-core.equality-dependent-pair-types open import foundation-core.equivalences open import foundation-core.homotopies @@ -21,7 +22,6 @@ open import foundation-core.identity-types open import foundation-core.propositions open import foundation-core.retractions open import foundation-core.truncation-levels -open import foundation-core.universe-levels ```
diff --git a/src/foundation-core/truncation-levels.lagda.md b/src/foundation-core/truncation-levels.lagda.md index 48d667a4f3..ded37e55c1 100644 --- a/src/foundation-core/truncation-levels.lagda.md +++ b/src/foundation-core/truncation-levels.lagda.md @@ -9,7 +9,7 @@ module foundation-core.truncation-levels where
Imports ```agda -open import foundation-core.universe-levels +open import foundation.universe-levels ```
diff --git a/src/foundation-core/type-arithmetic-cartesian-product-types.lagda.md b/src/foundation-core/type-arithmetic-cartesian-product-types.lagda.md deleted file mode 100644 index e556afcbc3..0000000000 --- a/src/foundation-core/type-arithmetic-cartesian-product-types.lagda.md +++ /dev/null @@ -1,161 +0,0 @@ -# Type arithmetic for cartesian product types - -```agda -module foundation-core.type-arithmetic-cartesian-product-types where -``` - -
Imports - -```agda -open import foundation-core.cartesian-product-types -open import foundation-core.contractible-types -open import foundation-core.dependent-pair-types -open import foundation-core.equality-cartesian-product-types -open import foundation-core.equivalences -open import foundation-core.functions -open import foundation-core.homotopies -open import foundation-core.identity-types -open import foundation-core.propositions -open import foundation-core.type-arithmetic-dependent-pair-types -open import foundation-core.universe-levels -``` - -
- -## Idea - -We prove laws for the manipulation of cartesian products with respect to -themselves and dependent pair types. - -## Laws - -### Commutativity of cartesian products - -```agda -module _ - {l1 l2 : Level} {A : UU l1} {B : UU l2} - where - - map-commutative-prod : A × B → B × A - pr1 (map-commutative-prod (pair a b)) = b - pr2 (map-commutative-prod (pair a b)) = a - - map-inv-commutative-prod : B × A → A × B - pr1 (map-inv-commutative-prod (pair b a)) = a - pr2 (map-inv-commutative-prod (pair b a)) = b - - issec-map-inv-commutative-prod : - (map-commutative-prod ∘ map-inv-commutative-prod) ~ id - issec-map-inv-commutative-prod (pair b a) = refl - - isretr-map-inv-commutative-prod : - (map-inv-commutative-prod ∘ map-commutative-prod) ~ id - isretr-map-inv-commutative-prod (pair a b) = refl - - is-equiv-map-commutative-prod : is-equiv map-commutative-prod - is-equiv-map-commutative-prod = - is-equiv-has-inverse - map-inv-commutative-prod - issec-map-inv-commutative-prod - isretr-map-inv-commutative-prod - - commutative-prod : (A × B) ≃ (B × A) - pr1 commutative-prod = map-commutative-prod - pr2 commutative-prod = is-equiv-map-commutative-prod -``` - -### Associativity of cartesian products - -```agda -module _ - {l1 l2 l3 : Level} (A : UU l1) (B : UU l2) (C : UU l3) - where - - map-associative-prod : (A × B) × C → A × (B × C) - map-associative-prod = map-associative-Σ A (λ x → B) (λ w → C) - - map-inv-associative-prod : A × (B × C) → (A × B) × C - map-inv-associative-prod = map-inv-associative-Σ A (λ x → B) (λ w → C) - - issec-map-inv-associative-prod : - (map-associative-prod ∘ map-inv-associative-prod) ~ id - issec-map-inv-associative-prod = - issec-map-inv-associative-Σ A (λ x → B) (λ w → C) - - isretr-map-inv-associative-prod : - (map-inv-associative-prod ∘ map-associative-prod) ~ id - isretr-map-inv-associative-prod = - isretr-map-inv-associative-Σ A (λ x → B) (λ w → C) - - is-equiv-map-associative-prod : is-equiv map-associative-prod - is-equiv-map-associative-prod = - is-equiv-map-associative-Σ A (λ x → B) (λ w → C) - - associative-prod : ((A × B) × C) ≃ (A × (B × C)) - associative-prod = associative-Σ A (λ x → B) (λ w → C) -``` - -### The unit laws of cartesian product types with respect to contractible types - -```agda -module _ - {l1 l2 : Level} {A : UU l1} {B : UU l2} (is-contr-B : is-contr B) - where - - right-unit-law-prod-is-contr : (A × B) ≃ A - right-unit-law-prod-is-contr = right-unit-law-Σ-is-contr (λ a → is-contr-B) - -module _ - {l1 l2 : Level} {A : UU l1} {B : UU l2} (is-contr-A : is-contr A) - where - - left-unit-law-prod-is-contr : (A × B) ≃ B - left-unit-law-prod-is-contr = - left-unit-law-Σ-is-contr is-contr-A (center is-contr-A) - - is-equiv-pr2-prod-is-contr : is-equiv (pr2 {B = λ a → B}) - is-equiv-pr2-prod-is-contr = - is-equiv-comp - ( pr1) - ( map-commutative-prod) - ( is-equiv-map-commutative-prod) - ( is-equiv-pr1-is-contr λ b → is-contr-A) - - equiv-pr2-prod-is-contr : (A × B) ≃ B - pr1 equiv-pr2-prod-is-contr = pr2 - pr2 equiv-pr2-prod-is-contr = is-equiv-pr2-prod-is-contr -``` - -### Adding redundant property - -```agda -equiv-add-redundant-prop : - {l1 l2 : Level} {A : UU l1} {B : UU l2} → - (is-prop B) → (f : A → B) → (A ≃ (A × B)) -pr1 (equiv-add-redundant-prop is-prop-B f) a = a , f a -pr2 (equiv-add-redundant-prop is-prop-B f) = - is-equiv-has-inverse - ( pr1) - ( λ p → eq-pair refl (eq-is-prop is-prop-B)) - ( λ a → refl) -``` - -## See also - -- Functorial properties of cartesian products are recorded in - [`foundation.functoriality-cartesian-product-types`](foundation.functoriality-cartesian-product-types.md). -- Equality proofs in cartesian product types are characterized in - [`foundation.equality-cartesian-product-types`](foundation.equality-cartesian-product-types.md). -- The universal property of cartesian product types is treated in - [`foundation.universal-property-cartesian-product-types`](foundation.universal-property-cartesian-product-types.md). - -- Arithmetical laws involving dependent pair types are recorded in - [`foundation.type-arithmetic-dependent-pair-types`](foundation.type-arithmetic-dependent-pair-types.md). - - Arithmetical laws involving dependent product types are recorded in - [`foundation.type-arithmetic-dependent-function-types`](foundation.type-arithmetic-dependent-function-types.md). -- Arithmetical laws involving coproduct types are recorded in - [`foundation.type-arithmetic-coproduct-types`](foundation.type-arithmetic-coproduct-types.md). -- Arithmetical laws involving the unit type are recorded in - [`foundation.type-arithmetic-unit-type`](foundation.type-arithmetic-unit-type.md). -- Arithmetical laws involving the empty type are recorded in - [`foundation.type-arithmetic-empty-type`](foundation.type-arithmetic-empty-type.md). diff --git a/src/foundation-core/type-arithmetic-dependent-pair-types.lagda.md b/src/foundation-core/type-arithmetic-dependent-pair-types.lagda.md deleted file mode 100644 index d1e412b200..0000000000 --- a/src/foundation-core/type-arithmetic-dependent-pair-types.lagda.md +++ /dev/null @@ -1,421 +0,0 @@ -# Type arithmetic for dependent pair types - -```agda -module foundation-core.type-arithmetic-dependent-pair-types where -``` - -
Imports - -```agda -open import foundation-core.cartesian-product-types -open import foundation-core.contractible-maps -open import foundation-core.contractible-types -open import foundation-core.dependent-pair-types -open import foundation-core.equality-dependent-pair-types -open import foundation-core.equivalences -open import foundation-core.fibers-of-maps -open import foundation-core.functions -open import foundation-core.homotopies -open import foundation-core.identity-types -open import foundation-core.singleton-induction -open import foundation-core.universe-levels -``` - -
- -## Idea - -We prove laws for the manipulation of dependent pair types with respect to -themselves and arithmetical laws with respect to contractible types. - -## Properties - -### The left unit law for Σ using a contractible base type - -```agda -module _ - {l1 l2 : Level} {A : UU l1} {B : A → UU l2} (C : is-contr A) (a : A) - where - - map-inv-left-unit-law-Σ-is-contr : B a → Σ A B - map-inv-left-unit-law-Σ-is-contr b = pair a b - - map-left-unit-law-Σ-is-contr : Σ A B → B a - map-left-unit-law-Σ-is-contr = - ind-Σ - ( ind-singleton-is-contr a C - ( λ x → B x → B a) - ( id)) - - issec-map-inv-left-unit-law-Σ-is-contr : - ( map-left-unit-law-Σ-is-contr ∘ map-inv-left-unit-law-Σ-is-contr) ~ id - issec-map-inv-left-unit-law-Σ-is-contr b = - ap - ( λ (f : B a → B a) → f b) - ( compute-ind-singleton-is-contr a C (λ x → B x → B a) id) - - isretr-map-inv-left-unit-law-Σ-is-contr : - ( map-inv-left-unit-law-Σ-is-contr ∘ map-left-unit-law-Σ-is-contr) ~ id - isretr-map-inv-left-unit-law-Σ-is-contr = - ind-Σ - ( ind-singleton-is-contr a C - ( λ x → - ( y : B x) → - Id - ( ( map-inv-left-unit-law-Σ-is-contr ∘ - map-left-unit-law-Σ-is-contr) - ( pair x y)) - ( pair x y)) - ( λ y → ap - ( map-inv-left-unit-law-Σ-is-contr) - ( ap - ( λ f → f y) - ( compute-ind-singleton-is-contr a C (λ x → B x → B a) id)))) - - is-equiv-map-left-unit-law-Σ-is-contr : - is-equiv map-left-unit-law-Σ-is-contr - is-equiv-map-left-unit-law-Σ-is-contr = - is-equiv-has-inverse - map-inv-left-unit-law-Σ-is-contr - issec-map-inv-left-unit-law-Σ-is-contr - isretr-map-inv-left-unit-law-Σ-is-contr - - left-unit-law-Σ-is-contr : Σ A B ≃ B a - pr1 left-unit-law-Σ-is-contr = map-left-unit-law-Σ-is-contr - pr2 left-unit-law-Σ-is-contr = is-equiv-map-left-unit-law-Σ-is-contr - - abstract - is-equiv-map-inv-left-unit-law-Σ-is-contr : - is-equiv map-inv-left-unit-law-Σ-is-contr - is-equiv-map-inv-left-unit-law-Σ-is-contr = - is-equiv-has-inverse - map-left-unit-law-Σ-is-contr - isretr-map-inv-left-unit-law-Σ-is-contr - issec-map-inv-left-unit-law-Σ-is-contr - - inv-left-unit-law-Σ-is-contr : B a ≃ Σ A B - pr1 inv-left-unit-law-Σ-is-contr = map-inv-left-unit-law-Σ-is-contr - pr2 inv-left-unit-law-Σ-is-contr = is-equiv-map-inv-left-unit-law-Σ-is-contr -``` - -### Right unit law for dependent pair types - -```agda -module _ - {l1 l2 : Level} {A : UU l1} {B : A → UU l2} - where - - abstract - is-equiv-pr1-is-contr : ((a : A) → is-contr (B a)) → is-equiv (pr1 {B = B}) - is-equiv-pr1-is-contr is-contr-B = - is-equiv-is-contr-map - ( λ x → is-contr-equiv - ( B x) - ( equiv-fib-pr1 B x) - ( is-contr-B x)) - - equiv-pr1 : ((a : A) → is-contr (B a)) → (Σ A B) ≃ A - pr1 (equiv-pr1 is-contr-B) = pr1 - pr2 (equiv-pr1 is-contr-B) = is-equiv-pr1-is-contr is-contr-B - - right-unit-law-Σ-is-contr : ((a : A) → is-contr (B a)) → (Σ A B) ≃ A - right-unit-law-Σ-is-contr = equiv-pr1 - - abstract - is-contr-is-equiv-pr1 : is-equiv (pr1 {B = B}) → ((a : A) → is-contr (B a)) - is-contr-is-equiv-pr1 is-equiv-pr1-B a = - is-contr-equiv' - ( fib pr1 a) - ( equiv-fib-pr1 B a) - ( is-contr-map-is-equiv is-equiv-pr1-B a) - - map-inv-right-unit-law-Σ-is-contr : - ((a : A) → is-contr (B a)) → A → Σ A B - map-inv-right-unit-law-Σ-is-contr H a = (a , center (H a)) - - issec-map-inv-right-unit-law-Σ-is-contr : - (H : (a : A) → is-contr (B a)) → - ( pr1 ∘ map-inv-right-unit-law-Σ-is-contr H) ~ id - issec-map-inv-right-unit-law-Σ-is-contr H = refl-htpy - - isretr-map-inv-right-unit-law-Σ-is-contr : - (H : (a : A) → is-contr (B a)) → - ( map-inv-right-unit-law-Σ-is-contr H ∘ pr1) ~ id - isretr-map-inv-right-unit-law-Σ-is-contr H (a , b) = - eq-pair-Σ refl (eq-is-contr (H a)) - - is-equiv-map-inv-right-unit-law-Σ-is-contr : - (H : (a : A) → is-contr (B a)) → - is-equiv (map-inv-right-unit-law-Σ-is-contr H) - is-equiv-map-inv-right-unit-law-Σ-is-contr H = - is-equiv-has-inverse - ( pr1) - ( isretr-map-inv-right-unit-law-Σ-is-contr H) - ( issec-map-inv-right-unit-law-Σ-is-contr H) - - inv-right-unit-law-Σ-is-contr : - (H : (a : A) → is-contr (B a)) → A ≃ Σ A B - pr1 (inv-right-unit-law-Σ-is-contr H) = map-inv-right-unit-law-Σ-is-contr H - pr2 (inv-right-unit-law-Σ-is-contr H) = - is-equiv-map-inv-right-unit-law-Σ-is-contr H -``` - -### Associativity of dependent pair types - -There are two ways to express associativity for dependent pair types. We -formalize both ways. - -```agda -module _ - {l1 l2 l3 : Level} (A : UU l1) (B : A → UU l2) (C : Σ A B → UU l3) - where - - map-associative-Σ : Σ (Σ A B) C → Σ A (λ x → Σ (B x) (λ y → C (pair x y))) - pr1 (map-associative-Σ ((x , y) , z)) = x - pr1 (pr2 (map-associative-Σ ((x , y) , z))) = y - pr2 (pr2 (map-associative-Σ ((x , y) , z))) = z - - map-inv-associative-Σ : Σ A (λ x → Σ (B x) (λ y → C (pair x y))) → Σ (Σ A B) C - pr1 (pr1 (map-inv-associative-Σ (x , y , z))) = x - pr2 (pr1 (map-inv-associative-Σ (x , y , z))) = y - pr2 (map-inv-associative-Σ (x , y , z)) = z - - isretr-map-inv-associative-Σ : - (map-inv-associative-Σ ∘ map-associative-Σ) ~ id - isretr-map-inv-associative-Σ (pair (pair x y) z) = refl - - issec-map-inv-associative-Σ : (map-associative-Σ ∘ map-inv-associative-Σ) ~ id - issec-map-inv-associative-Σ (pair x (pair y z)) = refl - - abstract - is-equiv-map-associative-Σ : is-equiv map-associative-Σ - is-equiv-map-associative-Σ = - is-equiv-has-inverse - map-inv-associative-Σ - issec-map-inv-associative-Σ - isretr-map-inv-associative-Σ - - associative-Σ : Σ (Σ A B) C ≃ Σ A (λ x → Σ (B x) (λ y → C (pair x y))) - pr1 associative-Σ = map-associative-Σ - pr2 associative-Σ = is-equiv-map-associative-Σ - - inv-associative-Σ : Σ A (λ x → Σ (B x) (λ y → C (pair x y))) ≃ Σ (Σ A B) C - pr1 inv-associative-Σ = map-inv-associative-Σ - pr2 inv-associative-Σ = - is-equiv-has-inverse - map-associative-Σ - isretr-map-inv-associative-Σ - issec-map-inv-associative-Σ -``` - -### Associativity, second formulation - -```agda -module _ - {l1 l2 l3 : Level} (A : UU l1) (B : A → UU l2) (C : (x : A) → B x → UU l3) - where - - map-associative-Σ' : - Σ (Σ A B) (λ w → C (pr1 w) (pr2 w)) → Σ A (λ x → Σ (B x) (C x)) - pr1 (map-associative-Σ' ((x , y) , z)) = x - pr1 (pr2 (map-associative-Σ' ((x , y) , z))) = y - pr2 (pr2 (map-associative-Σ' ((x , y) , z))) = z - - map-inv-associative-Σ' : - Σ A (λ x → Σ (B x) (C x)) → Σ (Σ A B) (λ w → C (pr1 w) (pr2 w)) - pr1 (pr1 (map-inv-associative-Σ' (x , y , z))) = x - pr2 (pr1 (map-inv-associative-Σ' (x , y , z))) = y - pr2 (map-inv-associative-Σ' (x , y , z)) = z - - issec-map-inv-associative-Σ' : - (map-associative-Σ' ∘ map-inv-associative-Σ') ~ id - issec-map-inv-associative-Σ' (pair x (pair y z)) = refl - - isretr-map-inv-associative-Σ' : - ( map-inv-associative-Σ' ∘ map-associative-Σ') ~ id - isretr-map-inv-associative-Σ' (pair (pair x y) z) = refl - - is-equiv-map-associative-Σ' : is-equiv map-associative-Σ' - is-equiv-map-associative-Σ' = - is-equiv-has-inverse - map-inv-associative-Σ' - issec-map-inv-associative-Σ' - isretr-map-inv-associative-Σ' - - associative-Σ' : - Σ (Σ A B) (λ w → C (pr1 w) (pr2 w)) ≃ Σ A (λ x → Σ (B x) (C x)) - pr1 associative-Σ' = map-associative-Σ' - pr2 associative-Σ' = is-equiv-map-associative-Σ' - - inv-associative-Σ' : - Σ A (λ x → Σ (B x) (C x)) ≃ Σ (Σ A B) (λ w → C (pr1 w) (pr2 w)) - pr1 inv-associative-Σ' = map-inv-associative-Σ' - pr2 inv-associative-Σ' = - is-equiv-has-inverse - map-associative-Σ' - isretr-map-inv-associative-Σ' - issec-map-inv-associative-Σ' -``` - -### The interchange law - -```agda -module _ - { l1 l2 l3 l4 : Level} { A : UU l1} {B : A → UU l2} {C : A → UU l3} - ( D : (x : A) → B x → C x → UU l4) - where - - map-interchange-Σ-Σ : - Σ (Σ A B) (λ t → Σ (C (pr1 t)) (D (pr1 t) (pr2 t))) → - Σ (Σ A C) (λ t → Σ (B (pr1 t)) (λ y → D (pr1 t) y (pr2 t))) - pr1 (pr1 (map-interchange-Σ-Σ t)) = pr1 (pr1 t) - pr2 (pr1 (map-interchange-Σ-Σ t)) = pr1 (pr2 t) - pr1 (pr2 (map-interchange-Σ-Σ t)) = pr2 (pr1 t) - pr2 (pr2 (map-interchange-Σ-Σ t)) = pr2 (pr2 t) - - map-inv-interchange-Σ-Σ : - Σ (Σ A C) (λ t → Σ (B (pr1 t)) (λ y → D (pr1 t) y (pr2 t))) → - Σ (Σ A B) (λ t → Σ (C (pr1 t)) (D (pr1 t) (pr2 t))) - pr1 (pr1 (map-inv-interchange-Σ-Σ t)) = pr1 (pr1 t) - pr2 (pr1 (map-inv-interchange-Σ-Σ t)) = pr1 (pr2 t) - pr1 (pr2 (map-inv-interchange-Σ-Σ t)) = pr2 (pr1 t) - pr2 (pr2 (map-inv-interchange-Σ-Σ t)) = pr2 (pr2 t) - - issec-map-inv-interchange-Σ-Σ : - ( map-interchange-Σ-Σ ∘ map-inv-interchange-Σ-Σ) ~ id - issec-map-inv-interchange-Σ-Σ (pair (pair a c) (pair b d)) = refl - - isretr-map-inv-interchange-Σ-Σ : - ( map-inv-interchange-Σ-Σ ∘ map-interchange-Σ-Σ) ~ id - isretr-map-inv-interchange-Σ-Σ (pair (pair a b) (pair c d)) = refl - - abstract - is-equiv-map-interchange-Σ-Σ : is-equiv map-interchange-Σ-Σ - is-equiv-map-interchange-Σ-Σ = - is-equiv-has-inverse - map-inv-interchange-Σ-Σ - issec-map-inv-interchange-Σ-Σ - isretr-map-inv-interchange-Σ-Σ - - interchange-Σ-Σ : - Σ (Σ A B) (λ t → Σ (C (pr1 t)) (D (pr1 t) (pr2 t))) ≃ - Σ (Σ A C) (λ t → Σ (B (pr1 t)) (λ y → D (pr1 t) y (pr2 t))) - pr1 interchange-Σ-Σ = map-interchange-Σ-Σ - pr2 interchange-Σ-Σ = is-equiv-map-interchange-Σ-Σ - - eq-interchange-Σ-Σ-is-contr : - {a : A} {b : B a} → is-contr (Σ (C a) (D a b)) → - {x y : Σ (C a) (D a b)} → - map-equiv interchange-Σ-Σ ((a , b) , x) ＝ - map-equiv interchange-Σ-Σ ((a , b) , y) - eq-interchange-Σ-Σ-is-contr H = - ap (map-equiv interchange-Σ-Σ) (ap (pair _) (eq-is-contr H)) -``` - -### Swapping the order of quantification in a Σ-type, on the left - -```agda -module _ - {l1 l2 l3 : Level} {A : UU l1} {B : UU l2} {C : A → B → UU l3} - where - - map-left-swap-Σ : Σ A (λ x → Σ B (C x)) → Σ B (λ y → Σ A (λ x → C x y)) - pr1 (map-left-swap-Σ (a , b , c)) = b - pr1 (pr2 (map-left-swap-Σ (a , b , c))) = a - pr2 (pr2 (map-left-swap-Σ (a , b , c))) = c - - map-inv-left-swap-Σ : - Σ B (λ y → Σ A (λ x → C x y)) → Σ A (λ x → Σ B (C x)) - pr1 (map-inv-left-swap-Σ (b , a , c)) = a - pr1 (pr2 (map-inv-left-swap-Σ (b , a , c))) = b - pr2 (pr2 (map-inv-left-swap-Σ (b , a , c))) = c - - isretr-map-inv-left-swap-Σ : (map-inv-left-swap-Σ ∘ map-left-swap-Σ) ~ id - isretr-map-inv-left-swap-Σ (pair a (pair b c)) = refl - - issec-map-inv-left-swap-Σ : (map-left-swap-Σ ∘ map-inv-left-swap-Σ) ~ id - issec-map-inv-left-swap-Σ (pair b (pair a c)) = refl - - abstract - is-equiv-map-left-swap-Σ : is-equiv map-left-swap-Σ - is-equiv-map-left-swap-Σ = - is-equiv-has-inverse - map-inv-left-swap-Σ - issec-map-inv-left-swap-Σ - isretr-map-inv-left-swap-Σ - - equiv-left-swap-Σ : Σ A (λ a → Σ B (C a)) ≃ Σ B (λ b → Σ A (λ a → C a b)) - pr1 equiv-left-swap-Σ = map-left-swap-Σ - pr2 equiv-left-swap-Σ = is-equiv-map-left-swap-Σ -``` - -### Swapping the order of quantification in a Σ-type, on the right - -```agda -module _ - {l1 l2 l3 : Level} {A : UU l1} {B : A → UU l2} {C : A → UU l3} - where - - map-right-swap-Σ : Σ (Σ A B) (C ∘ pr1) → Σ (Σ A C) (B ∘ pr1) - pr1 (pr1 (map-right-swap-Σ ((a , b) , c))) = a - pr2 (pr1 (map-right-swap-Σ ((a , b) , c))) = c - pr2 (map-right-swap-Σ ((a , b) , c)) = b - - map-inv-right-swap-Σ : Σ (Σ A C) (B ∘ pr1) → Σ (Σ A B) (C ∘ pr1) - pr1 (pr1 (map-inv-right-swap-Σ ((a , c) , b))) = a - pr2 (pr1 (map-inv-right-swap-Σ ((a , c) , b))) = b - pr2 (map-inv-right-swap-Σ ((a , c) , b)) = c - - issec-map-inv-right-swap-Σ : (map-right-swap-Σ ∘ map-inv-right-swap-Σ) ~ id - issec-map-inv-right-swap-Σ (pair (pair x y) z) = refl - - isretr-map-inv-right-swap-Σ : (map-inv-right-swap-Σ ∘ map-right-swap-Σ) ~ id - isretr-map-inv-right-swap-Σ (pair (pair x z) y) = refl - - is-equiv-map-right-swap-Σ : is-equiv map-right-swap-Σ - is-equiv-map-right-swap-Σ = - is-equiv-has-inverse - map-inv-right-swap-Σ - issec-map-inv-right-swap-Σ - isretr-map-inv-right-swap-Σ - - equiv-right-swap-Σ : Σ (Σ A B) (C ∘ pr1) ≃ Σ (Σ A C) (B ∘ pr1) - pr1 equiv-right-swap-Σ = map-right-swap-Σ - pr2 equiv-right-swap-Σ = is-equiv-map-right-swap-Σ -``` - -### Distributive laws of cartesian products over Σ - -```agda -left-distributive-prod-Σ : - {l1 l2 l3 : Level} {A : UU l1} {B : UU l2} {C : B → UU l3} → - (A × (Σ B C)) ≃ Σ B (λ b → A × (C b)) -left-distributive-prod-Σ = - equiv-left-swap-Σ - -right-distributive-prod-Σ : - {l1 l2 l3 : Level} {A : UU l1} {B : A → UU l2} {C : UU l3} → - ((Σ A B) × C) ≃ Σ A (λ a → B a × C) -right-distributive-prod-Σ {A} = - associative-Σ _ _ _ -``` - -## See also - -- Functorial properties of dependent pair types are recorded in - [`foundation.functoriality-dependent-pair-types`](foundation.functoriality-dependent-pair-types.md). -- Equality proofs in dependent pair types are characterized in - [`foundation.equality-dependent-pair-types`](foundation.equality-dependent-pair-types.md). -- The universal property of dependent pair types is treated in - [`foundation.universal-property-dependent-pair-types`](foundation.universal-property-dependent-pair-types.md). - -- Arithmetical laws involving cartesian product types are recorded in - [`foundation.type-arithmetic-cartesian-product-types`](foundation.type-arithmetic-cartesian-product-types.md). -- Arithmetical laws involving dependent product types are recorded in - [`foundation.type-arithmetic-dependent-function-types`](foundation.type-arithmetic-dependent-function-types.md). -- Arithmetical laws involving coproduct types are recorded in - [`foundation.type-arithmetic-coproduct-types`](foundation.type-arithmetic-coproduct-types.md). -- Arithmetical laws involving the unit type are recorded in - [`foundation.type-arithmetic-unit-type`](foundation.type-arithmetic-unit-type.md). -- Arithmetical laws involving the empty type are recorded in - [`foundation.type-arithmetic-empty-type`](foundation.type-arithmetic-empty-type.md). diff --git a/src/foundation-core/univalence.lagda.md b/src/foundation-core/univalence.lagda.md index 45082d613f..960320456e 100644 --- a/src/foundation-core/univalence.lagda.md +++ b/src/foundation-core/univalence.lagda.md @@ -7,12 +7,13 @@ module foundation-core.univalence where
Imports ```agda +open import foundation.dependent-pair-types +open import foundation.fundamental-theorem-of-identity-types +open import foundation.universe-levels + open import foundation-core.contractible-types -open import foundation-core.dependent-pair-types open import foundation-core.equivalences -open import foundation-core.fundamental-theorem-of-identity-types open import foundation-core.identity-types -open import foundation-core.universe-levels ```
diff --git a/src/foundation-core/universal-property-pullbacks.lagda.md b/src/foundation-core/universal-property-pullbacks.lagda.md index 62aafa9e7c..f115440a70 100644 --- a/src/foundation-core/universal-property-pullbacks.lagda.md +++ b/src/foundation-core/universal-property-pullbacks.lagda.md @@ -7,17 +7,18 @@ module foundation-core.universal-property-pullbacks where
Imports ```agda -open import foundation-core.cones-over-cospans +open import foundation.cones-over-cospans +open import foundation.dependent-pair-types +open import foundation.functions +open import foundation.universe-levels + open import foundation-core.contractible-maps open import foundation-core.contractible-types -open import foundation-core.dependent-pair-types open import foundation-core.equivalences -open import foundation-core.functions open import foundation-core.functoriality-dependent-pair-types open import foundation-core.functoriality-function-types open import foundation-core.homotopies open import foundation-core.identity-types -open import foundation-core.universe-levels ```
diff --git a/src/foundation-core/universal-property-truncation.lagda.md b/src/foundation-core/universal-property-truncation.lagda.md index 9329bf933b..264893bda5 100644 --- a/src/foundation-core/universal-property-truncation.lagda.md +++ b/src/foundation-core/universal-property-truncation.lagda.md @@ -7,21 +7,21 @@ module foundation-core.universal-property-truncation where
Imports ```agda +open import foundation.dependent-pair-types open import foundation.equivalences open import foundation.function-extensionality +open import foundation.functions open import foundation.type-theoretic-principle-of-choice +open import foundation.universe-levels open import foundation-core.contractible-maps open import foundation-core.contractible-types -open import foundation-core.dependent-pair-types -open import foundation-core.functions open import foundation-core.functoriality-dependent-pair-types open import foundation-core.homotopies open import foundation-core.identity-types open import foundation-core.sections open import foundation-core.truncated-types open import foundation-core.truncation-levels -open import foundation-core.universe-levels ```
diff --git a/src/foundation-core/universe-levels.lagda.md b/src/foundation-core/universe-levels.lagda.md deleted file mode 100644 index 3c8b7d972f..0000000000 --- a/src/foundation-core/universe-levels.lagda.md +++ /dev/null @@ -1,18 +0,0 @@ -# Universe levels - -```agda -{-# OPTIONS --safe --no-import-sorts #-} - -module foundation-core.universe-levels where - -open import Agda.Primitive - using (Level ; lzero ; lsuc ; _⊔_) - renaming (Set to UU ; Setω to UUω) - public -``` - -## Idea - -We import Agda's built in mechanism of universe levels. The universes are called -`UU`, which stands for `univalent universe`, although we will not immediately -assume that universes are univalent. diff --git a/src/foundation/0-connected-types.lagda.md b/src/foundation/0-connected-types.lagda.md index fef27a5ca3..c5444621bf 100644 --- a/src/foundation/0-connected-types.lagda.md +++ b/src/foundation/0-connected-types.lagda.md @@ -8,7 +8,9 @@ module foundation.0-connected-types where ```agda open import foundation.contractible-types +open import foundation.dependent-pair-types open import foundation.fiber-inclusions +open import foundation.functions open import foundation.functoriality-set-truncation open import foundation.inhabited-types open import foundation.mere-equality @@ -18,18 +20,16 @@ open import foundation.sets open import foundation.surjective-maps open import foundation.unit-type open import foundation.universal-property-unit-type +open import foundation.universe-levels open import foundation-core.cartesian-product-types -open import foundation-core.dependent-pair-types open import foundation-core.equivalences open import foundation-core.fibers-of-maps -open import foundation-core.functions open import foundation-core.identity-types open import foundation-core.propositions open import foundation-core.truncated-maps open import foundation-core.truncated-types open import foundation-core.truncation-levels -open import foundation-core.universe-levels ```
diff --git a/src/foundation/0-images-of-maps.lagda.md b/src/foundation/0-images-of-maps.lagda.md index b1593c8d72..611cf6cdc0 100644 --- a/src/foundation/0-images-of-maps.lagda.md +++ b/src/foundation/0-images-of-maps.lagda.md @@ -8,9 +8,9 @@ module foundation.0-images-of-maps where ```agda open import foundation.truncation-images-of-maps +open import foundation.universe-levels open import foundation-core.truncation-levels -open import foundation-core.universe-levels ``` diff --git a/src/foundation/0-maps.lagda.md b/src/foundation/0-maps.lagda.md index 73555fe537..7515655e6a 100644 --- a/src/foundation/0-maps.lagda.md +++ b/src/foundation/0-maps.lagda.md @@ -1,15 +1,158 @@ -# `0`-Maps +# 0-Maps ```agda module foundation.0-maps where - -open import foundation-core.0-maps public ```
Imports ```agda +open import foundation.dependent-pair-types +open import foundation.functions +open import foundation.universe-levels +open import foundation-core.fibers-of-maps +open import foundation-core.functoriality-dependent-pair-types +open import foundation-core.homotopies +open import foundation-core.sets +open import foundation-core.truncated-maps +open import foundation-core.truncation-levels ```
+ +## Definition + +Maps `f : A → B` of which the fibers are sets, i.e., 0-truncated types, are +called 0-maps. It is shown in +[`foundation.faithful-maps`](foundation.faithful-maps.md) that a map `f` is a +0-map if and only if `f` is faithful, i.e., `f` induces embeddings on identity +types. + +```agda +module _ + {l1 l2 : Level} + where + + is-0-map : {A : UU l1} {B : UU l2} → (A → B) → UU (l1 ⊔ l2) + is-0-map {A} {B} f = (y : B) → is-set (fib f y) + + 0-map : (A : UU l1) (B : UU l2) → UU (l1 ⊔ l2) + 0-map A B = Σ (A → B) is-0-map + + map-0-map : {A : UU l1} {B : UU l2} → 0-map A B → A → B + map-0-map = pr1 + + is-0-map-map-0-map : + {A : UU l1} {B : UU l2} (f : 0-map A B) → is-0-map (map-0-map f) + is-0-map-map-0-map = pr2 +``` + +## Properties + +### Projections of families of sets are `0`-maps + +```agda +module _ + {l1 l2 : Level} {A : UU l1} + where + + abstract + is-0-map-pr1 : + {B : A → UU l2} → ((x : A) → is-set (B x)) → is-0-map (pr1 {B = B}) + is-0-map-pr1 {B} H x = + is-set-equiv (B x) (equiv-fib-pr1 B x) (H x) + + pr1-0-map : + (B : A → Set l2) → 0-map (Σ A (λ x → type-Set (B x))) A + pr1 (pr1-0-map B) = pr1 + pr2 (pr1-0-map B) = is-0-map-pr1 (λ x → is-set-type-Set (B x)) +``` + +### `0`-maps are closed under homotopies + +```agda +module _ + {l1 l2 : Level} {A : UU l1} {B : UU l2} {f g : A → B} (H : f ~ g) + where + + is-0-map-htpy : is-0-map g → is-0-map f + is-0-map-htpy = is-trunc-map-htpy zero-𝕋 H +``` + +### `0`-maps are closed under composition + +```agda +module _ + {l1 l2 l3 : Level} {A : UU l1} {B : UU l2} {X : UU l3} + where + + is-0-map-comp : + (g : B → X) (h : A → B) → + is-0-map g → is-0-map h → is-0-map (g ∘ h) + is-0-map-comp = is-trunc-map-comp zero-𝕋 + + is-0-map-comp-htpy : + (f : A → X) (g : B → X) (h : A → B) (H : f ~ (g ∘ h)) → + is-0-map g → is-0-map h → is-0-map f + is-0-map-comp-htpy = is-trunc-map-comp-htpy zero-𝕋 +``` + +### If a composite is a 0-map, then so is its right factor + +```agda +module _ + {l1 l2 l3 : Level} {A : UU l1} {B : UU l2} {X : UU l3} + where + + is-0-map-right-factor : + (g : B → X) (h : A → B) → + is-0-map g → is-0-map (g ∘ h) → is-0-map h + is-0-map-right-factor = is-trunc-map-right-factor zero-𝕋 + + is-0-map-right-factor-htpy : + (f : A → X) (g : B → X) (h : A → B) (H : f ~ (g ∘ h)) → + is-0-map g → is-0-map f → is-0-map h + is-0-map-right-factor-htpy = is-trunc-map-right-factor-htpy zero-𝕋 +``` + +### A family of `0`-maps induces a `0`-map on total spaces + +```agda +module _ + {l1 l2 l3 : Level} {A : UU l1} {B : A → UU l2} {C : A → UU l3} + {f : (x : A) → B x → C x} + where + + abstract + is-0-map-tot : ((x : A) → is-0-map (f x)) → is-0-map (tot f) + is-0-map-tot = is-trunc-map-tot zero-𝕋 +``` + +### For any type family over the codomain, a `0`-map induces a `0`-map on total spaces + +In other words, `0`-maps are stable under pullbacks. We will come to this point +when we introduce homotopy pullbacks. + +```agda +module _ + {l1 l2 l3 : Level} {A : UU l1} {B : UU l2} {f : A → B} (C : B → UU l3) + where + + abstract + is-0-map-map-Σ-map-base : is-0-map f → is-0-map (map-Σ-map-base f C) + is-0-map-map-Σ-map-base = is-trunc-map-map-Σ-map-base zero-𝕋 C +``` + +### The functorial action of `Σ` preserves `0`-maps + +```agda +module _ + {l1 l2 l3 l4 : Level} {A : UU l1} {B : UU l2} {C : A → UU l3} + (D : B → UU l4) {f : A → B} {g : (x : A) → C x → D (f x)} + where + + is-0-map-map-Σ : + is-0-map f → ((x : A) → is-0-map (g x)) → is-0-map (map-Σ D f g) + is-0-map-map-Σ = is-trunc-map-map-Σ zero-𝕋 D +``` diff --git a/src/foundation/1-types.lagda.md b/src/foundation/1-types.lagda.md index 0bf28d7199..dda1ff537f 100644 --- a/src/foundation/1-types.lagda.md +++ b/src/foundation/1-types.lagda.md @@ -9,17 +9,17 @@ open import foundation-core.1-types public
Imports ```agda +open import foundation.dependent-pair-types open import foundation.subuniverses +open import foundation.universe-levels open import foundation-core.contractible-types -open import foundation-core.dependent-pair-types open import foundation-core.equivalences open import foundation-core.identity-types open import foundation-core.propositions open import foundation-core.subtypes open import foundation-core.truncated-types open import foundation-core.truncation-levels -open import foundation-core.universe-levels ```
diff --git a/src/foundation/2-types.lagda.md b/src/foundation/2-types.lagda.md index 70f062190d..1c2182a2eb 100644 --- a/src/foundation/2-types.lagda.md +++ b/src/foundation/2-types.lagda.md @@ -7,10 +7,11 @@ module foundation.2-types where
Imports ```agda -open import foundation-core.dependent-pair-types +open import foundation.dependent-pair-types +open import foundation.universe-levels + open import foundation-core.truncated-types open import foundation-core.truncation-levels -open import foundation-core.universe-levels ```
diff --git a/src/foundation/apartness-relations.lagda.md b/src/foundation/apartness-relations.lagda.md index 5c20454636..a751d5dff8 100644 --- a/src/foundation/apartness-relations.lagda.md +++ b/src/foundation/apartness-relations.lagda.md @@ -8,18 +8,18 @@ module foundation.apartness-relations where ```agda open import foundation.binary-relations +open import foundation.dependent-pair-types open import foundation.disjunction open import foundation.existential-quantification open import foundation.propositional-truncations +open import foundation.universe-levels open import foundation-core.cartesian-product-types open import foundation-core.coproduct-types -open import foundation-core.dependent-pair-types open import foundation-core.empty-types open import foundation-core.identity-types open import foundation-core.negation open import foundation-core.propositions -open import foundation-core.universe-levels ``` diff --git a/src/foundation/arithmetic-law-coproduct-and-sigma-decompositions.lagda.md b/src/foundation/arithmetic-law-coproduct-and-sigma-decompositions.lagda.md index 1b5384fe8f..3244ef5074 100644 --- a/src/foundation/arithmetic-law-coproduct-and-sigma-decompositions.lagda.md +++ b/src/foundation/arithmetic-law-coproduct-and-sigma-decompositions.lagda.md @@ -8,23 +8,23 @@ module foundation.arithmetic-law-coproduct-and-sigma-decompositions where ```agda open import foundation.coproduct-decompositions +open import foundation.dependent-pair-types open import foundation.equivalences +open import foundation.functions open import foundation.functoriality-coproduct-types open import foundation.relaxed-sigma-decompositions open import foundation.type-arithmetic-coproduct-types +open import foundation.type-arithmetic-dependent-pair-types open import foundation.univalence open import foundation.universal-property-coproduct-types +open import foundation.universe-levels open import foundation-core.cartesian-product-types open import foundation-core.contractible-types open import foundation-core.coproduct-types -open import foundation-core.dependent-pair-types -open import foundation-core.functions open import foundation-core.functoriality-dependent-pair-types open import foundation-core.homotopies open import foundation-core.identity-types -open import foundation-core.type-arithmetic-dependent-pair-types -open import foundation-core.universe-levels ``` diff --git a/src/foundation/arithmetic-law-product-and-pi-decompositions.lagda.md b/src/foundation/arithmetic-law-product-and-pi-decompositions.lagda.md index 7696e43a23..690421942d 100644 --- a/src/foundation/arithmetic-law-product-and-pi-decompositions.lagda.md +++ b/src/foundation/arithmetic-law-product-and-pi-decompositions.lagda.md @@ -8,7 +8,9 @@ module foundation.arithmetic-law-product-and-pi-decompositions where ```agda open import foundation.coproduct-decompositions +open import foundation.dependent-pair-types open import foundation.equivalences +open import foundation.functions open import foundation.functoriality-cartesian-product-types open import foundation.functoriality-coproduct-types open import foundation.functoriality-dependent-pair-types @@ -19,15 +21,13 @@ open import foundation.type-arithmetic-coproduct-types open import foundation.type-arithmetic-dependent-pair-types open import foundation.univalence open import foundation.universal-property-coproduct-types +open import foundation.universe-levels open import foundation-core.cartesian-product-types open import foundation-core.contractible-types open import foundation-core.coproduct-types -open import foundation-core.dependent-pair-types -open import foundation-core.functions open import foundation-core.homotopies open import foundation-core.identity-types -open import foundation-core.universe-levels ``` diff --git a/src/foundation/automorphisms.lagda.md b/src/foundation/automorphisms.lagda.md index 3eede0b718..bc5d12d351 100644 --- a/src/foundation/automorphisms.lagda.md +++ b/src/foundation/automorphisms.lagda.md @@ -1,15 +1,44 @@ -# Automorphisms of types +# Automorphisms ```agda module foundation.automorphisms where - -open import foundation-core.automorphisms public ```
Imports ```agda +open import foundation.dependent-pair-types +open import foundation.sets +open import foundation.universe-levels + +open import foundation-core.equivalences +open import structured-types.pointed-types ```
+ +## Idea + +An automorphism on a type `A` is an equivalence `A ≃ A`. We will just reuse the +infrastructure of equivalences for automorphisms. + +## Definitions + +### The type of automorphisms on a type + +```agda +Aut : {l : Level} → UU l → UU l +Aut Y = Y ≃ Y + +is-set-Aut : {l : Level} {A : UU l} → is-set A → is-set (Aut A) +is-set-Aut H = is-set-equiv-is-set H H + +Aut-Set : {l : Level} → Set l → Set l +pr1 (Aut-Set A) = Aut (type-Set A) +pr2 (Aut-Set A) = is-set-Aut (is-set-type-Set A) + +Aut-Pointed-Type : {l : Level} → UU l → Pointed-Type l +pr1 (Aut-Pointed-Type A) = Aut A +pr2 (Aut-Pointed-Type A) = id-equiv +``` diff --git a/src/foundation/axiom-l.lagda.md b/src/foundation/axiom-l.lagda.md index 4a01bbe40b..fae0bb36ab 100644 --- a/src/foundation/axiom-l.lagda.md +++ b/src/foundation/axiom-l.lagda.md @@ -7,26 +7,26 @@ module foundation.axiom-l where
Imports ```agda +open import foundation.dependent-pair-types open import foundation.embeddings open import foundation.equivalences open import foundation.full-subtypes open import foundation.function-extensionality +open import foundation.functions open import foundation.functoriality-dependent-function-types +open import foundation.fundamental-theorem-of-identity-types open import foundation.sets open import foundation.type-theoretic-principle-of-choice open import foundation.universal-property-identity-types +open import foundation.universe-levels open import foundation-core.contractible-types -open import foundation-core.dependent-pair-types open import foundation-core.fibers-of-maps -open import foundation-core.functions open import foundation-core.functoriality-dependent-pair-types -open import foundation-core.fundamental-theorem-of-identity-types open import foundation-core.identity-types open import foundation-core.injective-maps open import foundation-core.propositions open import foundation-core.univalence -open import foundation-core.universe-levels ```
diff --git a/src/foundation/axiom-of-choice.lagda.md b/src/foundation/axiom-of-choice.lagda.md index 5f221b231d..ea089242b8 100644 --- a/src/foundation/axiom-of-choice.lagda.md +++ b/src/foundation/axiom-of-choice.lagda.md @@ -7,23 +7,23 @@ module foundation.axiom-of-choice where
Imports ```agda +open import foundation.dependent-pair-types open import foundation.function-extensionality +open import foundation.functions open import foundation.functoriality-propositional-truncation open import foundation.projective-types open import foundation.propositional-truncations open import foundation.sections open import foundation.split-surjective-maps open import foundation.surjective-maps +open import foundation.universe-levels -open import foundation-core.dependent-pair-types open import foundation-core.equivalences open import foundation-core.fibers-of-maps -open import foundation-core.functions open import foundation-core.functoriality-dependent-pair-types open import foundation-core.homotopies open import foundation-core.identity-types open import foundation-core.sets -open import foundation-core.universe-levels ```
diff --git a/src/foundation/bands.lagda.md b/src/foundation/bands.lagda.md index 5a3d4f007a..014eb6117c 100644 --- a/src/foundation/bands.lagda.md +++ b/src/foundation/bands.lagda.md @@ -8,9 +8,9 @@ module foundation.bands where ```agda open import foundation.set-truncations +open import foundation.universe-levels open import foundation-core.equivalences -open import foundation-core.universe-levels ``` diff --git a/src/foundation/binary-embeddings.lagda.md b/src/foundation/binary-embeddings.lagda.md index 157cfa6c35..81566ec58e 100644 --- a/src/foundation/binary-embeddings.lagda.md +++ b/src/foundation/binary-embeddings.lagda.md @@ -8,12 +8,12 @@ module foundation.binary-embeddings where ```agda open import foundation.binary-equivalences +open import foundation.dependent-pair-types open import foundation.identity-types +open import foundation.universe-levels -open import foundation-core.dependent-pair-types open import foundation-core.embeddings open import foundation-core.equivalences -open import foundation-core.universe-levels ``` diff --git a/src/foundation/binary-equivalences-unordered-pairs-of-types.lagda.md b/src/foundation/binary-equivalences-unordered-pairs-of-types.lagda.md index c06562e463..5262453435 100644 --- a/src/foundation/binary-equivalences-unordered-pairs-of-types.lagda.md +++ b/src/foundation/binary-equivalences-unordered-pairs-of-types.lagda.md @@ -8,12 +8,12 @@ module foundation.binary-equivalences-unordered-pairs-of-types where ```agda open import foundation.binary-operations-unordered-pairs-of-types +open import foundation.functions open import foundation.products-unordered-pairs-of-types +open import foundation.universe-levels open import foundation.unordered-pairs open import foundation-core.equivalences -open import foundation-core.functions -open import foundation-core.universe-levels ``` diff --git a/src/foundation/binary-equivalences.lagda.md b/src/foundation/binary-equivalences.lagda.md index 10003b3b8b..2855ee5a99 100644 --- a/src/foundation/binary-equivalences.lagda.md +++ b/src/foundation/binary-equivalences.lagda.md @@ -7,10 +7,11 @@ module foundation.binary-equivalences where
Imports ```agda +open import foundation.dependent-pair-types +open import foundation.universe-levels + open import foundation-core.cartesian-product-types -open import foundation-core.dependent-pair-types open import foundation-core.equivalences -open import foundation-core.universe-levels ```
diff --git a/src/foundation/binary-functoriality-set-quotients.lagda.md b/src/foundation/binary-functoriality-set-quotients.lagda.md index 67251ca502..f554097a6c 100644 --- a/src/foundation/binary-functoriality-set-quotients.lagda.md +++ b/src/foundation/binary-functoriality-set-quotients.lagda.md @@ -10,29 +10,29 @@ module foundation.binary-functoriality-set-quotients where ```agda open import foundation.binary-homotopies +open import foundation.dependent-pair-types open import foundation.exponents-set-quotients open import foundation.function-extensionality +open import foundation.functions open import foundation.functoriality-set-quotients +open import foundation.fundamental-theorem-of-identity-types open import foundation.homotopies open import foundation.identity-types open import foundation.reflecting-maps-equivalence-relations open import foundation.set-quotients open import foundation.sets +open import foundation.subtype-identity-principle open import foundation.surjective-maps open import foundation.universal-property-set-quotients +open import foundation.universe-levels open import foundation-core.contractible-types -open import foundation-core.dependent-pair-types open import foundation-core.equivalence-relations open import foundation-core.equivalences -open import foundation-core.functions open import foundation-core.functoriality-dependent-function-types open import foundation-core.functoriality-dependent-pair-types -open import foundation-core.fundamental-theorem-of-identity-types open import foundation-core.propositions -open import foundation-core.subtype-identity-principle open import foundation-core.subtypes -open import foundation-core.universe-levels ``` diff --git a/src/foundation/binary-homotopies.lagda.md b/src/foundation/binary-homotopies.lagda.md index 994d8d160d..ea17025506 100644 --- a/src/foundation/binary-homotopies.lagda.md +++ b/src/foundation/binary-homotopies.lagda.md @@ -7,15 +7,15 @@ module foundation.binary-homotopies where
Imports ```agda +open import foundation.dependent-pair-types open import foundation.equality-dependent-function-types +open import foundation.fundamental-theorem-of-identity-types open import foundation.homotopies +open import foundation.universe-levels open import foundation-core.contractible-types -open import foundation-core.dependent-pair-types open import foundation-core.equivalences -open import foundation-core.fundamental-theorem-of-identity-types open import foundation-core.identity-types -open import foundation-core.universe-levels ```
diff --git a/src/foundation/binary-operations-unordered-pairs-of-types.lagda.md b/src/foundation/binary-operations-unordered-pairs-of-types.lagda.md index 6ddb84fbbb..2f5a31ac3c 100644 --- a/src/foundation/binary-operations-unordered-pairs-of-types.lagda.md +++ b/src/foundation/binary-operations-unordered-pairs-of-types.lagda.md @@ -8,9 +8,8 @@ module foundation.binary-operations-unordered-pairs-of-types where ```agda open import foundation.products-unordered-pairs-of-types +open import foundation.universe-levels open import foundation.unordered-pairs - -open import foundation-core.universe-levels ``` diff --git a/src/foundation/binary-reflecting-maps-equivalence-relations.lagda.md b/src/foundation/binary-reflecting-maps-equivalence-relations.lagda.md index e437a5c4b2..0df2751101 100644 --- a/src/foundation/binary-reflecting-maps-equivalence-relations.lagda.md +++ b/src/foundation/binary-reflecting-maps-equivalence-relations.lagda.md @@ -7,19 +7,19 @@ module foundation.binary-reflecting-maps-equivalence-relations where
Imports ```agda +open import foundation.dependent-pair-types open import foundation.equality-dependent-function-types +open import foundation.fundamental-theorem-of-identity-types open import foundation.homotopies +open import foundation.subtype-identity-principle +open import foundation.universe-levels open import foundation-core.contractible-types -open import foundation-core.dependent-pair-types open import foundation-core.equivalence-relations open import foundation-core.equivalences -open import foundation-core.fundamental-theorem-of-identity-types open import foundation-core.identity-types open import foundation-core.propositions open import foundation-core.sets -open import foundation-core.subtype-identity-principle -open import foundation-core.universe-levels ```
diff --git a/src/foundation/binary-relations.lagda.md b/src/foundation/binary-relations.lagda.md index 0e939f0446..ea164bc006 100644 --- a/src/foundation/binary-relations.lagda.md +++ b/src/foundation/binary-relations.lagda.md @@ -7,18 +7,18 @@ module foundation.binary-relations where
Imports ```agda +open import foundation.dependent-pair-types open import foundation.equality-dependent-function-types +open import foundation.fundamental-theorem-of-identity-types open import foundation.subtypes open import foundation.univalence +open import foundation.universe-levels open import foundation-core.cartesian-product-types open import foundation-core.contractible-types -open import foundation-core.dependent-pair-types open import foundation-core.equivalences -open import foundation-core.fundamental-theorem-of-identity-types open import foundation-core.identity-types open import foundation-core.propositions -open import foundation-core.universe-levels ```
diff --git a/src/foundation/binary-transport.lagda.md b/src/foundation/binary-transport.lagda.md index 1d4a0d8989..5a59f1332c 100644 --- a/src/foundation/binary-transport.lagda.md +++ b/src/foundation/binary-transport.lagda.md @@ -7,11 +7,12 @@ module foundation.binary-transport where
Imports ```agda -open import foundation-core.dependent-pair-types +open import foundation.dependent-pair-types +open import foundation.functions +open import foundation.universe-levels + open import foundation-core.equivalences -open import foundation-core.functions open import foundation-core.identity-types -open import foundation-core.universe-levels ```
diff --git a/src/foundation/booleans.lagda.md b/src/foundation/booleans.lagda.md index 094d865dbc..2c3354a503 100644 --- a/src/foundation/booleans.lagda.md +++ b/src/foundation/booleans.lagda.md @@ -7,22 +7,22 @@ module foundation.booleans where
Imports ```agda +open import foundation.dependent-pair-types +open import foundation.functions open import foundation.raising-universe-levels open import foundation.unit-type +open import foundation.universe-levels open import foundation-core.constant-maps open import foundation-core.coproduct-types -open import foundation-core.dependent-pair-types open import foundation-core.empty-types open import foundation-core.equivalences -open import foundation-core.functions open import foundation-core.homotopies open import foundation-core.identity-types open import foundation-core.injective-maps open import foundation-core.negation open import foundation-core.propositions open import foundation-core.sets -open import foundation-core.universe-levels open import univalent-combinatorics.finite-types open import univalent-combinatorics.standard-finite-types diff --git a/src/foundation/cantor-schroder-bernstein-escardo.lagda.md b/src/foundation/cantor-schroder-bernstein-escardo.lagda.md index 2a4f4d19cd..2359b19660 100644 --- a/src/foundation/cantor-schroder-bernstein-escardo.lagda.md +++ b/src/foundation/cantor-schroder-bernstein-escardo.lagda.md @@ -8,12 +8,13 @@ module foundation.cantor-schroder-bernstein-escardo where ```agda open import foundation.decidable-types +open import foundation.dependent-pair-types open import foundation.law-of-excluded-middle open import foundation.perfect-images open import foundation.split-surjective-maps +open import foundation.universe-levels open import foundation-core.coproduct-types -open import foundation-core.dependent-pair-types open import foundation-core.embeddings open import foundation-core.empty-types open import foundation-core.equivalences @@ -21,7 +22,6 @@ open import foundation-core.fibers-of-maps open import foundation-core.identity-types open import foundation-core.injective-maps open import foundation-core.negation -open import foundation-core.universe-levels ```
diff --git a/src/foundation/cantors-diagonal-argument.lagda.md b/src/foundation/cantors-diagonal-argument.lagda.md index 8455ba7416..8504154d1f 100644 --- a/src/foundation/cantors-diagonal-argument.lagda.md +++ b/src/foundation/cantors-diagonal-argument.lagda.md @@ -7,17 +7,17 @@ module foundation.cantors-diagonal-argument where
Imports ```agda +open import foundation.dependent-pair-types open import foundation.logical-equivalences open import foundation.negation open import foundation.propositional-truncations open import foundation.surjective-maps +open import foundation.universe-levels -open import foundation-core.dependent-pair-types open import foundation-core.empty-types open import foundation-core.fibers-of-maps open import foundation-core.function-extensionality open import foundation-core.propositions -open import foundation-core.universe-levels ```
diff --git a/src/foundation/cartesian-product-types.lagda.md b/src/foundation/cartesian-product-types.lagda.md index 6251b94a47..ee4be138ee 100644 --- a/src/foundation/cartesian-product-types.lagda.md +++ b/src/foundation/cartesian-product-types.lagda.md @@ -10,8 +10,7 @@ open import foundation-core.cartesian-product-types public ```agda open import foundation.subuniverses - -open import foundation-core.universe-levels +open import foundation.universe-levels ``` diff --git a/src/foundation/cartesian-products-set-quotients.lagda.md b/src/foundation/cartesian-products-set-quotients.lagda.md index 9d5411d1f5..c404d172aa 100644 --- a/src/foundation/cartesian-products-set-quotients.lagda.md +++ b/src/foundation/cartesian-products-set-quotients.lagda.md @@ -7,25 +7,25 @@ module foundation.cartesian-products-set-quotients where
Imports ```agda +open import foundation.dependent-pair-types +open import foundation.equality-cartesian-product-types open import foundation.function-extensionality +open import foundation.functions open import foundation.products-equivalence-relations open import foundation.reflecting-maps-equivalence-relations open import foundation.set-quotients open import foundation.sets open import foundation.uniqueness-set-quotients open import foundation.universal-property-set-quotients +open import foundation.universe-levels open import foundation-core.cartesian-product-types -open import foundation-core.dependent-pair-types -open import foundation-core.equality-cartesian-product-types open import foundation-core.equality-dependent-pair-types open import foundation-core.equivalence-relations open import foundation-core.equivalences -open import foundation-core.functions open import foundation-core.homotopies open import foundation-core.identity-types open import foundation-core.propositions -open import foundation-core.universe-levels ```
diff --git a/src/foundation/category-of-sets.lagda.md b/src/foundation/category-of-sets.lagda.md index 689e791127..553cd91dde 100644 --- a/src/foundation/category-of-sets.lagda.md +++ b/src/foundation/category-of-sets.lagda.md @@ -9,11 +9,11 @@ module foundation.category-of-sets where ```agda open import category-theory.large-precategories +open import foundation.functions open import foundation.sets +open import foundation.universe-levels -open import foundation-core.functions open import foundation-core.identity-types -open import foundation-core.universe-levels ``` diff --git a/src/foundation/choice-of-representatives-equivalence-relation.lagda.md b/src/foundation/choice-of-representatives-equivalence-relation.lagda.md index 0000b7e4a0..fe7543acef 100644 --- a/src/foundation/choice-of-representatives-equivalence-relation.lagda.md +++ b/src/foundation/choice-of-representatives-equivalence-relation.lagda.md @@ -7,23 +7,23 @@ module foundation.choice-of-representatives-equivalence-relation where
Imports ```agda +open import foundation.dependent-pair-types open import foundation.equivalence-classes +open import foundation.fundamental-theorem-of-identity-types open import foundation.propositional-truncations open import foundation.surjective-maps +open import foundation.type-arithmetic-dependent-pair-types +open import foundation.universe-levels open import foundation-core.cartesian-product-types open import foundation-core.contractible-types -open import foundation-core.dependent-pair-types open import foundation-core.embeddings open import foundation-core.equivalence-relations open import foundation-core.equivalences open import foundation-core.fibers-of-maps open import foundation-core.functoriality-dependent-pair-types -open import foundation-core.fundamental-theorem-of-identity-types open import foundation-core.identity-types open import foundation-core.logical-equivalences -open import foundation-core.type-arithmetic-dependent-pair-types -open import foundation-core.universe-levels ```
diff --git a/src/foundation/coherently-invertible-maps.lagda.md b/src/foundation/coherently-invertible-maps.lagda.md index c8ea1fe747..31c300027c 100644 --- a/src/foundation/coherently-invertible-maps.lagda.md +++ b/src/foundation/coherently-invertible-maps.lagda.md @@ -9,21 +9,21 @@ open import foundation-core.coherently-invertible-maps public
Imports ```agda +open import foundation.dependent-pair-types open import foundation.equivalences +open import foundation.functions open import foundation.identity-types +open import foundation.type-arithmetic-dependent-pair-types open import foundation.type-theoretic-principle-of-choice +open import foundation.universe-levels open import foundation-core.contractible-maps open import foundation-core.contractible-types -open import foundation-core.dependent-pair-types open import foundation-core.fibers-of-maps -open import foundation-core.functions open import foundation-core.functoriality-dependent-pair-types open import foundation-core.homotopies open import foundation-core.propositions open import foundation-core.sections -open import foundation-core.type-arithmetic-dependent-pair-types -open import foundation-core.universe-levels ```
diff --git a/src/foundation/commuting-3-simplices-of-homotopies.lagda.md b/src/foundation/commuting-3-simplices-of-homotopies.lagda.md index 05e548fae2..ea13bcc60e 100644 --- a/src/foundation/commuting-3-simplices-of-homotopies.lagda.md +++ b/src/foundation/commuting-3-simplices-of-homotopies.lagda.md @@ -1,15 +1,77 @@ # Commuting 3-simplices of homotopies ```agda -module foundation.commuting-3-simplices-of-homotopies where +{-# OPTIONS --safe #-} -open import foundation-core.commuting-3-simplices-of-homotopies public +module foundation.commuting-3-simplices-of-homotopies where ```
Imports ```agda +open import foundation.commuting-triangles-of-homotopies +open import foundation.universe-levels +open import foundation-core.homotopies ```
+ +## Idea + +A commuting 3-simplex of homotopies is a commuting diagram of the form + +```text + f ----------> g + | \ ^ | + | \ / | + | / | + | / \ | + V / v V + h ----------> i. +``` + +where f, g, h, and i are functions. + +## Definition + +```agda +module _ + {l1 l2 : Level} {A : UU l1} {B : A → UU l2} + {f g h i : (x : A) → B x} + (top : f ~ g) (left : f ~ h) (right : g ~ i) (bottom : h ~ i) + (diagonal-up : h ~ g) (diagonal-down : f ~ i) + (upper-left : coherence-triangle-homotopies top diagonal-up left) + (lower-right : coherence-triangle-homotopies bottom right diagonal-up) + (upper-right : coherence-triangle-homotopies diagonal-down right top) + (lower-left : coherence-triangle-homotopies diagonal-down bottom left) + where + + coherence-3-simplex-homotopies : UU (l1 ⊔ l2) + coherence-3-simplex-homotopies = + ( upper-right ∙h + left-whisk-htpy-coherence-triangle-homotopies + ( diagonal-up) + ( right) + ( upper-left)) ~ + ( ( lower-left ∙h + right-whisk-htpy-coherence-triangle-homotopies + ( right) + ( lower-right) + ( left)) ∙h + assoc-htpy left diagonal-up right) + + coherence-3-simplex-homotopies' : UU (l1 ⊔ l2) + coherence-3-simplex-homotopies' = + ( ( lower-left ∙h + right-whisk-htpy-coherence-triangle-homotopies + ( right) + ( lower-right) + ( left)) ∙h + assoc-htpy left diagonal-up right) ~ + ( upper-right ∙h + left-whisk-htpy-coherence-triangle-homotopies + ( diagonal-up) + ( right) + ( upper-left)) +``` diff --git a/src/foundation/commuting-3-simplices-of-maps.lagda.md b/src/foundation/commuting-3-simplices-of-maps.lagda.md index f44400745d..610a6d7ca7 100644 --- a/src/foundation/commuting-3-simplices-of-maps.lagda.md +++ b/src/foundation/commuting-3-simplices-of-maps.lagda.md @@ -1,15 +1,56 @@ # Commuting 3-simplices of maps ```agda -module foundation.commuting-3-simplices-of-maps where +{-# OPTIONS --safe #-} -open import foundation-core.commuting-3-simplices-of-maps public +module foundation.commuting-3-simplices-of-maps where ```
Imports ```agda +open import foundation.universe-levels +open import foundation-core.commuting-triangles-of-maps +open import foundation-core.homotopies ```
+ +## Idea + +A commuting 3-simplex is a commuting diagram of the form + +```text + A ----------> B + | \ ^ | + | \ / | + | / | + | / \ | + V / v V + X ----------> Y. +``` + +## Definition + +```agda +module _ + {l1 l2 l3 l4 : Level} {A : UU l1} {B : UU l2} {X : UU l3} {Y : UU l4} + (top : A → B) (left : A → X) (right : B → Y) (bottom : X → Y) + (diagonal-up : X → B) (diagonal-down : A → Y) + (upper-left : coherence-triangle-maps top diagonal-up left) + (lower-right : coherence-triangle-maps bottom right diagonal-up) + (upper-right : coherence-triangle-maps diagonal-down right top) + (lower-left : coherence-triangle-maps diagonal-down bottom left) + where + + coherence-3-simplex-maps : UU (l1 ⊔ l4) + coherence-3-simplex-maps = + ( upper-right ∙h (right ·l upper-left)) ~ + ( lower-left ∙h (lower-right ·r left)) + + coherence-3-simplex-maps' : UU (l1 ⊔ l4) + coherence-3-simplex-maps' = + ( lower-left ∙h (lower-right ·r left)) ~ + ( upper-right ∙h (right ·l upper-left)) +``` diff --git a/src/foundation/commuting-cubes-of-maps.lagda.md b/src/foundation/commuting-cubes-of-maps.lagda.md index de34cd2a32..7389275335 100644 --- a/src/foundation/commuting-cubes-of-maps.lagda.md +++ b/src/foundation/commuting-cubes-of-maps.lagda.md @@ -9,11 +9,12 @@ open import foundation-core.commuting-cubes-of-maps public
Imports ```agda -open import foundation-core.cones-over-cospans -open import foundation-core.dependent-pair-types -open import foundation-core.functions +open import foundation.cones-over-cospans +open import foundation.dependent-pair-types +open import foundation.functions +open import foundation.universe-levels + open import foundation-core.homotopies -open import foundation-core.universe-levels ```
diff --git a/src/foundation/commuting-squares-of-identifications.lagda.md b/src/foundation/commuting-squares-of-identifications.lagda.md index b62e0be093..1bf389320e 100644 --- a/src/foundation/commuting-squares-of-identifications.lagda.md +++ b/src/foundation/commuting-squares-of-identifications.lagda.md @@ -9,9 +9,10 @@ module foundation.commuting-squares-of-identifications where
Imports ```agda -open import foundation-core.functions +open import foundation.functions +open import foundation.universe-levels + open import foundation-core.identity-types -open import foundation-core.universe-levels ```
diff --git a/src/foundation/commuting-squares-of-maps.lagda.md b/src/foundation/commuting-squares-of-maps.lagda.md index 8c7023b084..b88a345213 100644 --- a/src/foundation/commuting-squares-of-maps.lagda.md +++ b/src/foundation/commuting-squares-of-maps.lagda.md @@ -10,11 +10,11 @@ open import foundation-core.commuting-squares-of-maps public ```agda open import foundation.equivalences +open import foundation.functions open import foundation.functoriality-function-types +open import foundation.universe-levels -open import foundation-core.functions open import foundation-core.identity-types -open import foundation-core.universe-levels ``` diff --git a/src/foundation/commuting-triangles-of-homotopies.lagda.md b/src/foundation/commuting-triangles-of-homotopies.lagda.md index 09651770e3..86d52cbdd5 100644 --- a/src/foundation/commuting-triangles-of-homotopies.lagda.md +++ b/src/foundation/commuting-triangles-of-homotopies.lagda.md @@ -1,15 +1,133 @@ # Commuting triangles of homotopies ```agda -module foundation.commuting-triangles-of-homotopies where +{-# OPTIONS --safe #-} -open import foundation-core.commuting-triangles-of-homotopies public +module foundation.commuting-triangles-of-homotopies where ```
Imports ```agda +open import foundation.functions +open import foundation.universe-levels +open import foundation-core.homotopies +open import foundation-core.identity-types ```
+ +## Idea + +A triangle of homotopies of maps + +```text + f ----> g + \ / + \ / + V V + h +``` + +is said to commute if there is a homotopy between the homotopy on the left +`f ~ h` and the composite homotopy `f ~ g ~ h`. + +## Definition + +```agda +module _ + {l1 l2 : Level} {A : UU l1} {B : A → UU l2} + {f g h : (x : A) → B x} + where + + coherence-triangle-homotopies : + (left : f ~ h) (right : g ~ h) (top : f ~ g) → UU (l1 ⊔ l2) + coherence-triangle-homotopies left right top = left ~ (top ∙h right) + + coherence-triangle-homotopies' : + (left : f ~ h) (right : g ~ h) (top : f ~ g) → UU (l1 ⊔ l2) + coherence-triangle-homotopies' left right top = (top ∙h right) ~ left +``` + +## Properties + +### Distributive law for left whiskering + +```agda +module _ + {l1 l2 : Level} {A : UU l1} {B : UU l2} + {f g h : A → B} + {l3 : Level} {X : UU l3} (i : B → X) + (left : f ~ h) (right : g ~ h) (top : f ~ g) + where + + distributivity-left-whisk : + coherence-triangle-homotopies left right top → + (i ·l left) ~ ((i ·l top) ∙h (i ·l right)) + distributivity-left-whisk T x = + ap-concat-eq i (top x) (right x) (left x) (T x) +``` + +### Left whiskering triangles of homotopies + +```agda +module _ + {l1 l2 : Level} {A : UU l1} {B : A → UU l2} + {f g h : (x : A) → B x} + {left : f ~ h} (right : g ~ h) {top : f ~ g} + where + + left-whisk-htpy-coherence-triangle-homotopies : + {i : (x : A) → B x} + (H : h ~ i) (T : coherence-triangle-homotopies left right top) → + coherence-triangle-homotopies {h = i} (left ∙h H) (right ∙h H) top + left-whisk-htpy-coherence-triangle-homotopies H T = + (λ x → ap (_∙ H x) (T x)) ∙h assoc-htpy top right H + +module _ + {l1 l2 : Level} {A : UU l1} {B : UU l2} + {f g h : A → B} + {left : f ~ h} (right : g ~ h) {top : f ~ g} + where + + left-whisk-coherence-triangle-homotopies : + {l3 : Level} {X : UU l3} (i : B → X) + (T : coherence-triangle-homotopies left right top) → + coherence-triangle-homotopies + {f = i ∘ f} {i ∘ g} {i ∘ h} + (i ·l left) (i ·l right) (i ·l top) + left-whisk-coherence-triangle-homotopies i = + distributivity-left-whisk i left right top +``` + +### Right whiskering triangles of homotopies + +```agda +module _ + {l1 l2 : Level} {A : UU l1} {B : A → UU l2} + {f g h : (x : A) → B x} + {left : f ~ h} (right : g ~ h) {top : f ~ g} + where + + right-whisk-htpy-coherence-triangle-homotopies : + {i : (x : A) → B x} + (T : coherence-triangle-homotopies left right top) (H : i ~ f) → + coherence-triangle-homotopies {f = i} (H ∙h left) right (H ∙h top) + right-whisk-htpy-coherence-triangle-homotopies T H = + (λ x → ap (H x ∙_) (T x)) ∙h (inv-htpy-assoc-htpy H top right) + +module _ + {l1 l2 : Level} {A : UU l1} {B : UU l2} + {f g h : A → B} + {left : f ~ h} (right : g ~ h) {top : f ~ g} + where + + right-whisk-coherence-triangle-homotopies : + {l3 : Level} {X : UU l3} + (T : coherence-triangle-homotopies left right top) (i : X → A) → + coherence-triangle-homotopies + {f = f ∘ i} {g ∘ i} {h ∘ i} + (left ·r i) (right ·r i) (top ·r i) + right-whisk-coherence-triangle-homotopies T i = T ∘ i +``` diff --git a/src/foundation/commuting-triangles-of-maps.lagda.md b/src/foundation/commuting-triangles-of-maps.lagda.md index 2059fad2e0..d280b8579f 100644 --- a/src/foundation/commuting-triangles-of-maps.lagda.md +++ b/src/foundation/commuting-triangles-of-maps.lagda.md @@ -11,9 +11,9 @@ open import foundation-core.commuting-triangles-of-maps public ```agda open import foundation.functoriality-dependent-function-types open import foundation.identity-types +open import foundation.universe-levels open import foundation-core.equivalences -open import foundation-core.universe-levels ``` diff --git a/src/foundation/complements-subtypes.lagda.md b/src/foundation/complements-subtypes.lagda.md index 121aa34d0f..4cfd41a52c 100644 --- a/src/foundation/complements-subtypes.lagda.md +++ b/src/foundation/complements-subtypes.lagda.md @@ -10,13 +10,13 @@ module foundation.complements-subtypes where open import foundation.decidable-propositions open import foundation.decidable-subtypes open import foundation.full-subtypes +open import foundation.functions open import foundation.negation open import foundation.propositional-truncations open import foundation.unions-subtypes +open import foundation.universe-levels -open import foundation-core.functions open import foundation-core.subtypes -open import foundation-core.universe-levels ``` diff --git a/src/foundation/complements.lagda.md b/src/foundation/complements.lagda.md index d3f9c99ab1..0f29fe5564 100644 --- a/src/foundation/complements.lagda.md +++ b/src/foundation/complements.lagda.md @@ -7,10 +7,11 @@ module foundation.complements where
Imports ```agda -open import foundation-core.dependent-pair-types +open import foundation.dependent-pair-types +open import foundation.functions +open import foundation.universe-levels + open import foundation-core.empty-types -open import foundation-core.functions -open import foundation-core.universe-levels ```
diff --git a/src/foundation/cones-over-cospans.lagda.md b/src/foundation/cones-over-cospans.lagda.md index 803d27c80d..230bd72e64 100644 --- a/src/foundation/cones-over-cospans.lagda.md +++ b/src/foundation/cones-over-cospans.lagda.md @@ -1,15 +1,270 @@ -# Cones on pullback diagrams +# Cones over cospans ```agda module foundation.cones-over-cospans where - -open import foundation-core.cones-over-cospans public ```
Imports ```agda +open import foundation.dependent-pair-types +open import foundation.functions +open import foundation.fundamental-theorem-of-identity-types +open import foundation.homotopies +open import foundation.structure-identity-principle +open import foundation.universe-levels +open import foundation-core.commuting-squares-of-maps +open import foundation-core.contractible-types +open import foundation-core.equivalences +open import foundation-core.identity-types ```
+ +## Idea + +A cone on a cospan `A --f--> X <--g-- B` with vertex `C` is a triple `(p,q,H)` +consisting of a map `p : C → A`, a map `q : C → B`, and a homotopy `H` +witnessing that the square + +```text + q + C -----> B + | | + p| |g + V V + A -----> X + f +``` + +commutes. + +## Definitions + +### Cones on cospans + +A cone on a cospan with a vertex C is a pair of functions from C into the +domains of the maps in the cospan, equipped with a homotopy witnessing that the +resulting square commutes. + +```agda +module _ + {l1 l2 l3 : Level} {A : UU l1} {B : UU l2} {X : UU l3} + (f : A → X) (g : B → X) + where + + cone : {l4 : Level} → UU l4 → UU (l1 ⊔ l2 ⊔ l3 ⊔ l4) + cone C = Σ (C → A) (λ p → Σ (C → B) (λ q → coherence-square-maps q p g f)) + +module _ + {l1 l2 l3 l4 : Level} {A : UU l1} {B : UU l2} {X : UU l3} {C : UU l4} + (f : A → X) (g : B → X) (c : cone f g C) + where + + vertical-map-cone : C → A + vertical-map-cone = pr1 c + + horizontal-map-cone : C → B + horizontal-map-cone = pr1 (pr2 c) + + coherence-square-cone : + coherence-square-maps horizontal-map-cone vertical-map-cone g f + coherence-square-cone = pr2 (pr2 c) +``` + +### Dependent cones + +```agda +cone-family : + {l1 l2 l3 l4 l5 l6 l7 l8 : Level} + {X : UU l1} {A : UU l2} {B : UU l3} {C : UU l4} + (PX : X → UU l5) {PA : A → UU l6} {PB : B → UU l7} + {f : A → X} {g : B → X} → + (f' : (a : A) → PA a → PX (f a)) (g' : (b : B) → PB b → PX (g b)) → + cone f g C → (C → UU l8) → UU (l4 ⊔ l5 ⊔ l6 ⊔ l7 ⊔ l8) +cone-family {C = C} PX {f = f} {g} f' g' c PC = + (x : C) → + cone + ( ( tr PX (coherence-square-cone f g c x)) ∘ + ( f' (vertical-map-cone f g c x))) + ( g' (horizontal-map-cone f g c x)) + ( PC x) +``` + +### Identifications of cones + +Next we characterize the identity type of the type of cones with a given vertex +C. Note that in the definition of htpy-cone we do not use pattern matching on +the cones c and c'. This is to ensure that the type htpy-cone f g c c' is a +Σ-type for any c and c', not just for c and c' of the form (pair p (pair q H)) +and (pair p' (pair q' H')) respectively. + +```agda +module _ + {l1 l2 l3 l4 : Level} {A : UU l1} {B : UU l2} {X : UU l3} + (f : A → X) (g : B → X) {C : UU l4} + where + + coherence-htpy-cone : + (c c' : cone f g C) (K : vertical-map-cone f g c ~ vertical-map-cone f g c') + (L : horizontal-map-cone f g c ~ horizontal-map-cone f g c') → UU (l4 ⊔ l3) + coherence-htpy-cone c c' K L = + ( coherence-square-cone f g c ∙h (g ·l L)) ~ + ( (f ·l K) ∙h coherence-square-cone f g c') + + htpy-cone : cone f g C → cone f g C → UU (l1 ⊔ l2 ⊔ l3 ⊔ l4) + htpy-cone c c' = + Σ ( vertical-map-cone f g c ~ vertical-map-cone f g c') + ( λ K → + Σ ( horizontal-map-cone f g c ~ horizontal-map-cone f g c') + ( coherence-htpy-cone c c' K)) + + refl-htpy-cone : (c : cone f g C) → htpy-cone c c + pr1 (refl-htpy-cone c) = refl-htpy + pr1 (pr2 (refl-htpy-cone c)) = refl-htpy + pr2 (pr2 (refl-htpy-cone c)) = right-unit-htpy + + htpy-eq-cone : (c c' : cone f g C) → c ＝ c' → htpy-cone c c' + htpy-eq-cone c .c refl = refl-htpy-cone c + + is-contr-total-htpy-cone : + (c : cone f g C) → is-contr (Σ (cone f g C) (htpy-cone c)) + is-contr-total-htpy-cone c = + is-contr-total-Eq-structure + ( λ p qH K → + Σ ( horizontal-map-cone f g c ~ pr1 qH) + ( coherence-htpy-cone c (pair p qH) K)) + ( is-contr-total-htpy (vertical-map-cone f g c)) + ( pair (vertical-map-cone f g c) refl-htpy) + ( is-contr-total-Eq-structure + ( λ q H → + coherence-htpy-cone c + ( pair (vertical-map-cone f g c) (pair q H)) + ( refl-htpy)) + ( is-contr-total-htpy (horizontal-map-cone f g c)) + ( pair (horizontal-map-cone f g c) refl-htpy) + ( is-contr-total-htpy (coherence-square-cone f g c ∙h refl-htpy))) + + is-equiv-htpy-eq-cone : (c c' : cone f g C) → is-equiv (htpy-eq-cone c c') + is-equiv-htpy-eq-cone c = + fundamental-theorem-id (is-contr-total-htpy-cone c) (htpy-eq-cone c) + + extensionality-cone : (c c' : cone f g C) → (c ＝ c') ≃ htpy-cone c c' + pr1 (extensionality-cone c c') = htpy-eq-cone c c' + pr2 (extensionality-cone c c') = is-equiv-htpy-eq-cone c c' + + eq-htpy-cone : (c c' : cone f g C) → htpy-cone c c' → (c ＝ c') + eq-htpy-cone c c' = map-inv-equiv (extensionality-cone c c') +``` + +### Precomposing cones + +```agda +module _ + {l1 l2 l3 : Level} {A : UU l1} {B : UU l2} {X : UU l3} + (f : A → X) (g : B → X) + where + + cone-map : + {l4 l5 : Level} {C : UU l4} {C' : UU l5} → + cone f g C → (C' → C) → cone f g C' + pr1 (cone-map c h) = vertical-map-cone f g c ∘ h + pr1 (pr2 (cone-map c h)) = horizontal-map-cone f g c ∘ h + pr2 (pr2 (cone-map c h)) = coherence-square-cone f g c ·r h +``` + +### Pasting cones horizontally + +```agda +module _ + {l1 l2 l3 l4 l5 l6 : Level} + {A : UU l1} {B : UU l2} {C : UU l3} {X : UU l4} {Y : UU l5} {Z : UU l6} + (i : X → Y) (j : Y → Z) (h : C → Z) + where + + pasting-horizontal-cone : + (c : cone j h B) → cone i (vertical-map-cone j h c) A → cone (j ∘ i) h A + pr1 (pasting-horizontal-cone c (pair f (pair p H))) = f + pr1 (pr2 (pasting-horizontal-cone c (pair f (pair p H)))) = + (horizontal-map-cone j h c) ∘ p + pr2 (pr2 (pasting-horizontal-cone c (pair f (pair p H)))) = + pasting-horizontal-coherence-square-maps p + ( horizontal-map-cone j h c) + ( f) + ( vertical-map-cone j h c) + ( h) + ( i) + ( j) + ( H) + ( coherence-square-cone j h c) +``` + +### Vertical composition of cones + +```agda +module _ + {l1 l2 l3 l4 l5 l6 : Level} + {A : UU l1} {B : UU l2} {C : UU l3} {X : UU l4} {Y : UU l5} {Z : UU l6} + (f : C → Z) (g : Y → Z) (h : X → Y) + where + + pasting-vertical-cone : + (c : cone f g B) → cone (horizontal-map-cone f g c) h A → cone f (g ∘ h) A + pr1 (pasting-vertical-cone c (pair p' (pair q' H'))) = + ( vertical-map-cone f g c) ∘ p' + pr1 (pr2 (pasting-vertical-cone c (pair p' (pair q' H')))) = q' + pr2 (pr2 (pasting-vertical-cone c (pair p' (pair q' H')))) = + pasting-vertical-coherence-square-maps q' p' h + ( horizontal-map-cone f g c) + ( vertical-map-cone f g c) + ( g) + ( f) + ( H') + ( coherence-square-cone f g c) +``` + +### The swapping function on cones + +```agda +swap-cone : + {l1 l2 l3 l4 : Level} {A : UU l1} {B : UU l2} {X : UU l3} {C : UU l4} + (f : A → X) (g : B → X) → cone f g C → cone g f C +pr1 (swap-cone f g c) = horizontal-map-cone f g c +pr1 (pr2 (swap-cone f g c)) = vertical-map-cone f g c +pr2 (pr2 (swap-cone f g c)) = inv-htpy (coherence-square-cone f g c) +``` + +### Parallel cones + +```agda +module _ + {l1 l2 l3 : Level} {A : UU l1} {B : UU l2} {X : UU l3} + {f f' : A → X} (Hf : f ~ f') {g g' : B → X} (Hg : g ~ g') + where + + coherence-htpy-parallel-cone : + {l4 : Level} {C : UU l4} (c : cone f g C) (c' : cone f' g' C) + (Hp : vertical-map-cone f g c ~ vertical-map-cone f' g' c') + (Hq : horizontal-map-cone f g c ~ horizontal-map-cone f' g' c') → + UU (l3 ⊔ l4) + coherence-htpy-parallel-cone c c' Hp Hq = + ( ( coherence-square-cone f g c) ∙h + ( (g ·l Hq) ∙h (Hg ·r horizontal-map-cone f' g' c'))) ~ + ( ( (f ·l Hp) ∙h (Hf ·r (vertical-map-cone f' g' c'))) ∙h + ( coherence-square-cone f' g' c')) + + fam-htpy-parallel-cone : + {l4 : Level} {C : UU l4} (c : cone f g C) → (c' : cone f' g' C) → + (vertical-map-cone f g c ~ vertical-map-cone f' g' c') → UU (l2 ⊔ l3 ⊔ l4) + fam-htpy-parallel-cone c c' Hp = + Σ ( horizontal-map-cone f g c ~ horizontal-map-cone f' g' c') + ( coherence-htpy-parallel-cone c c' Hp) + + htpy-parallel-cone : + {l4 : Level} {C : UU l4} → + cone f g C → cone f' g' C → UU (l1 ⊔ l2 ⊔ l3 ⊔ l4) + htpy-parallel-cone c c' = + Σ ( vertical-map-cone f g c ~ vertical-map-cone f' g' c') + ( fam-htpy-parallel-cone c c') +``` diff --git a/src/foundation/conjunction.lagda.md b/src/foundation/conjunction.lagda.md index d58a82f640..e51be853a1 100644 --- a/src/foundation/conjunction.lagda.md +++ b/src/foundation/conjunction.lagda.md @@ -8,14 +8,14 @@ module foundation.conjunction where ```agda open import foundation.decidable-types +open import foundation.dependent-pair-types +open import foundation.universe-levels open import foundation-core.cartesian-product-types open import foundation-core.decidable-propositions -open import foundation-core.dependent-pair-types open import foundation-core.equivalences open import foundation-core.logical-equivalences open import foundation-core.propositions -open import foundation-core.universe-levels ``` diff --git a/src/foundation/connected-components-universes.lagda.md b/src/foundation/connected-components-universes.lagda.md index 8a49f0484a..12a11e3726 100644 --- a/src/foundation/connected-components-universes.lagda.md +++ b/src/foundation/connected-components-universes.lagda.md @@ -8,22 +8,22 @@ module foundation.connected-components-universes where ```agda open import foundation.0-connected-types +open import foundation.dependent-pair-types open import foundation.empty-types open import foundation.functoriality-propositional-truncation +open import foundation.fundamental-theorem-of-identity-types open import foundation.mere-equivalences open import foundation.propositional-truncations open import foundation.raising-universe-levels +open import foundation.subtype-identity-principle open import foundation.subuniverses open import foundation.univalence +open import foundation.universe-levels open import foundation-core.contractible-types -open import foundation-core.dependent-pair-types open import foundation-core.equivalences -open import foundation-core.fundamental-theorem-of-identity-types open import foundation-core.identity-types -open import foundation-core.subtype-identity-principle open import foundation-core.subtypes -open import foundation-core.universe-levels ``` diff --git a/src/foundation/connected-components.lagda.md b/src/foundation/connected-components.lagda.md index 45e7031cba..d17d7a1006 100644 --- a/src/foundation/connected-components.lagda.md +++ b/src/foundation/connected-components.lagda.md @@ -8,15 +8,15 @@ module foundation.connected-components where ```agda open import foundation.0-connected-types +open import foundation.dependent-pair-types open import foundation.propositional-truncations open import foundation.propositions +open import foundation.universe-levels -open import foundation-core.dependent-pair-types open import foundation-core.equality-dependent-pair-types open import foundation-core.identity-types open import foundation-core.truncated-types open import foundation-core.truncation-levels -open import foundation-core.universe-levels open import higher-group-theory.higher-groups diff --git a/src/foundation/connected-maps.lagda.md b/src/foundation/connected-maps.lagda.md index c424bdcd1c..a2fd7c26ff 100644 --- a/src/foundation/connected-maps.lagda.md +++ b/src/foundation/connected-maps.lagda.md @@ -8,27 +8,27 @@ module foundation.connected-maps where ```agda open import foundation.connected-types +open import foundation.dependent-pair-types +open import foundation.functions open import foundation.functoriality-dependent-function-types +open import foundation.fundamental-theorem-of-identity-types open import foundation.homotopies open import foundation.structure-identity-principle +open import foundation.subtype-identity-principle open import foundation.truncated-types open import foundation.truncation-levels open import foundation.univalence +open import foundation.universe-levels open import foundation-core.contractible-maps open import foundation-core.contractible-types -open import foundation-core.dependent-pair-types open import foundation-core.embeddings open import foundation-core.equivalences open import foundation-core.fibers-of-maps -open import foundation-core.functions -open import foundation-core.fundamental-theorem-of-identity-types open import foundation-core.identity-types open import foundation-core.propositions -open import foundation-core.subtype-identity-principle open import foundation-core.subtypes open import foundation-core.truncated-maps -open import foundation-core.universe-levels ``` diff --git a/src/foundation/connected-types.lagda.md b/src/foundation/connected-types.lagda.md index f422aacc59..5c725a0a9b 100644 --- a/src/foundation/connected-types.lagda.md +++ b/src/foundation/connected-types.lagda.md @@ -8,15 +8,15 @@ module foundation.connected-types where ```agda open import foundation.contractible-types +open import foundation.functions open import foundation.truncations +open import foundation.universe-levels open import foundation-core.constant-maps open import foundation-core.equivalences -open import foundation-core.functions open import foundation-core.propositions open import foundation-core.truncated-types open import foundation-core.truncation-levels -open import foundation-core.universe-levels ``` diff --git a/src/foundation/constant-maps.lagda.md b/src/foundation/constant-maps.lagda.md index 1b7928aba8..0b45fca91f 100644 --- a/src/foundation/constant-maps.lagda.md +++ b/src/foundation/constant-maps.lagda.md @@ -9,19 +9,20 @@ open import foundation-core.constant-maps public
Imports ```agda +open import foundation.0-maps +open import foundation.dependent-pair-types +open import foundation.faithful-maps +open import foundation.functions open import foundation.type-arithmetic-unit-type open import foundation.unit-type +open import foundation.universe-levels -open import foundation-core.0-maps open import foundation-core.1-types open import foundation-core.contractible-maps -open import foundation-core.dependent-pair-types open import foundation-core.embeddings open import foundation-core.equivalences -open import foundation-core.faithful-maps open import foundation-core.fibers-of-maps open import foundation-core.function-extensionality -open import foundation-core.functions open import foundation-core.identity-types open import foundation-core.injective-maps open import foundation-core.propositional-maps @@ -30,7 +31,6 @@ open import foundation-core.sets open import foundation-core.truncated-maps open import foundation-core.truncated-types open import foundation-core.truncation-levels -open import foundation-core.universe-levels ```
diff --git a/src/foundation/contractible-maps.lagda.md b/src/foundation/contractible-maps.lagda.md index e766b06c92..18f0b23fb7 100644 --- a/src/foundation/contractible-maps.lagda.md +++ b/src/foundation/contractible-maps.lagda.md @@ -9,14 +9,14 @@ open import foundation-core.contractible-maps public
Imports ```agda +open import foundation.dependent-pair-types open import foundation.equivalences open import foundation.truncated-maps +open import foundation.universe-levels -open import foundation-core.dependent-pair-types open import foundation-core.logical-equivalences open import foundation-core.propositions open import foundation-core.truncation-levels -open import foundation-core.universe-levels ```
diff --git a/src/foundation/contractible-types.lagda.md b/src/foundation/contractible-types.lagda.md index 9a5b9eef02..d5ea7a0d8e 100644 --- a/src/foundation/contractible-types.lagda.md +++ b/src/foundation/contractible-types.lagda.md @@ -9,15 +9,16 @@ open import foundation-core.contractible-types public
Imports ```agda +open import foundation.dependent-pair-types open import foundation.function-extensionality +open import foundation.functions open import foundation.subuniverses open import foundation.unit-type +open import foundation.universe-levels open import foundation-core.constant-maps open import foundation-core.contractible-maps -open import foundation-core.dependent-pair-types open import foundation-core.equivalences -open import foundation-core.functions open import foundation-core.functoriality-dependent-pair-types open import foundation-core.identity-types open import foundation-core.propositions @@ -25,7 +26,6 @@ open import foundation-core.singleton-induction open import foundation-core.subtypes open import foundation-core.truncated-types open import foundation-core.truncation-levels -open import foundation-core.universe-levels ```
diff --git a/src/foundation/coproduct-decompositions-subuniverse.lagda.md b/src/foundation/coproduct-decompositions-subuniverse.lagda.md index 5525b0a865..ecb6f55b78 100644 --- a/src/foundation/coproduct-decompositions-subuniverse.lagda.md +++ b/src/foundation/coproduct-decompositions-subuniverse.lagda.md @@ -8,29 +8,29 @@ module foundation.coproduct-decompositions-subuniverse where ```agda open import foundation.coproduct-types +open import foundation.dependent-pair-types open import foundation.empty-types open import foundation.equivalence-extensionality open import foundation.equivalences +open import foundation.functions open import foundation.functoriality-coproduct-types +open import foundation.fundamental-theorem-of-identity-types open import foundation.structure-identity-principle open import foundation.subuniverses +open import foundation.type-arithmetic-cartesian-product-types open import foundation.type-arithmetic-coproduct-types +open import foundation.type-arithmetic-dependent-pair-types open import foundation.type-arithmetic-empty-type open import foundation.univalence +open import foundation.universe-levels open import foundation-core.cartesian-product-types open import foundation-core.contractible-types -open import foundation-core.dependent-pair-types open import foundation-core.equality-dependent-pair-types -open import foundation-core.functions open import foundation-core.functoriality-dependent-pair-types -open import foundation-core.fundamental-theorem-of-identity-types open import foundation-core.homotopies open import foundation-core.identity-types open import foundation-core.propositions -open import foundation-core.type-arithmetic-cartesian-product-types -open import foundation-core.type-arithmetic-dependent-pair-types -open import foundation-core.universe-levels ``` diff --git a/src/foundation/coproduct-decompositions.lagda.md b/src/foundation/coproduct-decompositions.lagda.md index 70b334af02..9af4b8750f 100644 --- a/src/foundation/coproduct-decompositions.lagda.md +++ b/src/foundation/coproduct-decompositions.lagda.md @@ -8,30 +8,30 @@ module foundation.coproduct-decompositions where ```agda open import foundation.coproduct-decompositions-subuniverse +open import foundation.dependent-pair-types open import foundation.equivalence-extensionality open import foundation.equivalences open import foundation.function-extensionality +open import foundation.functions open import foundation.functoriality-coproduct-types +open import foundation.fundamental-theorem-of-identity-types open import foundation.structure-identity-principle open import foundation.transport open import foundation.type-arithmetic-coproduct-types +open import foundation.type-arithmetic-dependent-pair-types open import foundation.type-arithmetic-empty-type open import foundation.unit-type open import foundation.univalence +open import foundation.universe-levels open import foundation-core.contractible-types open import foundation-core.coproduct-types -open import foundation-core.dependent-pair-types open import foundation-core.equality-dependent-pair-types open import foundation-core.fibers-of-maps -open import foundation-core.functions open import foundation-core.functoriality-dependent-pair-types -open import foundation-core.fundamental-theorem-of-identity-types open import foundation-core.homotopies open import foundation-core.identity-types open import foundation-core.propositions -open import foundation-core.type-arithmetic-dependent-pair-types -open import foundation-core.universe-levels open import univalent-combinatorics.equality-standard-finite-types open import univalent-combinatorics.standard-finite-types diff --git a/src/foundation/coproduct-types.lagda.md b/src/foundation/coproduct-types.lagda.md index 5e05033bb9..6d27e4124f 100644 --- a/src/foundation/coproduct-types.lagda.md +++ b/src/foundation/coproduct-types.lagda.md @@ -9,21 +9,21 @@ open import foundation-core.coproduct-types public
Imports ```agda +open import foundation.dependent-pair-types +open import foundation.functions open import foundation.noncontractible-types open import foundation.subuniverses open import foundation.unit-type +open import foundation.universe-levels open import foundation-core.contractible-types -open import foundation-core.dependent-pair-types open import foundation-core.empty-types open import foundation-core.equivalences -open import foundation-core.functions open import foundation-core.homotopies open import foundation-core.identity-types open import foundation-core.injective-maps open import foundation-core.negation open import foundation-core.propositions -open import foundation-core.universe-levels ```
diff --git a/src/foundation/coslice.lagda.md b/src/foundation/coslice.lagda.md index 76327b8166..e9038837b9 100644 --- a/src/foundation/coslice.lagda.md +++ b/src/foundation/coslice.lagda.md @@ -7,15 +7,15 @@ module foundation.coslice where
Imports ```agda +open import foundation.dependent-pair-types open import foundation.function-extensionality +open import foundation.functions open import foundation.structure-identity-principle +open import foundation.universe-levels -open import foundation-core.dependent-pair-types open import foundation-core.equivalences -open import foundation-core.functions open import foundation-core.homotopies open import foundation-core.identity-types -open import foundation-core.universe-levels ```
diff --git a/src/foundation/cospans.lagda.md b/src/foundation/cospans.lagda.md index cf0e05263e..64d4a9f3eb 100644 --- a/src/foundation/cospans.lagda.md +++ b/src/foundation/cospans.lagda.md @@ -1,15 +1,32 @@ # Cospans ```agda -module foundation.cospans where +{-# OPTIONS --safe #-} -open import foundation-core.cospans public +module foundation.cospans where ```
Imports ```agda +open import foundation.dependent-pair-types +open import foundation.universe-levels +open import foundation-core.cartesian-product-types ```
+ +## Definition + +### Cospans + +A cospan is a pair of functions with a common codomain + +```agda +cospan : + {l1 l2 : Level} (l : Level) (A : UU l1) (B : UU l2) → + UU (l1 ⊔ l2 ⊔ lsuc l) +cospan l A B = + Σ (UU l) (λ X → (A → X) × (B → X)) +``` diff --git a/src/foundation/decidable-dependent-function-types.lagda.md b/src/foundation/decidable-dependent-function-types.lagda.md index 823be37ea8..09cf70c1df 100644 --- a/src/foundation/decidable-dependent-function-types.lagda.md +++ b/src/foundation/decidable-dependent-function-types.lagda.md @@ -12,10 +12,10 @@ open import foundation.functoriality-dependent-function-types open import foundation.maybe open import foundation.universal-property-coproduct-types open import foundation.universal-property-maybe +open import foundation.universe-levels open import foundation-core.coproduct-types open import foundation-core.equivalences -open import foundation-core.universe-levels ``` diff --git a/src/foundation/decidable-dependent-pair-types.lagda.md b/src/foundation/decidable-dependent-pair-types.lagda.md index d0d6af97e2..6aebc9f8d2 100644 --- a/src/foundation/decidable-dependent-pair-types.lagda.md +++ b/src/foundation/decidable-dependent-pair-types.lagda.md @@ -8,16 +8,16 @@ module foundation.decidable-dependent-pair-types where ```agda open import foundation.decidable-types +open import foundation.dependent-pair-types +open import foundation.functions open import foundation.maybe open import foundation.type-arithmetic-coproduct-types open import foundation.type-arithmetic-unit-type +open import foundation.universe-levels open import foundation-core.coproduct-types -open import foundation-core.dependent-pair-types open import foundation-core.equivalences -open import foundation-core.functions open import foundation-core.functoriality-dependent-pair-types -open import foundation-core.universe-levels ``` diff --git a/src/foundation/decidable-embeddings.lagda.md b/src/foundation/decidable-embeddings.lagda.md index 2e688837cb..84a86a8088 100644 --- a/src/foundation/decidable-embeddings.lagda.md +++ b/src/foundation/decidable-embeddings.lagda.md @@ -10,30 +10,30 @@ module foundation.decidable-embeddings where open import foundation.decidable-maps open import foundation.decidable-subtypes open import foundation.decidable-types +open import foundation.dependent-pair-types open import foundation.embeddings open import foundation.equivalences +open import foundation.functions open import foundation.functoriality-cartesian-product-types +open import foundation.fundamental-theorem-of-identity-types open import foundation.homotopies open import foundation.identity-types open import foundation.propositional-maps open import foundation.structured-type-duality +open import foundation.subtype-identity-principle +open import foundation.type-arithmetic-dependent-pair-types open import foundation.type-theoretic-principle-of-choice +open import foundation.universe-levels open import foundation-core.cartesian-product-types open import foundation-core.contractible-maps open import foundation-core.contractible-types open import foundation-core.coproduct-types open import foundation-core.decidable-propositions -open import foundation-core.dependent-pair-types open import foundation-core.empty-types open import foundation-core.fibers-of-maps -open import foundation-core.functions open import foundation-core.functoriality-dependent-pair-types -open import foundation-core.fundamental-theorem-of-identity-types open import foundation-core.propositions -open import foundation-core.subtype-identity-principle -open import foundation-core.type-arithmetic-dependent-pair-types -open import foundation-core.universe-levels ``` diff --git a/src/foundation/decidable-equality.lagda.md b/src/foundation/decidable-equality.lagda.md index 5f95c34604..e05333ade2 100644 --- a/src/foundation/decidable-equality.lagda.md +++ b/src/foundation/decidable-equality.lagda.md @@ -9,13 +9,15 @@ module foundation.decidable-equality where ```agda open import foundation.coproduct-types open import foundation.decidable-types +open import foundation.dependent-pair-types open import foundation.double-negation open import foundation.negation open import foundation.sections +open import foundation.type-arithmetic-dependent-pair-types open import foundation.unit-type +open import foundation.universe-levels open import foundation-core.cartesian-product-types -open import foundation-core.dependent-pair-types open import foundation-core.empty-types open import foundation-core.equality-dependent-pair-types open import foundation-core.equivalences @@ -25,8 +27,6 @@ open import foundation-core.injective-maps open import foundation-core.propositions open import foundation-core.retractions open import foundation-core.sets -open import foundation-core.type-arithmetic-dependent-pair-types -open import foundation-core.universe-levels ``` diff --git a/src/foundation/decidable-equivalence-relations.lagda.md b/src/foundation/decidable-equivalence-relations.lagda.md index a6c773928a..c1bdfbcbaf 100644 --- a/src/foundation/decidable-equivalence-relations.lagda.md +++ b/src/foundation/decidable-equivalence-relations.lagda.md @@ -12,38 +12,38 @@ open import foundation.decidable-propositions open import foundation.decidable-relations open import foundation.decidable-subtypes open import foundation.decidable-types +open import foundation.dependent-pair-types open import foundation.effective-maps-equivalence-relations open import foundation.equivalence-classes open import foundation.equivalence-relations open import foundation.existential-quantification open import foundation.function-extensionality +open import foundation.functions open import foundation.functoriality-cartesian-product-types +open import foundation.fundamental-theorem-of-identity-types open import foundation.images open import foundation.propositional-truncations open import foundation.reflecting-maps-equivalence-relations open import foundation.sets open import foundation.slice open import foundation.surjective-maps +open import foundation.type-arithmetic-cartesian-product-types +open import foundation.type-arithmetic-dependent-pair-types open import foundation.universal-property-image +open import foundation.universe-levels open import foundation-core.cartesian-product-types open import foundation-core.contractible-types -open import foundation-core.dependent-pair-types open import foundation-core.embeddings open import foundation-core.equality-dependent-pair-types open import foundation-core.equivalences open import foundation-core.fibers-of-maps -open import foundation-core.functions open import foundation-core.functoriality-dependent-pair-types -open import foundation-core.fundamental-theorem-of-identity-types open import foundation-core.homotopies open import foundation-core.identity-types open import foundation-core.logical-equivalences open import foundation-core.propositions open import foundation-core.subtypes -open import foundation-core.type-arithmetic-cartesian-product-types -open import foundation-core.type-arithmetic-dependent-pair-types -open import foundation-core.universe-levels ``` diff --git a/src/foundation/decidable-maps.lagda.md b/src/foundation/decidable-maps.lagda.md index 85be938057..9425419115 100644 --- a/src/foundation/decidable-maps.lagda.md +++ b/src/foundation/decidable-maps.lagda.md @@ -9,15 +9,15 @@ module foundation.decidable-maps where ```agda open import foundation.decidable-equality open import foundation.decidable-types +open import foundation.dependent-pair-types +open import foundation.functions +open import foundation.universe-levels -open import foundation-core.dependent-pair-types open import foundation-core.equivalences open import foundation-core.fibers-of-maps -open import foundation-core.functions open import foundation-core.functoriality-dependent-pair-types open import foundation-core.identity-types open import foundation-core.retractions -open import foundation-core.universe-levels ``` diff --git a/src/foundation/decidable-propositions.lagda.md b/src/foundation/decidable-propositions.lagda.md index e17f83b1cf..a6fe35d9d5 100644 --- a/src/foundation/decidable-propositions.lagda.md +++ b/src/foundation/decidable-propositions.lagda.md @@ -11,19 +11,21 @@ open import foundation-core.decidable-propositions public ```agda open import foundation.booleans open import foundation.decidable-types +open import foundation.dependent-pair-types open import foundation.embeddings open import foundation.empty-types open import foundation.equivalences +open import foundation.functions open import foundation.negation open import foundation.propositional-extensionality open import foundation.raising-universe-levels open import foundation.type-arithmetic-coproduct-types +open import foundation.type-arithmetic-dependent-pair-types open import foundation.unit-type +open import foundation.universe-levels open import foundation-core.contractible-types open import foundation-core.coproduct-types -open import foundation-core.dependent-pair-types -open import foundation-core.functions open import foundation-core.homotopies open import foundation-core.identity-types open import foundation-core.logical-equivalences @@ -31,8 +33,6 @@ open import foundation-core.propositions open import foundation-core.sets open import foundation-core.small-types open import foundation-core.subtypes -open import foundation-core.type-arithmetic-dependent-pair-types -open import foundation-core.universe-levels open import univalent-combinatorics.counting open import univalent-combinatorics.finite-types diff --git a/src/foundation/decidable-relations.lagda.md b/src/foundation/decidable-relations.lagda.md index 373987f117..2259337ca0 100644 --- a/src/foundation/decidable-relations.lagda.md +++ b/src/foundation/decidable-relations.lagda.md @@ -9,13 +9,13 @@ module foundation.decidable-relations where ```agda open import foundation.binary-relations open import foundation.decidable-types +open import foundation.dependent-pair-types +open import foundation.universe-levels open import foundation-core.decidable-propositions -open import foundation-core.dependent-pair-types open import foundation-core.equivalences open import foundation-core.homotopies open import foundation-core.propositions -open import foundation-core.universe-levels ``` diff --git a/src/foundation/decidable-subtypes.lagda.md b/src/foundation/decidable-subtypes.lagda.md index abff817889..f3740a0a04 100644 --- a/src/foundation/decidable-subtypes.lagda.md +++ b/src/foundation/decidable-subtypes.lagda.md @@ -11,22 +11,22 @@ open import foundation.1-types open import foundation.coproduct-types open import foundation.decidable-propositions open import foundation.decidable-types +open import foundation.dependent-pair-types open import foundation.equality-dependent-function-types +open import foundation.functions open import foundation.functoriality-dependent-function-types open import foundation.sets open import foundation.subtypes +open import foundation.universe-levels -open import foundation-core.dependent-pair-types open import foundation-core.embeddings open import foundation-core.equivalences -open import foundation-core.functions open import foundation-core.identity-types open import foundation-core.injective-maps open import foundation-core.logical-equivalences open import foundation-core.propositions open import foundation-core.truncated-types open import foundation-core.truncation-levels -open import foundation-core.universe-levels ``` diff --git a/src/foundation/decidable-types.lagda.md b/src/foundation/decidable-types.lagda.md index 3e21cd02d5..42fda7ccc7 100644 --- a/src/foundation/decidable-types.lagda.md +++ b/src/foundation/decidable-types.lagda.md @@ -8,22 +8,22 @@ module foundation.decidable-types where ```agda open import foundation.coproduct-types +open import foundation.dependent-pair-types open import foundation.double-negation open import foundation.empty-types +open import foundation.functions open import foundation.hilberts-epsilon-operators open import foundation.negation open import foundation.propositional-truncations open import foundation.raising-universe-levels open import foundation.type-arithmetic-empty-type open import foundation.unit-type +open import foundation.universe-levels open import foundation-core.cartesian-product-types -open import foundation-core.dependent-pair-types open import foundation-core.equivalences -open import foundation-core.functions open import foundation-core.propositions open import foundation-core.retractions -open import foundation-core.universe-levels ``` diff --git a/src/foundation/dependent-binomial-theorem.lagda.md b/src/foundation/dependent-binomial-theorem.lagda.md index bcaca71bdb..ed58754757 100644 --- a/src/foundation/dependent-binomial-theorem.lagda.md +++ b/src/foundation/dependent-binomial-theorem.lagda.md @@ -9,6 +9,8 @@ module foundation.dependent-binomial-theorem where ```agda open import foundation.contractible-types open import foundation.coproduct-decompositions +open import foundation.dependent-pair-types +open import foundation.functions open import foundation.functoriality-cartesian-product-types open import foundation.functoriality-dependent-function-types open import foundation.identity-types @@ -16,16 +18,14 @@ open import foundation.raising-universe-levels open import foundation.type-theoretic-principle-of-choice open import foundation.universal-property-coproduct-types open import foundation.universal-property-dependent-pair-types +open import foundation.universe-levels open import foundation-core.cartesian-product-types open import foundation-core.coproduct-types -open import foundation-core.dependent-pair-types open import foundation-core.equivalences -open import foundation-core.functions open import foundation-core.functoriality-dependent-pair-types open import foundation-core.homotopies open import foundation-core.univalence -open import foundation-core.universe-levels open import univalent-combinatorics.equality-standard-finite-types open import univalent-combinatorics.standard-finite-types diff --git a/src/foundation/dependent-pair-types.lagda.md b/src/foundation/dependent-pair-types.lagda.md index 51da817145..6e1c81b293 100644 --- a/src/foundation/dependent-pair-types.lagda.md +++ b/src/foundation/dependent-pair-types.lagda.md @@ -4,14 +4,75 @@ {-# OPTIONS --safe #-} module foundation.dependent-pair-types where - -open import foundation-core.dependent-pair-types public ```
Imports ```agda - +open import foundation.universe-levels ```
+ +## Idea + +When `B` is a family of types over `A`, then we can form the type of pairs +`pair a b` consisting of an element `a : A` and an element `b : B a`. Such pairs +are called dependent pairs, since the type of the second component depends on +the first component. + +## Definition + +```agda +record Σ {l1 l2 : Level} (A : UU l1) (B : A → UU l2) : UU (l1 ⊔ l2) where + constructor pair + field + pr1 : A + pr2 : B pr1 + +open Σ public + +{-# BUILTIN SIGMA Σ #-} + +infixr 10 _,_ +pattern _,_ a b = pair a b +``` + +## Constructions + +```agda +ind-Σ : + {l1 l2 l3 : Level} {A : UU l1} {B : A → UU l2} {C : Σ A B → UU l3} → + ((x : A) (y : B x) → C (pair x y)) → ((t : Σ A B) → C t) +ind-Σ f (pair x y) = f x y + +ev-pair : + {l1 l2 l3 : Level} {A : UU l1} {B : A → UU l2} {C : Σ A B → UU l3} → + ((t : Σ A B) → C t) → (x : A) (y : B x) → C (pair x y) +ev-pair f x y = f (pair x y) + +triple : + {l1 l2 l3 : Level} {A : UU l1} {B : A → UU l2} {C : (x : A) → B x → UU l3} → + (a : A) (b : B a) → C a b → Σ A (λ x → Σ (B x) (C x)) +pr1 (triple a b c) = a +pr1 (pr2 (triple a b c)) = b +pr2 (pr2 (triple a b c)) = c + +triple' : + {l1 l2 l3 : Level} {A : UU l1} {B : A → UU l2} {C : Σ A B → UU l3} → + (a : A) (b : B a) → C (pair a b) → Σ (Σ A B) C +pr1 (pr1 (triple' a b c)) = a +pr2 (pr1 (triple' a b c)) = b +pr2 (triple' a b c) = c +``` + +### Families on dependent pair types + +```agda +module _ + {l1 l2 l3 : Level} {A : UU l1} {B : A → UU l2} + where + + fam-Σ : ((x : A) → B x → UU l3) → Σ A B → UU l3 + fam-Σ C (pair x y) = C x y +``` diff --git a/src/foundation/dependent-paths.lagda.md b/src/foundation/dependent-paths.lagda.md index d189c56479..9af20ccccf 100644 --- a/src/foundation/dependent-paths.lagda.md +++ b/src/foundation/dependent-paths.lagda.md @@ -11,14 +11,14 @@ module foundation.dependent-paths where
Imports ```agda +open import foundation.dependent-pair-types open import foundation.function-extensionality +open import foundation.functions open import foundation.identity-types +open import foundation.universe-levels -open import foundation-core.dependent-pair-types open import foundation-core.equivalences -open import foundation-core.functions open import foundation-core.homotopies -open import foundation-core.universe-levels ```
diff --git a/src/foundation/descent-coproduct-types.lagda.md b/src/foundation/descent-coproduct-types.lagda.md index 4c18a95486..017dd6a8db 100644 --- a/src/foundation/descent-coproduct-types.lagda.md +++ b/src/foundation/descent-coproduct-types.lagda.md @@ -7,19 +7,19 @@ module foundation.descent-coproduct-types where
Imports ```agda +open import foundation.cones-over-cospans +open import foundation.dependent-pair-types +open import foundation.functions open import foundation.functoriality-coproduct-types +open import foundation.functoriality-fibers-of-maps +open import foundation.universe-levels -open import foundation-core.cones-over-cospans open import foundation-core.coproduct-types -open import foundation-core.dependent-pair-types open import foundation-core.equality-dependent-pair-types open import foundation-core.equivalences -open import foundation-core.functions -open import foundation-core.functoriality-fibers-of-maps open import foundation-core.homotopies open import foundation-core.identity-types open import foundation-core.pullbacks -open import foundation-core.universe-levels ```
diff --git a/src/foundation/descent-dependent-pair-types.lagda.md b/src/foundation/descent-dependent-pair-types.lagda.md index 6c3e1f1853..ef4c7e05f8 100644 --- a/src/foundation/descent-dependent-pair-types.lagda.md +++ b/src/foundation/descent-dependent-pair-types.lagda.md @@ -7,16 +7,17 @@ module foundation.descent-dependent-pair-types where
Imports ```agda -open import foundation-core.cones-over-cospans -open import foundation-core.dependent-pair-types +open import foundation.cones-over-cospans +open import foundation.dependent-pair-types +open import foundation.functions +open import foundation.functoriality-fibers-of-maps +open import foundation.universe-levels + open import foundation-core.equivalences -open import foundation-core.functions open import foundation-core.functoriality-dependent-pair-types -open import foundation-core.functoriality-fibers-of-maps open import foundation-core.homotopies open import foundation-core.identity-types open import foundation-core.pullbacks -open import foundation-core.universe-levels ```
diff --git a/src/foundation/descent-empty-types.lagda.md b/src/foundation/descent-empty-types.lagda.md index 12794b99f4..e806d4e520 100644 --- a/src/foundation/descent-empty-types.lagda.md +++ b/src/foundation/descent-empty-types.lagda.md @@ -7,11 +7,12 @@ module foundation.descent-empty-types where
Imports ```agda -open import foundation-core.cones-over-cospans -open import foundation-core.dependent-pair-types +open import foundation.cones-over-cospans +open import foundation.dependent-pair-types +open import foundation.universe-levels + open import foundation-core.empty-types open import foundation-core.pullbacks -open import foundation-core.universe-levels ```
diff --git a/src/foundation/descent-equivalences.lagda.md b/src/foundation/descent-equivalences.lagda.md index a013c44593..60c172dc23 100644 --- a/src/foundation/descent-equivalences.lagda.md +++ b/src/foundation/descent-equivalences.lagda.md @@ -7,14 +7,14 @@ module foundation.descent-equivalences where
Imports ```agda +open import foundation.cones-over-cospans open import foundation.equivalences +open import foundation.functions +open import foundation.functoriality-fibers-of-maps +open import foundation.universe-levels -open import foundation-core.cones-over-cospans -open import foundation-core.functions open import foundation-core.functoriality-dependent-function-types -open import foundation-core.functoriality-fibers-of-maps open import foundation-core.pullbacks -open import foundation-core.universe-levels ```
diff --git a/src/foundation/diagonal-maps-of-types.lagda.md b/src/foundation/diagonal-maps-of-types.lagda.md index 516b912b02..16ee5f17da 100644 --- a/src/foundation/diagonal-maps-of-types.lagda.md +++ b/src/foundation/diagonal-maps-of-types.lagda.md @@ -9,13 +9,15 @@ open import foundation-core.diagonal-maps-of-types public
Imports ```agda -open import foundation-core.0-maps +open import foundation.0-maps +open import foundation.dependent-pair-types +open import foundation.faithful-maps +open import foundation.universe-levels + open import foundation-core.1-types open import foundation-core.cartesian-product-types open import foundation-core.contractible-maps -open import foundation-core.dependent-pair-types open import foundation-core.embeddings -open import foundation-core.faithful-maps open import foundation-core.fibers-of-maps open import foundation-core.identity-types open import foundation-core.propositional-maps @@ -24,7 +26,6 @@ open import foundation-core.sets open import foundation-core.truncated-maps open import foundation-core.truncated-types open import foundation-core.truncation-levels -open import foundation-core.universe-levels ```
diff --git a/src/foundation/diagonals-of-maps.lagda.md b/src/foundation/diagonals-of-maps.lagda.md index 931d48afc7..d30ee97f8e 100644 --- a/src/foundation/diagonals-of-maps.lagda.md +++ b/src/foundation/diagonals-of-maps.lagda.md @@ -7,13 +7,15 @@ module foundation.diagonals-of-maps where
Imports ```agda +open import foundation.dependent-pair-types +open import foundation.equality-fibers-of-maps +open import foundation.functions +open import foundation.universe-levels + open import foundation-core.contractible-maps -open import foundation-core.dependent-pair-types open import foundation-core.embeddings -open import foundation-core.equality-fibers-of-maps open import foundation-core.equivalences open import foundation-core.fibers-of-maps -open import foundation-core.functions open import foundation-core.homotopies open import foundation-core.identity-types open import foundation-core.propositional-maps @@ -21,7 +23,6 @@ open import foundation-core.pullbacks open import foundation-core.truncated-maps open import foundation-core.truncated-types open import foundation-core.truncation-levels -open import foundation-core.universe-levels ```
diff --git a/src/foundation/discrete-reflexive-relations.lagda.md b/src/foundation/discrete-reflexive-relations.lagda.md index 40fc63c05f..25fc505757 100644 --- a/src/foundation/discrete-reflexive-relations.lagda.md +++ b/src/foundation/discrete-reflexive-relations.lagda.md @@ -8,12 +8,12 @@ module foundation.discrete-reflexive-relations where ```agda open import foundation.contractible-types +open import foundation.dependent-pair-types open import foundation.reflexive-relations +open import foundation.universe-levels -open import foundation-core.dependent-pair-types open import foundation-core.identity-types open import foundation-core.propositions -open import foundation-core.universe-levels ``` diff --git a/src/foundation/discrete-relaxed-sigma-decompositions.lagda.md b/src/foundation/discrete-relaxed-sigma-decompositions.lagda.md index cc3d5a9376..9fd25d84b5 100644 --- a/src/foundation/discrete-relaxed-sigma-decompositions.lagda.md +++ b/src/foundation/discrete-relaxed-sigma-decompositions.lagda.md @@ -8,18 +8,18 @@ module foundation.discrete-relaxed-sigma-decompositions where ```agda open import foundation.contractible-types +open import foundation.dependent-pair-types open import foundation.equivalences +open import foundation.functions open import foundation.relaxed-sigma-decompositions +open import foundation.type-arithmetic-dependent-pair-types open import foundation.unit-type +open import foundation.universe-levels -open import foundation-core.dependent-pair-types open import foundation-core.equality-dependent-pair-types -open import foundation-core.functions open import foundation-core.identity-types open import foundation-core.propositions open import foundation-core.subtypes -open import foundation-core.type-arithmetic-dependent-pair-types -open import foundation-core.universe-levels ``` diff --git a/src/foundation/discrete-sigma-decompositions.lagda.md b/src/foundation/discrete-sigma-decompositions.lagda.md index 8a65930e69..4055e60dce 100644 --- a/src/foundation/discrete-sigma-decompositions.lagda.md +++ b/src/foundation/discrete-sigma-decompositions.lagda.md @@ -8,19 +8,19 @@ module foundation.discrete-sigma-decompositions where ```agda open import foundation.contractible-types +open import foundation.dependent-pair-types open import foundation.equivalences +open import foundation.functions open import foundation.propositional-truncations open import foundation.sigma-decompositions +open import foundation.type-arithmetic-dependent-pair-types open import foundation.unit-type +open import foundation.universe-levels -open import foundation-core.dependent-pair-types open import foundation-core.equality-dependent-pair-types -open import foundation-core.functions open import foundation-core.identity-types open import foundation-core.propositions open import foundation-core.subtypes -open import foundation-core.type-arithmetic-dependent-pair-types -open import foundation-core.universe-levels ``` diff --git a/src/foundation/discrete-types.lagda.md b/src/foundation/discrete-types.lagda.md index c6a95477d9..734915f32a 100644 --- a/src/foundation/discrete-types.lagda.md +++ b/src/foundation/discrete-types.lagda.md @@ -12,15 +12,15 @@ open import foundation-core.discrete-types public open import foundation.apartness-relations open import foundation.binary-relations open import foundation.decidable-types +open import foundation.dependent-pair-types open import foundation.negation open import foundation.propositional-truncations open import foundation.tight-apartness-relations +open import foundation.universe-levels open import foundation-core.coproduct-types -open import foundation-core.dependent-pair-types open import foundation-core.identity-types open import foundation-core.propositions -open import foundation-core.universe-levels ``` diff --git a/src/foundation/disjunction.lagda.md b/src/foundation/disjunction.lagda.md index a06f1c133a..4b176414df 100644 --- a/src/foundation/disjunction.lagda.md +++ b/src/foundation/disjunction.lagda.md @@ -9,15 +9,15 @@ module foundation.disjunction where ```agda open import foundation.conjunction open import foundation.decidable-types +open import foundation.dependent-pair-types +open import foundation.functions open import foundation.propositional-truncations +open import foundation.universe-levels open import foundation-core.coproduct-types open import foundation-core.decidable-propositions -open import foundation-core.dependent-pair-types open import foundation-core.equivalences -open import foundation-core.functions open import foundation-core.propositions -open import foundation-core.universe-levels ``` diff --git a/src/foundation/double-negation.lagda.md b/src/foundation/double-negation.lagda.md index a623e0aee0..645301205d 100644 --- a/src/foundation/double-negation.lagda.md +++ b/src/foundation/double-negation.lagda.md @@ -7,16 +7,16 @@ module foundation.double-negation where
Imports ```agda +open import foundation.dependent-pair-types +open import foundation.functions open import foundation.negation open import foundation.propositional-truncations +open import foundation.universe-levels open import foundation-core.cartesian-product-types open import foundation-core.coproduct-types -open import foundation-core.dependent-pair-types open import foundation-core.empty-types -open import foundation-core.functions open import foundation-core.propositions -open import foundation-core.universe-levels ```
diff --git a/src/foundation/double-powersets.lagda.md b/src/foundation/double-powersets.lagda.md index db2fa08402..9788978ec3 100644 --- a/src/foundation/double-powersets.lagda.md +++ b/src/foundation/double-powersets.lagda.md @@ -7,14 +7,14 @@ module foundation.double-powersets where
Imports ```agda +open import foundation.dependent-pair-types open import foundation.existential-quantification open import foundation.powersets open import foundation.propositional-truncations +open import foundation.universe-levels -open import foundation-core.dependent-pair-types open import foundation-core.propositions open import foundation-core.subtypes -open import foundation-core.universe-levels open import order-theory.large-posets open import order-theory.posets diff --git a/src/foundation/dubuc-penon-compact-types.lagda.md b/src/foundation/dubuc-penon-compact-types.lagda.md index 1e9545dc7b..fd422c89a8 100644 --- a/src/foundation/dubuc-penon-compact-types.lagda.md +++ b/src/foundation/dubuc-penon-compact-types.lagda.md @@ -8,10 +8,10 @@ module foundation.dubuc-penon-compact-types where ```agda open import foundation.disjunction +open import foundation.universe-levels open import foundation-core.propositions open import foundation-core.subtypes -open import foundation-core.universe-levels ```
diff --git a/src/foundation/effective-maps-equivalence-relations.lagda.md b/src/foundation/effective-maps-equivalence-relations.lagda.md index 21dae9d8a2..2063e9e857 100644 --- a/src/foundation/effective-maps-equivalence-relations.lagda.md +++ b/src/foundation/effective-maps-equivalence-relations.lagda.md @@ -8,12 +8,12 @@ module foundation.effective-maps-equivalence-relations where ```agda open import foundation.surjective-maps +open import foundation.universe-levels open import foundation-core.cartesian-product-types open import foundation-core.equivalence-relations open import foundation-core.equivalences open import foundation-core.identity-types -open import foundation-core.universe-levels ``` diff --git a/src/foundation/embeddings.lagda.md b/src/foundation/embeddings.lagda.md index 89af8d2961..41629acf37 100644 --- a/src/foundation/embeddings.lagda.md +++ b/src/foundation/embeddings.lagda.md @@ -10,24 +10,24 @@ open import foundation-core.embeddings public ```agda open import foundation.commuting-squares-of-maps +open import foundation.cones-over-cospans +open import foundation.dependent-pair-types open import foundation.equivalences +open import foundation.functions open import foundation.functoriality-cartesian-product-types +open import foundation.fundamental-theorem-of-identity-types open import foundation.identity-types open import foundation.truncated-maps +open import foundation.universe-levels open import foundation-core.cartesian-product-types -open import foundation-core.cones-over-cospans -open import foundation-core.dependent-pair-types -open import foundation-core.functions open import foundation-core.functoriality-dependent-pair-types -open import foundation-core.fundamental-theorem-of-identity-types open import foundation-core.homotopies open import foundation-core.propositional-maps open import foundation-core.propositions open import foundation-core.pullbacks open import foundation-core.sections open import foundation-core.truncation-levels -open import foundation-core.universe-levels ``` diff --git a/src/foundation/empty-types.lagda.md b/src/foundation/empty-types.lagda.md index 7eb6528d83..7f2ba7f8cd 100644 --- a/src/foundation/empty-types.lagda.md +++ b/src/foundation/empty-types.lagda.md @@ -9,19 +9,19 @@ open import foundation-core.empty-types public
Imports ```agda +open import foundation.dependent-pair-types open import foundation.embeddings open import foundation.equivalences +open import foundation.functions open import foundation.propositional-truncations open import foundation.raising-universe-levels open import foundation.subuniverses open import foundation.univalence +open import foundation.universe-levels open import foundation-core.contractible-types -open import foundation-core.dependent-pair-types open import foundation-core.equality-dependent-pair-types -open import foundation-core.functions open import foundation-core.propositions -open import foundation-core.universe-levels ```
diff --git a/src/foundation/endomorphisms.lagda.md b/src/foundation/endomorphisms.lagda.md index 64a209f79e..a455eff47a 100644 --- a/src/foundation/endomorphisms.lagda.md +++ b/src/foundation/endomorphisms.lagda.md @@ -9,13 +9,13 @@ open import foundation-core.endomorphisms public
Imports ```agda +open import foundation.dependent-pair-types +open import foundation.functions open import foundation.unit-type +open import foundation.universe-levels -open import foundation-core.dependent-pair-types -open import foundation-core.functions open import foundation-core.identity-types open import foundation-core.sets -open import foundation-core.universe-levels open import group-theory.monoids open import group-theory.semigroups diff --git a/src/foundation/epimorphisms-with-respect-to-sets.lagda.md b/src/foundation/epimorphisms-with-respect-to-sets.lagda.md index ffaee41cfc..b4e68a027c 100644 --- a/src/foundation/epimorphisms-with-respect-to-sets.lagda.md +++ b/src/foundation/epimorphisms-with-respect-to-sets.lagda.md @@ -7,6 +7,7 @@ module foundation.epimorphisms-with-respect-to-sets where
Imports ```agda +open import foundation.dependent-pair-types open import foundation.epimorphisms-with-respect-to-truncated-types open import foundation.existential-quantification open import foundation.function-extensionality @@ -15,15 +16,14 @@ open import foundation.propositional-truncations open import foundation.sets open import foundation.surjective-maps open import foundation.unit-type +open import foundation.universe-levels -open import foundation-core.dependent-pair-types open import foundation-core.equivalences open import foundation-core.identity-types open import foundation-core.injective-maps open import foundation-core.propositions open import foundation-core.truncation-levels open import foundation-core.univalence -open import foundation-core.universe-levels ```
diff --git a/src/foundation/epimorphisms-with-respect-to-truncated-types.lagda.md b/src/foundation/epimorphisms-with-respect-to-truncated-types.lagda.md index 113971539c..d4960f19a1 100644 --- a/src/foundation/epimorphisms-with-respect-to-truncated-types.lagda.md +++ b/src/foundation/epimorphisms-with-respect-to-truncated-types.lagda.md @@ -9,17 +9,17 @@ module foundation.epimorphisms-with-respect-to-truncated-types where ```agda open import foundation.commuting-squares-of-maps open import foundation.embeddings +open import foundation.functions open import foundation.functoriality-truncation open import foundation.truncation-equivalences open import foundation.truncations +open import foundation.universe-levels open import foundation-core.equivalences -open import foundation-core.functions open import foundation-core.homotopies open import foundation-core.propositions open import foundation-core.truncated-types open import foundation-core.truncation-levels -open import foundation-core.universe-levels ```
diff --git a/src/foundation/epimorphisms.lagda.md b/src/foundation/epimorphisms.lagda.md index db87b50a46..4c841a5fc7 100644 --- a/src/foundation/epimorphisms.lagda.md +++ b/src/foundation/epimorphisms.lagda.md @@ -7,9 +7,10 @@ module foundation.epimorphisms where
Imports ```agda +open import foundation.functions +open import foundation.universe-levels + open import foundation-core.embeddings -open import foundation-core.functions -open import foundation-core.universe-levels ```
diff --git a/src/foundation/equality-cartesian-product-types.lagda.md b/src/foundation/equality-cartesian-product-types.lagda.md index 906fb34d9f..dfe39e941a 100644 --- a/src/foundation/equality-cartesian-product-types.lagda.md +++ b/src/foundation/equality-cartesian-product-types.lagda.md @@ -1,15 +1,148 @@ # Equality of cartesian product types ```agda -module foundation.equality-cartesian-product-types where +{-# OPTIONS --safe #-} -open import foundation-core.equality-cartesian-product-types public +module foundation.equality-cartesian-product-types where ```
Imports ```agda +open import foundation.dependent-pair-types +open import foundation.functions +open import foundation.universe-levels +open import foundation-core.cartesian-product-types +open import foundation-core.equivalences +open import foundation-core.homotopies +open import foundation-core.identity-types ```
+ +## Idea + +Identifications `Id (pair x y) (pair x' y')` in a cartesian product are +equivalently described as pairs of identifications `Id x x'` and `Id y y'`. This +provides us with a characterization of the identity type of cartesian product +types. + +## Definition + +```agda +module _ + {l1 l2 : Level} {A : UU l1} {B : UU l2} + where + + Eq-prod : (s t : A × B) → UU (l1 ⊔ l2) + Eq-prod s t = ((pr1 s) ＝ (pr1 t)) × ((pr2 s) ＝ (pr2 t)) +``` + +## Properties + +### The type `Eq-prod s t` is equivalent to `Id s t` + +```agda +module _ + {l1 l2 : Level} {A : UU l1} {B : UU l2} + where + + eq-pair' : {s t : A × B} → Eq-prod s t → s ＝ t + eq-pair' {pair x y} {pair .x .y} (pair refl refl) = refl + + eq-pair : + {s t : A × B} → (pr1 s) ＝ (pr1 t) → (pr2 s) ＝ (pr2 t) → s ＝ t + eq-pair p q = eq-pair' (pair p q) + + pair-eq : {s t : A × B} → s ＝ t → Eq-prod s t + pr1 (pair-eq α) = ap pr1 α + pr2 (pair-eq α) = ap pr2 α + + isretr-pair-eq : + {s t : A × B} → ((pair-eq {s} {t}) ∘ (eq-pair' {s} {t})) ~ id + isretr-pair-eq {pair x y} {pair .x .y} (pair refl refl) = refl + + issec-pair-eq : + {s t : A × B} → ((eq-pair' {s} {t}) ∘ (pair-eq {s} {t})) ~ id + issec-pair-eq {pair x y} {pair .x .y} refl = refl + + abstract + is-equiv-eq-pair : + (s t : A × B) → is-equiv (eq-pair' {s} {t}) + is-equiv-eq-pair s t = + is-equiv-has-inverse pair-eq issec-pair-eq isretr-pair-eq + + equiv-eq-pair : + (s t : A × B) → Eq-prod s t ≃ (s ＝ t) + pr1 (equiv-eq-pair s t) = eq-pair' + pr2 (equiv-eq-pair s t) = is-equiv-eq-pair s t + + abstract + is-equiv-pair-eq : + (s t : A × B) → is-equiv (pair-eq {s} {t}) + is-equiv-pair-eq s t = + is-equiv-has-inverse eq-pair' isretr-pair-eq issec-pair-eq + + equiv-pair-eq : + (s t : A × B) → (s ＝ t) ≃ Eq-prod s t + pr1 (equiv-pair-eq s t) = pair-eq + pr2 (equiv-pair-eq s t) = is-equiv-pair-eq s t +``` + +## Properties + +### Commuting triangles for `eq-pair` + +```agda +module _ + {l1 l2 : Level} {A : UU l1} {B : UU l2} + where + + triangle-eq-pair : + {a0 a1 : A} {b0 b1 : B} (p : a0 ＝ a1) (q : b0 ＝ b1) → + eq-pair p q ＝ ((eq-pair p refl) ∙ (eq-pair refl q)) + triangle-eq-pair refl refl = refl + + triangle-eq-pair' : + {a0 a1 : A} {b0 b1 : B} (p : a0 ＝ a1) (q : b0 ＝ b1) → + eq-pair p q ＝ ((eq-pair refl q) ∙ (eq-pair p refl)) + triangle-eq-pair' refl refl = refl +``` + +### `eq-pair` preserves concatenation + +```agda +eq-pair-concat : + {l1 l2 : Level} {A : UU l1} {B : UU l2} {x x' x'' : A} {y y' y'' : B} + (p : x ＝ x') (p' : x' ＝ x'') (q : y ＝ y') (q' : y' ＝ y'') → + ( eq-pair {s = pair x y} {t = pair x'' y''} (p ∙ p') (q ∙ q')) ＝ + ( ( eq-pair {s = pair x y} {t = pair x' y'} p q) ∙ + ( eq-pair p' q')) +eq-pair-concat refl p' refl q' = refl +``` + +### `eq-pair` computes in the expected way when the action on paths of the projections is applies + +```agda +ap-pr1-eq-pair : + {l1 l2 : Level} {A : UU l1} {B : UU l2} + {x x' : A} (p : x ＝ x') {y y' : B} (q : y ＝ y') → + ap pr1 (eq-pair {s = pair x y} {pair x' y'} p q) ＝ p +ap-pr1-eq-pair refl refl = refl + +ap-pr2-eq-pair : + {l1 l2 : Level} {A : UU l1} {B : UU l2} + {x x' : A} (p : x ＝ x') {y y' : B} (q : y ＝ y') → + ap pr2 (eq-pair {s = pair x y} {pair x' y'} p q) ＝ q +ap-pr2-eq-pair refl refl = refl +``` + +## See also + +- Equality proofs in dependent pair types are characterized in + [`foundation.equality-dependent-pair-types`](foundation.equality-dependent-pair-types.md). +- Equality proofs in dependent product types are characterized in + [`foundation.equality-dependent-function-types`](foundation.equality-dependent-function-types.md). +- Equality proofs in coproduct types are characterized in + [`foundation.equality-coproduct-types`](foundation.equality-coproduct-types.md). diff --git a/src/foundation/equality-coproduct-types.lagda.md b/src/foundation/equality-coproduct-types.lagda.md index 6519fe9d0f..8f9ba09235 100644 --- a/src/foundation/equality-coproduct-types.lagda.md +++ b/src/foundation/equality-coproduct-types.lagda.md @@ -7,22 +7,23 @@ module foundation.equality-coproduct-types where
Imports ```agda +open import foundation.dependent-pair-types +open import foundation.functions +open import foundation.fundamental-theorem-of-identity-types +open import foundation.universe-levels + open import foundation-core.contractible-types open import foundation-core.coproduct-types -open import foundation-core.dependent-pair-types open import foundation-core.embeddings open import foundation-core.empty-types open import foundation-core.equivalences -open import foundation-core.functions open import foundation-core.functoriality-dependent-pair-types -open import foundation-core.fundamental-theorem-of-identity-types open import foundation-core.homotopies open import foundation-core.identity-types open import foundation-core.negation open import foundation-core.sets open import foundation-core.truncated-types open import foundation-core.truncation-levels -open import foundation-core.universe-levels ```
diff --git a/src/foundation/equality-dependent-function-types.lagda.md b/src/foundation/equality-dependent-function-types.lagda.md index 8f7bb5da17..51f9c151e2 100644 --- a/src/foundation/equality-dependent-function-types.lagda.md +++ b/src/foundation/equality-dependent-function-types.lagda.md @@ -7,14 +7,14 @@ module foundation.equality-dependent-function-types where
Imports ```agda +open import foundation.dependent-pair-types +open import foundation.fundamental-theorem-of-identity-types open import foundation.type-theoretic-principle-of-choice +open import foundation.universe-levels open import foundation-core.contractible-types -open import foundation-core.dependent-pair-types open import foundation-core.equivalences -open import foundation-core.fundamental-theorem-of-identity-types open import foundation-core.identity-types -open import foundation-core.universe-levels ```
diff --git a/src/foundation/equality-dependent-pair-types.lagda.md b/src/foundation/equality-dependent-pair-types.lagda.md index 3020f2eb5e..3c10dde70f 100644 --- a/src/foundation/equality-dependent-pair-types.lagda.md +++ b/src/foundation/equality-dependent-pair-types.lagda.md @@ -9,10 +9,9 @@ open import foundation-core.equality-dependent-pair-types public
Imports ```agda +open import foundation.dependent-pair-types open import foundation.identity-types - -open import foundation-core.dependent-pair-types -open import foundation-core.universe-levels +open import foundation.universe-levels ```
diff --git a/src/foundation/equality-fibers-of-maps.lagda.md b/src/foundation/equality-fibers-of-maps.lagda.md index 3aecd39b52..b2fcec3b60 100644 --- a/src/foundation/equality-fibers-of-maps.lagda.md +++ b/src/foundation/equality-fibers-of-maps.lagda.md @@ -2,14 +2,119 @@ ```agda module foundation.equality-fibers-of-maps where - -open import foundation-core.equality-fibers-of-maps public ```
Imports ```agda +open import foundation.dependent-pair-types +open import foundation.functions +open import foundation.identity-types +open import foundation.universe-levels +open import foundation-core.equality-dependent-pair-types +open import foundation-core.equivalences +open import foundation-core.fibers-of-maps +open import foundation-core.functoriality-dependent-pair-types +open import foundation-core.homotopies ```
+ +## Idea + +In the file +[`foundation-core.fibers-of-maps`](foundation-core.fibers-of-maps.md) we already +gave one characterization of the identity type of `fib f b`, for an arbitrary +map `f : A → B`. Here we give a second characterization, using the fibers of the +action on identifications of `f`. + +## Theorem + +For any map `f : A → B` any `b : B` and any `x y : fib f b`, there is an +equivalence + +```text +(x ＝ y) ≃ fib (ap f) ((pr2 x) ∙ (inv (pr2 y))) +``` + +### Proof + +```agda +module _ + {l1 l2 : Level} {A : UU l1} {B : UU l2} (f : A → B) {b : B} + where + + fib-ap-eq-fib-fiberwise : + (s t : fib f b) (p : (pr1 s) ＝ (pr1 t)) → + ((tr (λ (a : A) → (f a) ＝ b) p (pr2 s)) ＝ (pr2 t)) → + (ap f p ＝ ((pr2 s) ∙ (inv (pr2 t)))) + fib-ap-eq-fib-fiberwise (pair .x' p) (pair x' refl) refl = + inv ∘ (concat right-unit refl) + + abstract + is-fiberwise-equiv-fib-ap-eq-fib-fiberwise : + (s t : fib f b) → is-fiberwise-equiv (fib-ap-eq-fib-fiberwise s t) + is-fiberwise-equiv-fib-ap-eq-fib-fiberwise (pair x y) (pair .x refl) refl = + is-equiv-comp + ( inv) + ( concat right-unit refl) + ( is-equiv-concat right-unit refl) + ( is-equiv-inv (y ∙ refl) refl) + + fib-ap-eq-fib : + (s t : fib f b) → s ＝ t → + fib (ap f {x = pr1 s} {y = pr1 t}) ((pr2 s) ∙ (inv (pr2 t))) + pr1 (fib-ap-eq-fib s .s refl) = refl + pr2 (fib-ap-eq-fib s .s refl) = inv (right-inv (pr2 s)) + + triangle-fib-ap-eq-fib : + (s t : fib f b) → + ( fib-ap-eq-fib s t) ~ + ( (tot (fib-ap-eq-fib-fiberwise s t)) ∘ (pair-eq-Σ {s = s} {t})) + triangle-fib-ap-eq-fib (pair x refl) .(pair x refl) refl = refl + + abstract + is-equiv-fib-ap-eq-fib : (s t : fib f b) → is-equiv (fib-ap-eq-fib s t) + is-equiv-fib-ap-eq-fib s t = + is-equiv-comp-htpy + ( fib-ap-eq-fib s t) + ( tot (fib-ap-eq-fib-fiberwise s t)) + ( pair-eq-Σ {s = s} {t}) + ( triangle-fib-ap-eq-fib s t) + ( is-equiv-pair-eq-Σ s t) + ( is-equiv-tot-is-fiberwise-equiv + ( is-fiberwise-equiv-fib-ap-eq-fib-fiberwise s t)) + + equiv-fib-ap-eq-fib : + (s t : fib f b) → + (s ＝ t) ≃ fib (ap f {x = pr1 s} {y = pr1 t}) ((pr2 s) ∙ (inv (pr2 t))) + pr1 (equiv-fib-ap-eq-fib s t) = fib-ap-eq-fib s t + pr2 (equiv-fib-ap-eq-fib s t) = is-equiv-fib-ap-eq-fib s t + +module _ + {l1 l2 : Level} {A : UU l1} {B : UU l2} (f : A → B) (x y : A) + where + + eq-fib-fib-ap : + (q : f x ＝ f y) → (pair x q) ＝ (pair y refl) → fib (ap f {x} {y}) q + eq-fib-fib-ap q = + (tr (fib (ap f)) right-unit) ∘ (fib-ap-eq-fib f (pair x q) (pair y refl)) + + abstract + is-equiv-eq-fib-fib-ap : + (q : (f x) ＝ (f y)) → is-equiv (eq-fib-fib-ap q) + is-equiv-eq-fib-fib-ap q = + is-equiv-comp + ( tr (fib (ap f)) right-unit) + ( fib-ap-eq-fib f (pair x q) (pair y refl)) + ( is-equiv-fib-ap-eq-fib f (pair x q) (pair y refl)) + ( is-equiv-tr (fib (ap f)) right-unit) +``` + +## See also + +- Equality proofs in dependent pair types are characterized in + [`foundation.equality-dependent-pair-types`](foundation.equality-dependent-pair-types.md). +- Equality proofs in dependent function types are characterized in + [`foundation.equality-dependent-function-types`](foundation.equality-dependent-function-types.md). diff --git a/src/foundation/equivalence-classes.lagda.md b/src/foundation/equivalence-classes.lagda.md index 4f2e68340e..780c13d057 100644 --- a/src/foundation/equivalence-classes.lagda.md +++ b/src/foundation/equivalence-classes.lagda.md @@ -7,33 +7,33 @@ module foundation.equivalence-classes where
Imports ```agda +open import foundation.dependent-pair-types open import foundation.effective-maps-equivalence-relations open import foundation.existential-quantification open import foundation.functoriality-propositional-truncation +open import foundation.fundamental-theorem-of-identity-types open import foundation.inhabited-subtypes open import foundation.locally-small-types open import foundation.propositional-truncations open import foundation.reflecting-maps-equivalence-relations open import foundation.slice open import foundation.small-types +open import foundation.subtype-identity-principle open import foundation.subtypes open import foundation.surjective-maps open import foundation.universal-property-image +open import foundation.universe-levels open import foundation-core.cartesian-product-types open import foundation-core.contractible-types -open import foundation-core.dependent-pair-types open import foundation-core.embeddings open import foundation-core.equivalence-relations open import foundation-core.equivalences open import foundation-core.functoriality-dependent-pair-types -open import foundation-core.fundamental-theorem-of-identity-types open import foundation-core.identity-types open import foundation-core.logical-equivalences open import foundation-core.propositions open import foundation-core.sets -open import foundation-core.subtype-identity-principle -open import foundation-core.universe-levels ```
diff --git a/src/foundation/equivalence-extensionality.lagda.md b/src/foundation/equivalence-extensionality.lagda.md index e86b875528..9be466d333 100644 --- a/src/foundation/equivalence-extensionality.lagda.md +++ b/src/foundation/equivalence-extensionality.lagda.md @@ -7,23 +7,23 @@ module foundation.equivalence-extensionality where
Imports ```agda +open import foundation.dependent-pair-types open import foundation.function-extensionality +open import foundation.functions +open import foundation.fundamental-theorem-of-identity-types +open import foundation.subtype-identity-principle open import foundation.type-theoretic-principle-of-choice +open import foundation.universe-levels open import foundation-core.contractible-maps open import foundation-core.contractible-types -open import foundation-core.dependent-pair-types open import foundation-core.equivalences open import foundation-core.fibers-of-maps -open import foundation-core.functions open import foundation-core.functoriality-dependent-function-types open import foundation-core.functoriality-dependent-pair-types -open import foundation-core.fundamental-theorem-of-identity-types open import foundation-core.homotopies open import foundation-core.identity-types open import foundation-core.propositions -open import foundation-core.subtype-identity-principle -open import foundation-core.universe-levels ```
diff --git a/src/foundation/equivalence-induction.lagda.md b/src/foundation/equivalence-induction.lagda.md index e991341b1c..a0932212d4 100644 --- a/src/foundation/equivalence-induction.lagda.md +++ b/src/foundation/equivalence-induction.lagda.md @@ -9,13 +9,13 @@ open import foundation-core.equivalence-induction public
Imports ```agda +open import foundation.dependent-pair-types +open import foundation.functions open import foundation.univalence +open import foundation.universe-levels -open import foundation-core.dependent-pair-types open import foundation-core.equivalences -open import foundation-core.functions open import foundation-core.sections -open import foundation-core.universe-levels ```
diff --git a/src/foundation/equivalence-relations.lagda.md b/src/foundation/equivalence-relations.lagda.md index 1e13d984f5..cef100b595 100644 --- a/src/foundation/equivalence-relations.lagda.md +++ b/src/foundation/equivalence-relations.lagda.md @@ -10,34 +10,34 @@ open import foundation-core.equivalence-relations public ```agda open import foundation.binary-relations +open import foundation.dependent-pair-types open import foundation.effective-maps-equivalence-relations open import foundation.equivalence-classes open import foundation.full-subtypes +open import foundation.functions +open import foundation.fundamental-theorem-of-identity-types open import foundation.inhabited-subtypes open import foundation.partitions open import foundation.propositional-truncations open import foundation.reflecting-maps-equivalence-relations open import foundation.set-quotients open import foundation.sigma-decompositions +open import foundation.subtype-identity-principle open import foundation.subtypes open import foundation.surjective-maps +open import foundation.type-arithmetic-dependent-pair-types open import foundation.uniqueness-set-quotients open import foundation.universal-property-set-quotients +open import foundation.universe-levels open import foundation-core.cartesian-product-types open import foundation-core.contractible-types -open import foundation-core.dependent-pair-types open import foundation-core.equivalences -open import foundation-core.functions open import foundation-core.functoriality-dependent-pair-types -open import foundation-core.fundamental-theorem-of-identity-types open import foundation-core.identity-types open import foundation-core.logical-equivalences open import foundation-core.propositions open import foundation-core.sets -open import foundation-core.subtype-identity-principle -open import foundation-core.type-arithmetic-dependent-pair-types -open import foundation-core.universe-levels ``` diff --git a/src/foundation/equivalences-maybe.lagda.md b/src/foundation/equivalences-maybe.lagda.md index 84ebe155c6..24360563fa 100644 --- a/src/foundation/equivalences-maybe.lagda.md +++ b/src/foundation/equivalences-maybe.lagda.md @@ -7,26 +7,26 @@ module foundation.equivalences-maybe where
Imports ```agda +open import foundation.dependent-pair-types open import foundation.equality-coproduct-types open import foundation.equivalence-extensionality open import foundation.equivalences +open import foundation.functions open import foundation.functoriality-coproduct-types open import foundation.maybe open import foundation.unit-type open import foundation.universal-property-maybe +open import foundation.universe-levels open import foundation-core.coproduct-types -open import foundation-core.dependent-pair-types open import foundation-core.embeddings open import foundation-core.empty-types open import foundation-core.equality-dependent-pair-types -open import foundation-core.functions open import foundation-core.homotopies open import foundation-core.identity-types open import foundation-core.injective-maps open import foundation-core.propositions open import foundation-core.sets -open import foundation-core.universe-levels ```
diff --git a/src/foundation/equivalences.lagda.md b/src/foundation/equivalences.lagda.md index a3e8c522f6..0e937c0a61 100644 --- a/src/foundation/equivalences.lagda.md +++ b/src/foundation/equivalences.lagda.md @@ -9,23 +9,24 @@ open import foundation-core.equivalences public
Imports ```agda +open import foundation.cones-over-cospans +open import foundation.dependent-pair-types open import foundation.equivalence-extensionality open import foundation.function-extensionality +open import foundation.functions +open import foundation.functoriality-fibers-of-maps +open import foundation.identity-systems open import foundation.identity-types open import foundation.truncated-maps open import foundation.type-theoretic-principle-of-choice +open import foundation.universe-levels -open import foundation-core.cones-over-cospans open import foundation-core.contractible-maps open import foundation-core.contractible-types -open import foundation-core.dependent-pair-types open import foundation-core.embeddings open import foundation-core.fibers-of-maps -open import foundation-core.functions open import foundation-core.functoriality-dependent-function-types open import foundation-core.functoriality-dependent-pair-types -open import foundation-core.functoriality-fibers-of-maps -open import foundation-core.identity-systems open import foundation-core.propositions open import foundation-core.pullbacks open import foundation-core.retractions @@ -34,7 +35,6 @@ open import foundation-core.sets open import foundation-core.subtypes open import foundation-core.truncated-types open import foundation-core.truncation-levels -open import foundation-core.universe-levels ```
diff --git a/src/foundation/exclusive-disjunction.lagda.md b/src/foundation/exclusive-disjunction.lagda.md index fc9b0d4998..1661ee5068 100644 --- a/src/foundation/exclusive-disjunction.lagda.md +++ b/src/foundation/exclusive-disjunction.lagda.md @@ -11,18 +11,20 @@ open import foundation.conjunction open import foundation.contractible-types open import foundation.coproduct-types open import foundation.decidable-types +open import foundation.dependent-pair-types open import foundation.equality-coproduct-types open import foundation.functoriality-coproduct-types open import foundation.negation open import foundation.propositional-extensionality open import foundation.symmetric-operations +open import foundation.type-arithmetic-cartesian-product-types open import foundation.type-arithmetic-coproduct-types open import foundation.universal-property-coproduct-types +open import foundation.universe-levels open import foundation.unordered-pairs open import foundation-core.cartesian-product-types open import foundation-core.decidable-propositions -open import foundation-core.dependent-pair-types open import foundation-core.embeddings open import foundation-core.empty-types open import foundation-core.equality-dependent-pair-types @@ -31,8 +33,6 @@ open import foundation-core.functoriality-dependent-function-types open import foundation-core.functoriality-dependent-pair-types open import foundation-core.identity-types open import foundation-core.propositions -open import foundation-core.type-arithmetic-cartesian-product-types -open import foundation-core.universe-levels open import univalent-combinatorics.2-element-types open import univalent-combinatorics.equality-finite-types diff --git a/src/foundation/existential-quantification.lagda.md b/src/foundation/existential-quantification.lagda.md index 4b0e38437f..cfaefb2c24 100644 --- a/src/foundation/existential-quantification.lagda.md +++ b/src/foundation/existential-quantification.lagda.md @@ -8,15 +8,15 @@ module foundation.existential-quantification where ```agda open import foundation.conjunction +open import foundation.dependent-pair-types open import foundation.propositional-extensionality open import foundation.propositional-truncations +open import foundation.universe-levels -open import foundation-core.dependent-pair-types open import foundation-core.equivalences open import foundation-core.identity-types open import foundation-core.logical-equivalences open import foundation-core.propositions -open import foundation-core.universe-levels ``` diff --git a/src/foundation/exponents-set-quotients.lagda.md b/src/foundation/exponents-set-quotients.lagda.md index 7ccdc0fb06..f6cfa3cbf5 100644 --- a/src/foundation/exponents-set-quotients.lagda.md +++ b/src/foundation/exponents-set-quotients.lagda.md @@ -10,22 +10,22 @@ module foundation.exponents-set-quotients where ```agda open import foundation.binary-relations +open import foundation.dependent-pair-types open import foundation.function-extensionality +open import foundation.functions open import foundation.functoriality-set-quotients open import foundation.reflecting-maps-equivalence-relations open import foundation.set-quotients open import foundation.sets open import foundation.universal-property-set-quotients +open import foundation.universe-levels open import foundation-core.contractible-types -open import foundation-core.dependent-pair-types open import foundation-core.embeddings open import foundation-core.equivalence-relations -open import foundation-core.functions open import foundation-core.homotopies open import foundation-core.identity-types open import foundation-core.propositions -open import foundation-core.universe-levels ``` diff --git a/src/foundation/faithful-maps.lagda.md b/src/foundation/faithful-maps.lagda.md index 54d292e0a5..b61b90d1a3 100644 --- a/src/foundation/faithful-maps.lagda.md +++ b/src/foundation/faithful-maps.lagda.md @@ -2,14 +2,260 @@ ```agda module foundation.faithful-maps where - -open import foundation-core.faithful-maps public ```
Imports ```agda +open import foundation.0-maps +open import foundation.dependent-pair-types +open import foundation.functions +open import foundation.universe-levels +open import foundation-core.embeddings +open import foundation-core.equivalences +open import foundation-core.functoriality-dependent-pair-types +open import foundation-core.homotopies +open import foundation-core.identity-types +open import foundation-core.propositional-maps +open import foundation-core.sets +open import foundation-core.truncated-maps +open import foundation-core.truncation-levels ```
+ +## Idea + +Since we sometimes think of types as ∞-groupoids, with the groupoid structure +provided implicitly by the identity type and its induction principle, we can +think of maps as functors of ∞-groupoids. We borrow some terminology of +functors, and call a map faithful if it induces embeddings on identity types. + +## Definition + +```agda +module _ + {l1 l2 : Level} {A : UU l1} {B : UU l2} + where + + is-faithful : (A → B) → UU (l1 ⊔ l2) + is-faithful f = (x y : A) → is-emb (ap f {x} {y}) + +faithful-map : {l1 l2 : Level} → UU l1 → UU l2 → UU (l1 ⊔ l2) +faithful-map A B = Σ (A → B) is-faithful + +module _ + {l1 l2 : Level} {A : UU l1} {B : UU l2} + where + + map-faithful-map : faithful-map A B → A → B + map-faithful-map = pr1 + + is-faithful-map-faithful-map : + (f : faithful-map A B) → is-faithful (map-faithful-map f) + is-faithful-map-faithful-map = pr2 + + emb-ap-faithful-map : + (f : faithful-map A B) {x y : A} → + (x ＝ y) ↪ (map-faithful-map f x ＝ map-faithful-map f y) + pr1 (emb-ap-faithful-map f {x} {y}) = ap (map-faithful-map f) + pr2 (emb-ap-faithful-map f {x} {y}) = is-faithful-map-faithful-map f x y + + is-faithful-is-emb : {f : A → B} → is-emb f → is-faithful f + is-faithful-is-emb {f} H x y = is-emb-is-equiv (H x y) + + faithful-map-emb : (A ↪ B) → faithful-map A B + pr1 (faithful-map-emb f) = map-emb f + pr2 (faithful-map-emb f) = is-faithful-is-emb (is-emb-map-emb f) + + is-faithful-is-equiv : {f : A → B} → is-equiv f → is-faithful f + is-faithful-is-equiv H = is-faithful-is-emb (is-emb-is-equiv H) + + faithful-map-equiv : (A ≃ B) → faithful-map A B + pr1 (faithful-map-equiv e) = map-equiv e + pr2 (faithful-map-equiv e) = is-faithful-is-equiv (is-equiv-map-equiv e) + + emb-ap : (f : A ↪ B) (x y : A) → (x ＝ y) ↪ (map-emb f x ＝ map-emb f y) + pr1 (emb-ap f x y) = ap (map-emb f) {x} {y} + pr2 (emb-ap f x y) = is-faithful-is-emb (is-emb-map-emb f) x y +``` + +## Examples + +### The identity map is faithful + +```agda +module _ + {l : Level} {A : UU l} + where + + id-faithful-map : faithful-map A A + id-faithful-map = faithful-map-emb id-emb + + is-faithful-id-faithful-map : is-faithful (id {A = A}) + is-faithful-id-faithful-map = is-faithful-map-faithful-map id-faithful-map +``` + +### Any `0`-map is faithful + +```agda +module _ + {l1 l2 : Level} {A : UU l1} {B : UU l2} {f : A → B} + where + + is-0-map-is-faithful : is-faithful f → is-0-map f + is-0-map-is-faithful H = + is-trunc-map-is-trunc-map-ap neg-one-𝕋 f + ( λ x y → is-prop-map-is-emb (H x y)) + + is-faithful-is-0-map : is-0-map f → is-faithful f + is-faithful-is-0-map H x y = + is-emb-is-prop-map (is-trunc-map-ap-is-trunc-map neg-one-𝕋 f H x y) +``` + +## Properties + +### The projection map of a family of sets is faithful + +```agda +module _ + {l1 l2 : Level} {A : UU l1} + where + + abstract + is-faithful-pr1 : + {B : A → UU l2} → ((x : A) → is-set (B x)) → is-faithful (pr1 {B = B}) + is-faithful-pr1 H = is-faithful-is-0-map (is-0-map-pr1 H) + + pr1-faithful-map : + (B : A → Set l2) → faithful-map (Σ A (λ x → type-Set (B x))) A + pr1 (pr1-faithful-map B) = pr1 + pr2 (pr1-faithful-map B) = is-faithful-pr1 (λ x → is-set-type-Set (B x)) +``` + +### Faithful maps are closed under homotopies + +```agda +module _ + {l1 l2 : Level} {A : UU l1} {B : UU l2} {f g : A → B} (H : f ~ g) + where + + abstract + is-faithful-htpy : is-faithful g → is-faithful f + is-faithful-htpy K = + is-faithful-is-0-map (is-0-map-htpy H (is-0-map-is-faithful K)) +``` + +### Faithful maps are closed under composition + +```agda +module _ + {l1 l2 l3 : Level} {A : UU l1} {B : UU l2} {X : UU l3} + where + + abstract + is-faithful-comp : + (g : B → X) (h : A → B) → + is-faithful g → is-faithful h → is-faithful (g ∘ h) + is-faithful-comp g h is-faithful-g is-faithful-h = + is-faithful-is-0-map + ( is-0-map-comp g h + ( is-0-map-is-faithful is-faithful-g) + ( is-0-map-is-faithful is-faithful-h)) + + abstract + is-faithful-comp-htpy : + (f : A → X) (g : B → X) (h : A → B) (H : f ~ (g ∘ h)) → + is-faithful g → is-faithful h → is-faithful f + is-faithful-comp-htpy f g h H is-faithful-g is-faithful-h = + is-faithful-is-0-map + ( is-0-map-comp-htpy f g h H + ( is-0-map-is-faithful is-faithful-g) + ( is-0-map-is-faithful is-faithful-h)) +``` + +### If a composite is faithful, then its right factor is faithful + +```agda +module _ + {l1 l2 l3 : Level} {A : UU l1} {B : UU l2} {X : UU l3} + where + + is-faithful-right-factor : + (g : B → X) (h : A → B) → + is-faithful g → is-faithful (g ∘ h) → is-faithful h + is-faithful-right-factor g h is-faithful-g is-faithful-gh = + is-faithful-is-0-map + ( is-0-map-right-factor g h + ( is-0-map-is-faithful is-faithful-g) + ( is-0-map-is-faithful is-faithful-gh)) + + is-faithful-right-factor-htpy : + (f : A → X) (g : B → X) (h : A → B) (H : f ~ (g ∘ h)) → + is-faithful g → is-faithful f → is-faithful h + is-faithful-right-factor-htpy f g h H is-faithful-g is-faithful-f = + is-faithful-is-0-map + ( is-0-map-right-factor-htpy f g h H + ( is-0-map-is-faithful is-faithful-g) + ( is-0-map-is-faithful is-faithful-f)) +``` + +### The map on total spaces induced by a family of truncated maps is truncated + +```agda +module _ + {l1 l2 l3 : Level} {A : UU l1} {B : A → UU l2} {C : A → UU l3} + {f : (x : A) → B x → C x} + where + + is-faithful-tot : ((x : A) → is-faithful (f x)) → is-faithful (tot f) + is-faithful-tot H = + is-faithful-is-0-map (is-0-map-tot (λ x → is-0-map-is-faithful (H x))) + +module _ + {l1 l2 l3 : Level} {A : UU l1} {B : A → UU l2} {C : A → UU l3} + where + + tot-faithful-map : + ((x : A) → faithful-map (B x) (C x)) → faithful-map (Σ A B) (Σ A C) + pr1 (tot-faithful-map f) = tot (λ x → map-faithful-map (f x)) + pr2 (tot-faithful-map f) = + is-faithful-tot (λ x → is-faithful-map-faithful-map (f x)) + +module _ + {l1 l2 l3 : Level} {A : UU l1} {B : UU l2} + where + + module _ + {f : A → B} (C : B → UU l3) + where + + abstract + is-faithful-map-Σ-map-base : + is-faithful f → is-faithful (map-Σ-map-base f C) + is-faithful-map-Σ-map-base H = + is-faithful-is-0-map + ( is-0-map-map-Σ-map-base C (is-0-map-is-faithful H)) + + faithful-map-Σ-faithful-map-base : + (f : faithful-map A B) (C : B → UU l3) → + faithful-map (Σ A (λ a → C (map-faithful-map f a))) (Σ B C) + pr1 (faithful-map-Σ-faithful-map-base f C) = + map-Σ-map-base (map-faithful-map f) C + pr2 (faithful-map-Σ-faithful-map-base f C) = + is-faithful-map-Σ-map-base C (is-faithful-map-faithful-map f) + +module _ + {l1 l2 l3 l4 : Level} {A : UU l1} {B : UU l2} {C : A → UU l3} + (D : B → UU l4) {f : A → B} {g : (x : A) → C x → D (f x)} + where + + is-faithful-map-Σ : + is-faithful f → ((x : A) → is-faithful (g x)) → is-faithful (map-Σ D f g) + is-faithful-map-Σ H K = + is-faithful-is-0-map + ( is-0-map-map-Σ D + ( is-0-map-is-faithful H) + ( λ x → is-0-map-is-faithful (K x))) +``` diff --git a/src/foundation/fiber-inclusions.lagda.md b/src/foundation/fiber-inclusions.lagda.md index 5b56649253..5e6781bb93 100644 --- a/src/foundation/fiber-inclusions.lagda.md +++ b/src/foundation/fiber-inclusions.lagda.md @@ -7,19 +7,21 @@ module foundation.fiber-inclusions where
Imports ```agda +open import foundation.0-maps +open import foundation.cones-over-cospans +open import foundation.dependent-pair-types +open import foundation.faithful-maps open import foundation.fibers-of-maps open import foundation.identity-types +open import foundation.type-arithmetic-dependent-pair-types open import foundation.unit-type +open import foundation.universe-levels -open import foundation-core.0-maps open import foundation-core.1-types -open import foundation-core.cones-over-cospans open import foundation-core.contractible-maps -open import foundation-core.dependent-pair-types open import foundation-core.embeddings open import foundation-core.equality-dependent-pair-types open import foundation-core.equivalences -open import foundation-core.faithful-maps open import foundation-core.functoriality-dependent-pair-types open import foundation-core.homotopies open import foundation-core.propositional-maps @@ -29,8 +31,6 @@ open import foundation-core.sets open import foundation-core.truncated-maps open import foundation-core.truncated-types open import foundation-core.truncation-levels -open import foundation-core.type-arithmetic-dependent-pair-types -open import foundation-core.universe-levels ```
diff --git a/src/foundation/fibered-equivalences.lagda.md b/src/foundation/fibered-equivalences.lagda.md index d870a689c2..693664444c 100644 --- a/src/foundation/fibered-equivalences.lagda.md +++ b/src/foundation/fibered-equivalences.lagda.md @@ -7,21 +7,21 @@ module foundation.fibered-equivalences where
Imports ```agda +open import foundation.dependent-pair-types open import foundation.embeddings open import foundation.equivalences open import foundation.fibered-maps +open import foundation.functions open import foundation.pullbacks open import foundation.slice +open import foundation.universe-levels open import foundation-core.cartesian-product-types -open import foundation-core.dependent-pair-types open import foundation-core.fibers-of-maps -open import foundation-core.functions open import foundation-core.homotopies open import foundation-core.identity-types open import foundation-core.propositions open import foundation-core.subtypes -open import foundation-core.universe-levels ```
diff --git a/src/foundation/fibered-involutions.lagda.md b/src/foundation/fibered-involutions.lagda.md index 0e31181ed6..fc0e8297b7 100644 --- a/src/foundation/fibered-involutions.lagda.md +++ b/src/foundation/fibered-involutions.lagda.md @@ -7,15 +7,15 @@ module foundation.fibered-involutions where
Imports ```agda +open import foundation.dependent-pair-types open import foundation.fibered-maps +open import foundation.functions +open import foundation.universe-levels open import foundation-core.cartesian-product-types -open import foundation-core.dependent-pair-types open import foundation-core.equivalences -open import foundation-core.functions open import foundation-core.homotopies open import foundation-core.involutions -open import foundation-core.universe-levels ```
diff --git a/src/foundation/fibered-maps.lagda.md b/src/foundation/fibered-maps.lagda.md index 324bf7ee38..e565dc079d 100644 --- a/src/foundation/fibered-maps.lagda.md +++ b/src/foundation/fibered-maps.lagda.md @@ -7,25 +7,25 @@ module foundation.fibered-maps where
Imports ```agda +open import foundation.cones-over-cospans +open import foundation.dependent-pair-types open import foundation.function-extensionality +open import foundation.functions +open import foundation.fundamental-theorem-of-identity-types open import foundation.homotopies open import foundation.slice open import foundation.structure-identity-principle +open import foundation.universe-levels open import foundation-core.commuting-squares-of-maps -open import foundation-core.cones-over-cospans open import foundation-core.contractible-types -open import foundation-core.dependent-pair-types open import foundation-core.equality-dependent-pair-types open import foundation-core.equivalences open import foundation-core.fibers-of-maps -open import foundation-core.functions -open import foundation-core.fundamental-theorem-of-identity-types open import foundation-core.identity-types open import foundation-core.small-types open import foundation-core.truncated-types open import foundation-core.truncation-levels -open import foundation-core.universe-levels ```
diff --git a/src/foundation/fibers-of-maps.lagda.md b/src/foundation/fibers-of-maps.lagda.md index 80c5dbd59d..78d282ebb1 100644 --- a/src/foundation/fibers-of-maps.lagda.md +++ b/src/foundation/fibers-of-maps.lagda.md @@ -9,19 +9,19 @@ open import foundation-core.fibers-of-maps public
Imports ```agda +open import foundation.cones-over-cospans +open import foundation.dependent-pair-types +open import foundation.functions open import foundation.type-arithmetic-unit-type open import foundation.unit-type +open import foundation.universe-levels -open import foundation-core.cones-over-cospans open import foundation-core.constant-maps -open import foundation-core.dependent-pair-types -open import foundation-core.functions open import foundation-core.functoriality-dependent-pair-types open import foundation-core.homotopies open import foundation-core.identity-types open import foundation-core.pullbacks open import foundation-core.universal-property-pullbacks -open import foundation-core.universe-levels ```
diff --git a/src/foundation/full-subtypes.lagda.md b/src/foundation/full-subtypes.lagda.md index 3a6092378b..5039a9ff7a 100644 --- a/src/foundation/full-subtypes.lagda.md +++ b/src/foundation/full-subtypes.lagda.md @@ -8,15 +8,15 @@ module foundation.full-subtypes where ```agda open import foundation.decidable-subtypes +open import foundation.dependent-pair-types +open import foundation.type-arithmetic-dependent-pair-types open import foundation.unit-type +open import foundation.universe-levels -open import foundation-core.dependent-pair-types open import foundation-core.equivalences open import foundation-core.identity-types open import foundation-core.propositions open import foundation-core.subtypes -open import foundation-core.type-arithmetic-dependent-pair-types -open import foundation-core.universe-levels ``` diff --git a/src/foundation/function-extensionality.lagda.md b/src/foundation/function-extensionality.lagda.md index d7e5aeef46..242c0a5d3b 100644 --- a/src/foundation/function-extensionality.lagda.md +++ b/src/foundation/function-extensionality.lagda.md @@ -9,12 +9,13 @@ open import foundation-core.function-extensionality public
Imports ```agda -open import foundation-core.dependent-pair-types +open import foundation.dependent-pair-types +open import foundation.functions +open import foundation.universe-levels + open import foundation-core.equivalences -open import foundation-core.functions open import foundation-core.homotopies open import foundation-core.identity-types -open import foundation-core.universe-levels ```
diff --git a/src/foundation/functional-correspondences.lagda.md b/src/foundation/functional-correspondences.lagda.md index 4c9d7b828f..250df866ed 100644 --- a/src/foundation/functional-correspondences.lagda.md +++ b/src/foundation/functional-correspondences.lagda.md @@ -8,18 +8,18 @@ module foundation.functional-correspondences where ```agda open import foundation.contractible-types +open import foundation.dependent-pair-types open import foundation.equality-dependent-function-types open import foundation.function-extensionality +open import foundation.fundamental-theorem-of-identity-types +open import foundation.subtype-identity-principle open import foundation.univalence +open import foundation.universe-levels -open import foundation-core.dependent-pair-types open import foundation-core.equivalences -open import foundation-core.fundamental-theorem-of-identity-types open import foundation-core.identity-types open import foundation-core.propositions -open import foundation-core.subtype-identity-principle open import foundation-core.subtypes -open import foundation-core.universe-levels ``` diff --git a/src/foundation/functions.lagda.md b/src/foundation/functions.lagda.md index 6f6d0732b7..5a16a153f1 100644 --- a/src/foundation/functions.lagda.md +++ b/src/foundation/functions.lagda.md @@ -4,14 +4,84 @@ {-# OPTIONS --safe #-} module foundation.functions where - -open import foundation-core.functions public ```
Imports ```agda - +open import foundation.universe-levels ```
+ +## Idea + +Functions are primitive in Agda. Here we construct some basic functions + +## Examples + +### The identity function + +```agda +id : {l : Level} {A : UU l} → A → A +id a = a + +idω : {A : UUω} → A → A +idω a = a +``` + +### Dependent composition of functions + +```agda +_∘_ : + {l1 l2 l3 : Level} {A : UU l1} {B : A → UU l2} {C : (a : A) → B a → UU l3} → + ({a : A} → (b : B a) → C a b) → (f : (a : A) → B a) → (a : A) → C a (f a) +(g ∘ f) a = g (f a) +``` + +### Evaluation at a point + +```agda +ev-point : + {l1 l2 : Level} {A : UU l1} (a : A) {P : A → UU l2} → ((x : A) → P x) → P a +ev-point a f = f a + +ev-point' : + {l1 l2 : Level} {A : UU l1} (a : A) {X : UU l2} → (A → X) → X +ev-point' a f = f a +``` + +### Precomposition functions + +```agda +precomp-Π : + {l1 l2 l3 : Level} {A : UU l1} {B : UU l2} (f : A → B) (C : B → UU l3) → + ((b : B) → C b) → ((a : A) → C (f a)) +precomp-Π f C h a = h (f a) + +precomp : + {l1 l2 l3 : Level} {A : UU l1} {B : UU l2} (f : A → B) (C : UU l3) → + (B → C) → (A → C) +precomp f C = precomp-Π f (λ b → C) +``` + +### Postcomposition functions + +```agda +postcomp : + {l1 l2 l3 : Level} {X : UU l1} {Y : UU l2} (A : UU l3) → + (X → Y) → (A → X) → (A → Y) +postcomp A f h = f ∘ h + +map-Π : + {l1 l2 l3 : Level} {I : UU l1} {A : I → UU l2} {B : I → UU l3} + (f : (i : I) → A i → B i) → + ((i : I) → A i) → ((i : I) → B i) +map-Π f h i = f i (h i) + +map-Π' : + {l1 l2 l3 l4 : Level} {I : UU l1} {A : I → UU l2} {B : I → UU l3} + {J : UU l4} (α : J → I) → + ((i : I) → A i → B i) → ((j : J) → A (α j)) → ((j : J) → B (α j)) +map-Π' α f = map-Π (λ j → f (α j)) +``` diff --git a/src/foundation/functoriality-cartesian-product-types.lagda.md b/src/foundation/functoriality-cartesian-product-types.lagda.md index 1b80be7acc..ad4e66e6ff 100644 --- a/src/foundation/functoriality-cartesian-product-types.lagda.md +++ b/src/foundation/functoriality-cartesian-product-types.lagda.md @@ -7,17 +7,18 @@ module foundation.functoriality-cartesian-product-types where
Imports ```agda +open import foundation.dependent-pair-types +open import foundation.equality-cartesian-product-types +open import foundation.functions +open import foundation.universe-levels + open import foundation-core.cartesian-product-types open import foundation-core.contractible-maps open import foundation-core.contractible-types -open import foundation-core.dependent-pair-types -open import foundation-core.equality-cartesian-product-types open import foundation-core.equivalences open import foundation-core.fibers-of-maps -open import foundation-core.functions open import foundation-core.homotopies open import foundation-core.identity-types -open import foundation-core.universe-levels ```
diff --git a/src/foundation/functoriality-coproduct-types.lagda.md b/src/foundation/functoriality-coproduct-types.lagda.md index 59da9d1657..12317065f0 100644 --- a/src/foundation/functoriality-coproduct-types.lagda.md +++ b/src/foundation/functoriality-coproduct-types.lagda.md @@ -8,10 +8,13 @@ module foundation.functoriality-coproduct-types where ```agda open import foundation.coproduct-types +open import foundation.dependent-pair-types +open import foundation.equality-cartesian-product-types open import foundation.equality-coproduct-types open import foundation.equivalence-extensionality open import foundation.equivalences open import foundation.function-extensionality +open import foundation.functions open import foundation.functoriality-cartesian-product-types open import foundation.homotopies open import foundation.propositional-truncations @@ -19,21 +22,18 @@ open import foundation.structure-identity-principle open import foundation.surjective-maps open import foundation.unit-type open import foundation.universal-property-coproduct-types +open import foundation.universe-levels open import foundation-core.cartesian-product-types open import foundation-core.contractible-types -open import foundation-core.dependent-pair-types open import foundation-core.empty-types -open import foundation-core.equality-cartesian-product-types open import foundation-core.fibers-of-maps -open import foundation-core.functions open import foundation-core.functoriality-dependent-function-types open import foundation-core.functoriality-dependent-pair-types open import foundation-core.identity-types open import foundation-core.injective-maps open import foundation-core.negation open import foundation-core.propositions -open import foundation-core.universe-levels ``` diff --git a/src/foundation/functoriality-dependent-function-types.lagda.md b/src/foundation/functoriality-dependent-function-types.lagda.md index d0f5ca29e0..4cd8c6bf74 100644 --- a/src/foundation/functoriality-dependent-function-types.lagda.md +++ b/src/foundation/functoriality-dependent-function-types.lagda.md @@ -9,27 +9,27 @@ open import foundation-core.functoriality-dependent-function-types public
Imports ```agda +open import foundation.dependent-pair-types open import foundation.equivalence-extensionality open import foundation.equivalences open import foundation.function-extensionality +open import foundation.functions open import foundation.identity-types open import foundation.type-theoretic-principle-of-choice open import foundation.unit-type open import foundation.universal-property-unit-type +open import foundation.universe-levels open import foundation-core.commuting-squares-of-maps open import foundation-core.constant-maps -open import foundation-core.dependent-pair-types open import foundation-core.embeddings open import foundation-core.fibers-of-maps -open import foundation-core.functions open import foundation-core.functoriality-dependent-pair-types open import foundation-core.homotopies open import foundation-core.propositional-maps open import foundation-core.truncated-maps open import foundation-core.truncated-types open import foundation-core.truncation-levels -open import foundation-core.universe-levels ```
diff --git a/src/foundation/functoriality-dependent-pair-types.lagda.md b/src/foundation/functoriality-dependent-pair-types.lagda.md index 729c6f2286..abf13bbd73 100644 --- a/src/foundation/functoriality-dependent-pair-types.lagda.md +++ b/src/foundation/functoriality-dependent-pair-types.lagda.md @@ -9,16 +9,17 @@ open import foundation-core.functoriality-dependent-pair-types public
Imports ```agda -open import foundation-core.cones-over-cospans -open import foundation-core.dependent-pair-types +open import foundation.cones-over-cospans +open import foundation.dependent-pair-types +open import foundation.functions +open import foundation.type-arithmetic-dependent-pair-types +open import foundation.universe-levels + open import foundation-core.equality-dependent-pair-types open import foundation-core.equivalences -open import foundation-core.functions open import foundation-core.homotopies open import foundation-core.identity-types open import foundation-core.pullbacks -open import foundation-core.type-arithmetic-dependent-pair-types -open import foundation-core.universe-levels ```
diff --git a/src/foundation/functoriality-fibers-of-maps.lagda.md b/src/foundation/functoriality-fibers-of-maps.lagda.md index b5c88446ca..6928a1dcfc 100644 --- a/src/foundation/functoriality-fibers-of-maps.lagda.md +++ b/src/foundation/functoriality-fibers-of-maps.lagda.md @@ -1,15 +1,121 @@ -# Functoriality of `fib` +# The functoriality of `fib` ```agda module foundation.functoriality-fibers-of-maps where - -open import foundation-core.functoriality-fibers-of-maps public ```
Imports ```agda +open import foundation.cones-over-cospans +open import foundation.dependent-pair-types +open import foundation.functions +open import foundation.universe-levels +open import foundation-core.equality-dependent-pair-types +open import foundation-core.fibers-of-maps +open import foundation-core.functoriality-dependent-pair-types +open import foundation-core.homotopies +open import foundation-core.identity-types ```
+ +## Idea + +Any commuting square + +induces a map between the fibers of the vertical maps + +## Definitions + +### Any cone induces a family of maps between the fibers of the vertical maps + +```agda +module _ + {l1 l2 l3 l4 : Level} {A : UU l1} {B : UU l2} {C : UU l3} {X : UU l4} + (f : A → X) (g : B → X) (c : cone f g C) + where + + map-fib-cone : (x : A) → fib (pr1 c) x → fib g (f x) + pr1 (map-fib-cone x t) = pr1 (pr2 c) (pr1 t) + pr2 (map-fib-cone x t) = (inv (pr2 (pr2 c) (pr1 t))) ∙ (ap f (pr2 t)) + +map-fib-cone-id : + {l1 l2 : Level} {B : UU l1} {X : UU l2} (g : B → X) (x : X) → + map-fib-cone id g (triple g id refl-htpy) x ~ id +map-fib-cone-id g .(g b) (pair b refl) = + refl +``` + +## Properties + +### Computing `map-fib-cone` of a horizontal pasting of cones + +```agda +module _ + {l1 l2 l3 l4 l5 l6 : Level} + {A : UU l1} {B : UU l2} {C : UU l3} {X : UU l4} {Y : UU l5} {Z : UU l6} + (i : X → Y) (j : Y → Z) (h : C → Z) + where + + map-fib-pasting-horizontal-cone : + (c : cone j h B) (d : cone i (pr1 c) A) → (x : X) → + ( map-fib-cone (j ∘ i) h (pasting-horizontal-cone i j h c d) x) ~ + ( (map-fib-cone j h c (i x)) ∘ (map-fib-cone i (pr1 c) d x)) + map-fib-pasting-horizontal-cone + (pair g (pair q K)) (pair f (pair p H)) .(f a) (pair a refl) = + eq-pair-Σ + ( refl) + ( ( ap + ( concat' (h (q (p a))) refl) + ( distributive-inv-concat (ap j (H a)) (K (p a)))) ∙ + ( ( assoc (inv (K (p a))) (inv (ap j (H a))) refl) ∙ + ( ap + ( concat (inv (K (p a))) (j (i (f a)))) + ( ( ap (concat' (j (g (p a))) refl) (inv (ap-inv j (H a)))) ∙ + ( inv (ap-concat j (inv (H a)) refl)))))) +``` + +### Computing `map-fib-cone` of a horizontal pasting of cones + +```agda +module _ + {l1 l2 l3 l4 l5 l6 : Level} + {A : UU l1} {B : UU l2} {C : UU l3} {X : UU l4} {Y : UU l5} {Z : UU l6} + (f : C → Z) (g : Y → Z) (h : X → Y) + where + + map-fib-pasting-vertical-cone : + (c : cone f g B) (d : cone (pr1 (pr2 c)) h A) (x : C) → + ( ( map-fib-cone f (g ∘ h) (pasting-vertical-cone f g h c d) x) ∘ + ( inv-map-compute-fib-comp (pr1 c) (pr1 d) x)) ~ + ( ( inv-map-compute-fib-comp g h (f x)) ∘ + ( map-Σ + ( λ t → fib h (pr1 t)) + ( map-fib-cone f g c x) + ( λ t → map-fib-cone (pr1 (pr2 c)) h d (pr1 t)))) + map-fib-pasting-vertical-cone + (pair p (pair q H)) (pair p' (pair q' H')) .(p (p' a)) + (pair (pair .(p' a) refl) (pair a refl)) = + eq-pair-Σ refl + ( ( right-unit) ∙ + ( ( distributive-inv-concat (H (p' a)) (ap g (H' a))) ∙ + ( ( ap + ( concat (inv (ap g (H' a))) (f (p (p' a)))) + ( inv right-unit)) ∙ + ( ap + ( concat' (g (h (q' a))) + ( pr2 + ( map-fib-cone f g + ( triple p q H) + ( p (p' a)) + ( pair (p' a) refl)))) + ( ( inv (ap-inv g (H' a))) ∙ + ( ap (ap g) (inv right-unit))))))) +``` + +## See also + +- Equality proofs in the fiber of a map are characterized in + [`foundation.equality-fibers-of-maps`](foundation.equality-fibers-of-maps.md). diff --git a/src/foundation/functoriality-function-types.lagda.md b/src/foundation/functoriality-function-types.lagda.md index ceb8a71927..1013de3cf3 100644 --- a/src/foundation/functoriality-function-types.lagda.md +++ b/src/foundation/functoriality-function-types.lagda.md @@ -10,15 +10,15 @@ open import foundation-core.functoriality-function-types public ```agda open import foundation.function-extensionality +open import foundation.functions open import foundation.functoriality-dependent-function-types open import foundation.unit-type +open import foundation.universe-levels open import foundation-core.constant-maps -open import foundation-core.functions open import foundation-core.homotopies open import foundation-core.truncated-maps open import foundation-core.truncation-levels -open import foundation-core.universe-levels ``` diff --git a/src/foundation/functoriality-propositional-truncation.lagda.md b/src/foundation/functoriality-propositional-truncation.lagda.md index e939ce6544..2df9e4a5c2 100644 --- a/src/foundation/functoriality-propositional-truncation.lagda.md +++ b/src/foundation/functoriality-propositional-truncation.lagda.md @@ -7,17 +7,17 @@ module foundation.functoriality-propositional-truncation where
Imports ```agda +open import foundation.dependent-pair-types +open import foundation.functions open import foundation.propositional-truncations +open import foundation.universe-levels open import foundation-core.contractible-types -open import foundation-core.dependent-pair-types open import foundation-core.equivalences open import foundation-core.function-extensionality -open import foundation-core.functions open import foundation-core.homotopies open import foundation-core.identity-types open import foundation-core.propositions -open import foundation-core.universe-levels ```
diff --git a/src/foundation/functoriality-set-quotients.lagda.md b/src/foundation/functoriality-set-quotients.lagda.md index cc5441b1cb..3c3875f0d1 100644 --- a/src/foundation/functoriality-set-quotients.lagda.md +++ b/src/foundation/functoriality-set-quotients.lagda.md @@ -9,28 +9,28 @@ module foundation.functoriality-set-quotients where
Imports ```agda +open import foundation.dependent-pair-types open import foundation.equivalence-extensionality +open import foundation.functions +open import foundation.fundamental-theorem-of-identity-types open import foundation.homotopies open import foundation.reflecting-maps-equivalence-relations open import foundation.set-quotients +open import foundation.subtype-identity-principle open import foundation.surjective-maps open import foundation.uniqueness-set-quotients open import foundation.universal-property-set-quotients +open import foundation.universe-levels open import foundation-core.commuting-squares-of-maps open import foundation-core.contractible-types -open import foundation-core.dependent-pair-types open import foundation-core.equivalence-relations open import foundation-core.equivalences -open import foundation-core.functions -open import foundation-core.fundamental-theorem-of-identity-types open import foundation-core.identity-types open import foundation-core.logical-equivalences open import foundation-core.propositions open import foundation-core.sets -open import foundation-core.subtype-identity-principle open import foundation-core.subtypes -open import foundation-core.universe-levels ```
diff --git a/src/foundation/functoriality-set-truncation.lagda.md b/src/foundation/functoriality-set-truncation.lagda.md index 776209c685..bfec82c1d6 100644 --- a/src/foundation/functoriality-set-truncation.lagda.md +++ b/src/foundation/functoriality-set-truncation.lagda.md @@ -7,6 +7,8 @@ module foundation.functoriality-set-truncation where
Imports ```agda +open import foundation.dependent-pair-types +open import foundation.functions open import foundation.functoriality-truncation open import foundation.images open import foundation.propositional-truncations @@ -17,13 +19,12 @@ open import foundation.uniqueness-image open import foundation.uniqueness-set-truncations open import foundation.universal-property-image open import foundation.universal-property-set-truncation +open import foundation.universe-levels open import foundation-core.contractible-types -open import foundation-core.dependent-pair-types open import foundation-core.embeddings open import foundation-core.equivalences open import foundation-core.fibers-of-maps -open import foundation-core.functions open import foundation-core.functoriality-dependent-pair-types open import foundation-core.homotopies open import foundation-core.identity-types @@ -31,7 +32,6 @@ open import foundation-core.injective-maps open import foundation-core.propositions open import foundation-core.sets open import foundation-core.truncation-levels -open import foundation-core.universe-levels ```
diff --git a/src/foundation/functoriality-truncation.lagda.md b/src/foundation/functoriality-truncation.lagda.md index 0ec10dc66b..9a9cbda3a1 100644 --- a/src/foundation/functoriality-truncation.lagda.md +++ b/src/foundation/functoriality-truncation.lagda.md @@ -7,18 +7,18 @@ module foundation.functoriality-truncation where
Imports ```agda +open import foundation.dependent-pair-types +open import foundation.functions open import foundation.truncations +open import foundation.universe-levels open import foundation-core.commuting-squares-of-maps open import foundation-core.contractible-types -open import foundation-core.dependent-pair-types open import foundation-core.equivalences open import foundation-core.function-extensionality -open import foundation-core.functions open import foundation-core.homotopies open import foundation-core.identity-types open import foundation-core.truncation-levels -open import foundation-core.universe-levels ```
diff --git a/src/foundation/fundamental-theorem-of-identity-types.lagda.md b/src/foundation/fundamental-theorem-of-identity-types.lagda.md index 149c730bf7..500ed759d8 100644 --- a/src/foundation/fundamental-theorem-of-identity-types.lagda.md +++ b/src/foundation/fundamental-theorem-of-identity-types.lagda.md @@ -2,14 +2,131 @@ ```agda module foundation.fundamental-theorem-of-identity-types where - -open import foundation-core.fundamental-theorem-of-identity-types public ```
Imports ```agda +open import foundation.dependent-pair-types +open import foundation.universe-levels +open import foundation-core.contractible-types +open import foundation-core.equivalences +open import foundation-core.functoriality-dependent-pair-types +open import foundation-core.homotopies +open import foundation-core.identity-types +open import foundation-core.retractions +open import foundation-core.sections ```
+ +## Idea + +The fundamental theorem of identity type provides a way to characterize identity +types. It uses the fact that a family of maps `f : (x : A) → a ＝ x → B x` is a +family of equivalences if and only if it induces an equivalence +`Σ A (Id a) → Σ A B` on total spaces. Note that the total space `Σ A (Id a)` is +contractible. Therefore, any map `Σ A (Id a) → Σ A B` is an equivalence if and +only if `Σ A B` is contractible. + +## Theorem + +For any family of maps `f : (x : A) → a ＝ x → B x`, the following are +equivalent: + +1. Each `f x` is an equivalence +2. The total space `Σ A B` is contractible. + +```agda +module _ + {l1 l2 : Level} {A : UU l1} {B : A → UU l2} {a : A} + where + + abstract + fundamental-theorem-id : + is-contr (Σ A B) → (f : (x : A) → a ＝ x → B x) → is-fiberwise-equiv f + fundamental-theorem-id is-contr-AB f = + is-fiberwise-equiv-is-equiv-tot + ( is-equiv-is-contr (tot f) (is-contr-total-path a) is-contr-AB) + + abstract + fundamental-theorem-id' : + (f : (x : A) → a ＝ x → B x) → is-fiberwise-equiv f → is-contr (Σ A B) + fundamental-theorem-id' f is-fiberwise-equiv-f = + is-contr-is-equiv' + ( Σ A (Id a)) + ( tot f) + ( is-equiv-tot-is-fiberwise-equiv is-fiberwise-equiv-f) + ( is-contr-total-path a) +``` + +## Corollaries + +```agda +module _ + {l1 l2 : Level} {A : UU l1} {B : A → UU l2} (a : A) (b : B a) + where + + abstract + fundamental-theorem-id-J : + is-contr (Σ A B) → is-fiberwise-equiv (ind-Id a (λ x p → B x) b) + fundamental-theorem-id-J is-contr-AB = + fundamental-theorem-id is-contr-AB (ind-Id a (λ x p → B x) b) + + abstract + fundamental-theorem-id-J' : + (is-fiberwise-equiv (ind-Id a (λ x p → B x) b)) → is-contr (Σ A B) + fundamental-theorem-id-J' H = + is-contr-is-equiv' + ( Σ A (Id a)) + ( tot (ind-Id a (λ x p → B x) b)) + ( is-equiv-tot-is-fiberwise-equiv H) + ( is-contr-total-path a) +``` + +### Retracts of the identity type are equivalent to the identity type + +```agda +module _ + {l1 l2 : Level} {A : UU l1} {B : A → UU l2} (a : A) + where + + abstract + fundamental-theorem-id-retr : + (i : (x : A) → B x → a ＝ x) → (R : (x : A) → retr (i x)) → + is-fiberwise-equiv i + fundamental-theorem-id-retr i R = + is-fiberwise-equiv-is-equiv-tot + ( is-equiv-is-contr (tot i) + ( is-contr-retract-of (Σ _ (λ y → a ＝ y)) + ( pair (tot i) + ( pair (tot λ x → pr1 (R x)) + ( ( inv-htpy (preserves-comp-tot i (λ x → pr1 (R x)))) ∙h + ( ( tot-htpy λ x → pr2 (R x)) ∙h (tot-id B))))) + ( is-contr-total-path a)) + ( is-contr-total-path a)) +``` + +### The fundamental theorem of identity types, using sections + +```agda +module _ + {l1 l2 : Level} {A : UU l1} {B : A → UU l2} (a : A) + where + + abstract + fundamental-theorem-id-sec : + (f : (x : A) → a ＝ x → B x) → ((x : A) → sec (f x)) → + is-fiberwise-equiv f + fundamental-theorem-id-sec f sec-f x = + is-equiv-sec-is-equiv (f x) (sec-f x) (is-fiberwise-equiv-i x) + where + i : (x : A) → B x → a ＝ x + i = λ x → pr1 (sec-f x) + retr-i : (x : A) → retr (i x) + pr1 (retr-i x) = f x + pr2 (retr-i x) = pr2 (sec-f x) + is-fiberwise-equiv-i : is-fiberwise-equiv i + is-fiberwise-equiv-i = fundamental-theorem-id-retr a i retr-i +``` diff --git a/src/foundation/global-choice.lagda.md b/src/foundation/global-choice.lagda.md index c0cb4ef001..f249ded323 100644 --- a/src/foundation/global-choice.lagda.md +++ b/src/foundation/global-choice.lagda.md @@ -7,13 +7,13 @@ module foundation.global-choice where
Imports ```agda +open import foundation.dependent-pair-types open import foundation.functoriality-propositional-truncation open import foundation.hilberts-epsilon-operators +open import foundation.universe-levels -open import foundation-core.dependent-pair-types open import foundation-core.equivalences open import foundation-core.negation -open import foundation-core.universe-levels open import univalent-combinatorics.2-element-types open import univalent-combinatorics.standard-finite-types diff --git a/src/foundation/hexagons-of-identifications.lagda.md b/src/foundation/hexagons-of-identifications.lagda.md index bf519d444e..3f5290432c 100644 --- a/src/foundation/hexagons-of-identifications.lagda.md +++ b/src/foundation/hexagons-of-identifications.lagda.md @@ -7,8 +7,9 @@ module foundation.hexagons-of-identifications where
Imports ```agda +open import foundation.universe-levels + open import foundation-core.identity-types -open import foundation-core.universe-levels ```
diff --git a/src/foundation/hilberts-epsilon-operators.lagda.md b/src/foundation/hilberts-epsilon-operators.lagda.md index 648037737b..9ed96c4e0d 100644 --- a/src/foundation/hilberts-epsilon-operators.lagda.md +++ b/src/foundation/hilberts-epsilon-operators.lagda.md @@ -7,12 +7,12 @@ module foundation.hilberts-epsilon-operators where
Imports ```agda +open import foundation.functions open import foundation.functoriality-propositional-truncation open import foundation.propositional-truncations +open import foundation.universe-levels open import foundation-core.equivalences -open import foundation-core.functions -open import foundation-core.universe-levels ```
diff --git a/src/foundation/homotopies.lagda.md b/src/foundation/homotopies.lagda.md index 1d5d175ca4..3e2945d33b 100644 --- a/src/foundation/homotopies.lagda.md +++ b/src/foundation/homotopies.lagda.md @@ -9,18 +9,18 @@ open import foundation-core.homotopies public
Imports ```agda +open import foundation.dependent-pair-types open import foundation.function-extensionality +open import foundation.functions +open import foundation.identity-systems open import foundation.identity-types +open import foundation.universe-levels open import foundation-core.contractible-types -open import foundation-core.dependent-pair-types open import foundation-core.equivalences -open import foundation-core.functions open import foundation-core.functoriality-dependent-function-types open import foundation-core.functoriality-dependent-pair-types -open import foundation-core.identity-systems open import foundation-core.sections -open import foundation-core.universe-levels ```
diff --git a/src/foundation/identity-systems.lagda.md b/src/foundation/identity-systems.lagda.md index b71c7a55f5..84f02d7e18 100644 --- a/src/foundation/identity-systems.lagda.md +++ b/src/foundation/identity-systems.lagda.md @@ -2,14 +2,81 @@ ```agda module foundation.identity-systems where - -open import foundation-core.identity-systems public ```
Imports ```agda +open import foundation.dependent-pair-types +open import foundation.fundamental-theorem-of-identity-types +open import foundation.universe-levels +open import foundation-core.contractible-types +open import foundation-core.equivalences +open import foundation-core.identity-types +open import foundation-core.sections ```
+ +## Idea + +A unary identity system on a type `A` equipped with a point `a : A` consists of +a type family `B` over `A` equipped with a point `b : B a` that satisfies an +induction principle analogous to the induction principle of the identity type at +`a`. + +```agda +module _ + {l1 l2 : Level} (l : Level) {A : UU l1} (B : A → UU l2) (a : A) (b : B a) + where + + IND-identity-system : UU (l1 ⊔ l2 ⊔ lsuc l) + IND-identity-system = + ( P : (x : A) (y : B x) → UU l) → + sec (λ (h : (x : A) (y : B x) → P x y) → h a b) +``` + +## Properties + +### A type family over `A` is an identity system if and only if it is equivalent to the identity type + +```agda +module _ + {l1 l2 : Level} {A : UU l1} {B : A → UU l2} (a : A) (b : B a) + where + + abstract + Ind-identity-system : + (is-contr-AB : is-contr (Σ A B)) → + {l : Level} → IND-identity-system l B a b + pr1 (Ind-identity-system is-contr-AB P) p x y = + tr + ( fam-Σ P) + ( eq-is-contr is-contr-AB) + ( p) + pr2 (Ind-identity-system is-contr-AB P) p = + ap + ( λ t → tr (fam-Σ P) t p) + ( eq-is-contr' + ( is-prop-is-contr is-contr-AB (pair a b) (pair a b)) + ( eq-is-contr is-contr-AB) + ( refl)) + + abstract + is-contr-total-space-IND-identity-system : + ({l : Level} → IND-identity-system l B a b) → is-contr (Σ A B) + pr1 (pr1 (is-contr-total-space-IND-identity-system ind)) = a + pr2 (pr1 (is-contr-total-space-IND-identity-system ind)) = b + pr2 (is-contr-total-space-IND-identity-system ind) (pair x y) = + pr1 (ind (λ x' y' → (pair a b) ＝ (pair x' y'))) refl x y + + abstract + fundamental-theorem-id-IND-identity-system : + ({l : Level} → IND-identity-system l B a b) → + (f : (x : A) → a ＝ x → B x) → (x : A) → is-equiv (f x) + fundamental-theorem-id-IND-identity-system ind f = + fundamental-theorem-id + ( is-contr-total-space-IND-identity-system ind) + ( f) +``` diff --git a/src/foundation/identity-truncated-types.lagda.md b/src/foundation/identity-truncated-types.lagda.md index 3ce239bc5c..3da3145597 100644 --- a/src/foundation/identity-truncated-types.lagda.md +++ b/src/foundation/identity-truncated-types.lagda.md @@ -8,12 +8,12 @@ module foundation.identity-truncated-types where ```agda open import foundation.univalence +open import foundation.universe-levels open import foundation-core.equivalences open import foundation-core.identity-types open import foundation-core.truncated-types open import foundation-core.truncation-levels -open import foundation-core.universe-levels ```
diff --git a/src/foundation/identity-types.lagda.md b/src/foundation/identity-types.lagda.md index 994eb01515..ade1454bcb 100644 --- a/src/foundation/identity-types.lagda.md +++ b/src/foundation/identity-types.lagda.md @@ -10,14 +10,14 @@ open import foundation-core.identity-types public ```agda open import foundation.binary-equivalences +open import foundation.dependent-pair-types open import foundation.equivalence-extensionality open import foundation.function-extensionality +open import foundation.functions +open import foundation.universe-levels -open import foundation-core.dependent-pair-types open import foundation-core.equivalences -open import foundation-core.functions open import foundation-core.homotopies -open import foundation-core.universe-levels ``` diff --git a/src/foundation/images-subtypes.lagda.md b/src/foundation/images-subtypes.lagda.md index fece81644c..3e247a71db 100644 --- a/src/foundation/images-subtypes.lagda.md +++ b/src/foundation/images-subtypes.lagda.md @@ -7,11 +7,11 @@ module foundation.images-subtypes where
Imports ```agda +open import foundation.functions open import foundation.images +open import foundation.universe-levels -open import foundation-core.functions open import foundation-core.subtypes -open import foundation-core.universe-levels ```
diff --git a/src/foundation/images.lagda.md b/src/foundation/images.lagda.md index 4954fb6bbb..93346c8b46 100644 --- a/src/foundation/images.lagda.md +++ b/src/foundation/images.lagda.md @@ -7,28 +7,28 @@ module foundation.images where
Imports ```agda +open import foundation.dependent-pair-types +open import foundation.functions +open import foundation.fundamental-theorem-of-identity-types open import foundation.propositional-truncations open import foundation.slice +open import foundation.subtype-identity-principle open import foundation.surjective-maps +open import foundation.universe-levels open import foundation-core.1-types open import foundation-core.contractible-types -open import foundation-core.dependent-pair-types open import foundation-core.embeddings open import foundation-core.equivalences open import foundation-core.fibers-of-maps -open import foundation-core.functions -open import foundation-core.fundamental-theorem-of-identity-types open import foundation-core.homotopies open import foundation-core.identity-types open import foundation-core.injective-maps open import foundation-core.propositions open import foundation-core.sets -open import foundation-core.subtype-identity-principle open import foundation-core.subtypes open import foundation-core.truncated-types open import foundation-core.truncation-levels -open import foundation-core.universe-levels ```
diff --git a/src/foundation/impredicative-encodings.lagda.md b/src/foundation/impredicative-encodings.lagda.md index 9fa4522d50..6bea18ab02 100644 --- a/src/foundation/impredicative-encodings.lagda.md +++ b/src/foundation/impredicative-encodings.lagda.md @@ -8,20 +8,20 @@ module foundation.impredicative-encodings where ```agda open import foundation.conjunction +open import foundation.dependent-pair-types open import foundation.disjunction open import foundation.existential-quantification open import foundation.negation open import foundation.propositional-truncations +open import foundation.universe-levels open import foundation-core.coproduct-types -open import foundation-core.dependent-pair-types open import foundation-core.empty-types open import foundation-core.equivalences open import foundation-core.identity-types open import foundation-core.logical-equivalences open import foundation-core.propositions open import foundation-core.sets -open import foundation-core.universe-levels ``` diff --git a/src/foundation/impredicative-universes.lagda.md b/src/foundation/impredicative-universes.lagda.md index f994a01b1b..8699e535dc 100644 --- a/src/foundation/impredicative-universes.lagda.md +++ b/src/foundation/impredicative-universes.lagda.md @@ -7,9 +7,10 @@ module foundation.impredicative-universes where
Imports ```agda +open import foundation.universe-levels + open import foundation-core.propositions open import foundation-core.small-types -open import foundation-core.universe-levels ```
diff --git a/src/foundation/induction-principle-propositional-truncation.lagda.md b/src/foundation/induction-principle-propositional-truncation.lagda.md index 7fe3e955bc..91269e4fb7 100644 --- a/src/foundation/induction-principle-propositional-truncation.lagda.md +++ b/src/foundation/induction-principle-propositional-truncation.lagda.md @@ -7,10 +7,11 @@ module foundation.induction-principle-propositional-truncation where
Imports ```agda -open import foundation-core.dependent-pair-types +open import foundation.dependent-pair-types +open import foundation.universe-levels + open import foundation-core.identity-types open import foundation-core.propositions -open import foundation-core.universe-levels ```
diff --git a/src/foundation/inhabited-subtypes.lagda.md b/src/foundation/inhabited-subtypes.lagda.md index 0cb17603b0..ceb1aa49d5 100644 --- a/src/foundation/inhabited-subtypes.lagda.md +++ b/src/foundation/inhabited-subtypes.lagda.md @@ -7,18 +7,18 @@ module foundation.inhabited-subtypes where
Imports ```agda +open import foundation.dependent-pair-types +open import foundation.functions open import foundation.inhabited-types open import foundation.propositional-truncations +open import foundation.subtype-identity-principle open import foundation.subtypes +open import foundation.universe-levels open import foundation-core.contractible-types -open import foundation-core.dependent-pair-types open import foundation-core.equivalences -open import foundation-core.functions open import foundation-core.identity-types open import foundation-core.propositions -open import foundation-core.subtype-identity-principle -open import foundation-core.universe-levels ```
diff --git a/src/foundation/inhabited-types.lagda.md b/src/foundation/inhabited-types.lagda.md index 516334cce9..5892516b3c 100644 --- a/src/foundation/inhabited-types.lagda.md +++ b/src/foundation/inhabited-types.lagda.md @@ -7,19 +7,19 @@ module foundation.inhabited-types where
Imports ```agda +open import foundation.dependent-pair-types open import foundation.equality-dependent-function-types open import foundation.functoriality-propositional-truncation +open import foundation.fundamental-theorem-of-identity-types open import foundation.propositional-truncations +open import foundation.subtype-identity-principle open import foundation.univalence +open import foundation.universe-levels open import foundation-core.contractible-types -open import foundation-core.dependent-pair-types open import foundation-core.equivalences -open import foundation-core.fundamental-theorem-of-identity-types open import foundation-core.identity-types open import foundation-core.propositions -open import foundation-core.subtype-identity-principle -open import foundation-core.universe-levels ```
diff --git a/src/foundation/injective-maps.lagda.md b/src/foundation/injective-maps.lagda.md index 92e6bd7fbf..fbf753ab58 100644 --- a/src/foundation/injective-maps.lagda.md +++ b/src/foundation/injective-maps.lagda.md @@ -9,9 +9,10 @@ open import foundation-core.injective-maps public
Imports ```agda +open import foundation.universe-levels + open import foundation-core.empty-types open import foundation-core.negation -open import foundation-core.universe-levels ```
diff --git a/src/foundation/interchange-law.lagda.md b/src/foundation/interchange-law.lagda.md index 2c80b7c0e2..71fa15950c 100644 --- a/src/foundation/interchange-law.lagda.md +++ b/src/foundation/interchange-law.lagda.md @@ -7,8 +7,9 @@ module foundation.interchange-law where
Imports ```agda +open import foundation.universe-levels + open import foundation-core.identity-types -open import foundation-core.universe-levels ```
diff --git a/src/foundation/intersections-subtypes.lagda.md b/src/foundation/intersections-subtypes.lagda.md index 995111348d..57f1f72645 100644 --- a/src/foundation/intersections-subtypes.lagda.md +++ b/src/foundation/intersections-subtypes.lagda.md @@ -10,10 +10,10 @@ module foundation.intersections-subtypes where open import foundation.conjunction open import foundation.decidable-subtypes open import foundation.large-locale-of-subtypes +open import foundation.universe-levels open import foundation-core.propositions open import foundation-core.subtypes -open import foundation-core.universe-levels ``` diff --git a/src/foundation/involutions.lagda.md b/src/foundation/involutions.lagda.md index 59f7ec80be..6985c8820a 100644 --- a/src/foundation/involutions.lagda.md +++ b/src/foundation/involutions.lagda.md @@ -9,15 +9,15 @@ open import foundation-core.involutions public
Imports ```agda +open import foundation.automorphisms open import foundation.equivalence-extensionality open import foundation.equivalences +open import foundation.functions +open import foundation.universe-levels -open import foundation-core.automorphisms -open import foundation-core.functions open import foundation-core.homotopies open import foundation-core.identity-types open import foundation-core.injective-maps -open import foundation-core.universe-levels ```
diff --git a/src/foundation/isolated-points.lagda.md b/src/foundation/isolated-points.lagda.md index 603f1db837..2ce05a796a 100644 --- a/src/foundation/isolated-points.lagda.md +++ b/src/foundation/isolated-points.lagda.md @@ -12,28 +12,28 @@ open import foundation.decidable-embeddings open import foundation.decidable-equality open import foundation.decidable-maps open import foundation.decidable-types +open import foundation.dependent-pair-types open import foundation.embeddings +open import foundation.functions +open import foundation.fundamental-theorem-of-identity-types open import foundation.identity-types open import foundation.maybe open import foundation.negation open import foundation.type-arithmetic-unit-type open import foundation.unit-type +open import foundation.universe-levels open import foundation-core.contractible-types open import foundation-core.coproduct-types open import foundation-core.decidable-propositions -open import foundation-core.dependent-pair-types open import foundation-core.empty-types open import foundation-core.equivalences -open import foundation-core.functions open import foundation-core.functoriality-dependent-pair-types -open import foundation-core.fundamental-theorem-of-identity-types open import foundation-core.homotopies open import foundation-core.injective-maps open import foundation-core.propositions open import foundation-core.sets open import foundation-core.subtypes -open import foundation-core.universe-levels ``` diff --git a/src/foundation/isomorphisms-of-sets.lagda.md b/src/foundation/isomorphisms-of-sets.lagda.md index 6c551d1c60..62624eca2e 100644 --- a/src/foundation/isomorphisms-of-sets.lagda.md +++ b/src/foundation/isomorphisms-of-sets.lagda.md @@ -7,17 +7,17 @@ module foundation.isomorphisms-of-sets where
Imports ```agda +open import foundation.dependent-pair-types open import foundation.equivalences open import foundation.function-extensionality +open import foundation.functions open import foundation.sets +open import foundation.universe-levels open import foundation-core.cartesian-product-types -open import foundation-core.dependent-pair-types -open import foundation-core.functions open import foundation-core.identity-types open import foundation-core.propositions open import foundation-core.subtypes -open import foundation-core.universe-levels ```
diff --git a/src/foundation/iterated-cartesian-product-types.lagda.md b/src/foundation/iterated-cartesian-product-types.lagda.md index 8d277d2a77..fb6cba73d9 100644 --- a/src/foundation/iterated-cartesian-product-types.lagda.md +++ b/src/foundation/iterated-cartesian-product-types.lagda.md @@ -11,23 +11,23 @@ open import elementary-number-theory.natural-numbers open import finite-group-theory.permutations-standard-finite-types +open import foundation.dependent-pair-types +open import foundation.functions open import foundation.functoriality-cartesian-product-types open import foundation.functoriality-dependent-function-types +open import foundation.type-arithmetic-cartesian-product-types open import foundation.type-arithmetic-dependent-function-types open import foundation.unit-type open import foundation.univalence open import foundation.universal-property-coproduct-types open import foundation.universal-property-empty-type +open import foundation.universe-levels open import foundation-core.cartesian-product-types open import foundation-core.contractible-types open import foundation-core.coproduct-types -open import foundation-core.dependent-pair-types open import foundation-core.equivalences -open import foundation-core.functions open import foundation-core.identity-types -open import foundation-core.type-arithmetic-cartesian-product-types -open import foundation-core.universe-levels open import lists.arrays open import lists.concatenation-lists diff --git a/src/foundation/iterating-automorphisms.lagda.md b/src/foundation/iterating-automorphisms.lagda.md index 0dd82dd8d6..ce820fefd3 100644 --- a/src/foundation/iterating-automorphisms.lagda.md +++ b/src/foundation/iterating-automorphisms.lagda.md @@ -11,15 +11,15 @@ open import elementary-number-theory.addition-integers open import elementary-number-theory.integers open import elementary-number-theory.natural-numbers +open import foundation.automorphisms open import foundation.equivalence-extensionality +open import foundation.functions open import foundation.iterating-functions +open import foundation.universe-levels -open import foundation-core.automorphisms open import foundation-core.coproduct-types open import foundation-core.equivalences -open import foundation-core.functions open import foundation-core.homotopies -open import foundation-core.universe-levels ``` diff --git a/src/foundation/iterating-functions.lagda.md b/src/foundation/iterating-functions.lagda.md index e4c27ac83c..e26856a943 100644 --- a/src/foundation/iterating-functions.lagda.md +++ b/src/foundation/iterating-functions.lagda.md @@ -12,15 +12,15 @@ open import elementary-number-theory.exponentiation-natural-numbers open import elementary-number-theory.multiplication-natural-numbers open import elementary-number-theory.natural-numbers +open import foundation.dependent-pair-types open import foundation.function-extensionality +open import foundation.universe-levels open import foundation-core.commuting-squares-of-maps -open import foundation-core.dependent-pair-types open import foundation-core.endomorphisms open import foundation-core.homotopies open import foundation-core.identity-types open import foundation-core.sets -open import foundation-core.universe-levels open import group-theory.monoid-actions ``` diff --git a/src/foundation/iterating-involutions.lagda.md b/src/foundation/iterating-involutions.lagda.md index 58b42fca1b..3599144e30 100644 --- a/src/foundation/iterating-involutions.lagda.md +++ b/src/foundation/iterating-involutions.lagda.md @@ -11,11 +11,11 @@ open import elementary-number-theory.modular-arithmetic-standard-finite-types open import elementary-number-theory.natural-numbers open import foundation.iterating-functions +open import foundation.universe-levels open import foundation-core.coproduct-types open import foundation-core.identity-types open import foundation-core.involutions -open import foundation-core.universe-levels open import univalent-combinatorics.standard-finite-types ``` diff --git a/src/foundation/large-dependent-pair-types.lagda.md b/src/foundation/large-dependent-pair-types.lagda.md index c6e709dc3b..7187ba599c 100644 --- a/src/foundation/large-dependent-pair-types.lagda.md +++ b/src/foundation/large-dependent-pair-types.lagda.md @@ -9,7 +9,7 @@ module foundation.large-dependent-pair-types where
Imports ```agda -open import foundation-core.universe-levels +open import foundation.universe-levels ```
diff --git a/src/foundation/large-homotopies.lagda.md b/src/foundation/large-homotopies.lagda.md index 9e05534556..9ab08d5e30 100644 --- a/src/foundation/large-homotopies.lagda.md +++ b/src/foundation/large-homotopies.lagda.md @@ -10,8 +10,7 @@ module foundation.large-homotopies where ```agda open import foundation.large-identity-types - -open import foundation-core.universe-levels +open import foundation.universe-levels ``` diff --git a/src/foundation/large-identity-types.lagda.md b/src/foundation/large-identity-types.lagda.md index ddfd9c0fe9..5fc4ca3fdf 100644 --- a/src/foundation/large-identity-types.lagda.md +++ b/src/foundation/large-identity-types.lagda.md @@ -9,7 +9,7 @@ module foundation.large-identity-types where
Imports ```agda -open import foundation-core.universe-levels +open import foundation.universe-levels ```
diff --git a/src/foundation/large-locale-of-propositions.lagda.md b/src/foundation/large-locale-of-propositions.lagda.md index 93cbab1c5e..997d796ed0 100644 --- a/src/foundation/large-locale-of-propositions.lagda.md +++ b/src/foundation/large-locale-of-propositions.lagda.md @@ -9,12 +9,12 @@ module foundation.large-locale-of-propositions where ```agda open import foundation.conjunction open import foundation.existential-quantification +open import foundation.functions open import foundation.propositional-extensionality open import foundation.unit-type +open import foundation.universe-levels -open import foundation-core.functions open import foundation-core.propositions -open import foundation-core.universe-levels open import order-theory.large-frames open import order-theory.large-locales diff --git a/src/foundation/large-locale-of-subtypes.lagda.md b/src/foundation/large-locale-of-subtypes.lagda.md index 9942fc5003..6ee1b1d72d 100644 --- a/src/foundation/large-locale-of-subtypes.lagda.md +++ b/src/foundation/large-locale-of-subtypes.lagda.md @@ -8,11 +8,11 @@ module foundation.large-locale-of-subtypes where ```agda open import foundation.large-locale-of-propositions +open import foundation.universe-levels open import foundation-core.identity-types open import foundation-core.propositions open import foundation-core.sets -open import foundation-core.universe-levels open import order-theory.greatest-lower-bounds-large-posets open import order-theory.large-locales diff --git a/src/foundation/law-of-excluded-middle.lagda.md b/src/foundation/law-of-excluded-middle.lagda.md index 352a4a7d1b..a082e117c8 100644 --- a/src/foundation/law-of-excluded-middle.lagda.md +++ b/src/foundation/law-of-excluded-middle.lagda.md @@ -8,12 +8,12 @@ module foundation.law-of-excluded-middle where ```agda open import foundation.decidable-types +open import foundation.dependent-pair-types +open import foundation.universe-levels open import foundation-core.decidable-propositions -open import foundation-core.dependent-pair-types open import foundation-core.negation open import foundation-core.propositions -open import foundation-core.universe-levels open import univalent-combinatorics.2-element-types ``` diff --git a/src/foundation/lawveres-fixed-point-theorem.lagda.md b/src/foundation/lawveres-fixed-point-theorem.lagda.md index 02ac3ea0bb..749b5aeac7 100644 --- a/src/foundation/lawveres-fixed-point-theorem.lagda.md +++ b/src/foundation/lawveres-fixed-point-theorem.lagda.md @@ -7,14 +7,14 @@ module foundation.lawveres-fixed-point-theorem where
Imports ```agda +open import foundation.dependent-pair-types open import foundation.existential-quantification open import foundation.propositional-truncations open import foundation.surjective-maps +open import foundation.universe-levels -open import foundation-core.dependent-pair-types open import foundation-core.function-extensionality open import foundation-core.identity-types -open import foundation-core.universe-levels ```
diff --git a/src/foundation/lesser-limited-principle-of-omniscience.lagda.md b/src/foundation/lesser-limited-principle-of-omniscience.lagda.md index 1ac7dfec34..a97ed1c165 100644 --- a/src/foundation/lesser-limited-principle-of-omniscience.lagda.md +++ b/src/foundation/lesser-limited-principle-of-omniscience.lagda.md @@ -11,11 +11,11 @@ open import elementary-number-theory.natural-numbers open import elementary-number-theory.parity-natural-numbers open import foundation.disjunction +open import foundation.universe-levels open import foundation-core.fibers-of-maps open import foundation-core.propositions open import foundation-core.sets -open import foundation-core.universe-levels open import univalent-combinatorics.standard-finite-types ``` diff --git a/src/foundation/limited-principle-of-omniscience.lagda.md b/src/foundation/limited-principle-of-omniscience.lagda.md index 6d394b644e..fcf98e1c1f 100644 --- a/src/foundation/limited-principle-of-omniscience.lagda.md +++ b/src/foundation/limited-principle-of-omniscience.lagda.md @@ -11,11 +11,11 @@ open import elementary-number-theory.natural-numbers open import foundation.disjunction open import foundation.existential-quantification +open import foundation.universe-levels open import foundation-core.identity-types open import foundation-core.propositions open import foundation-core.sets -open import foundation-core.universe-levels open import univalent-combinatorics.standard-finite-types ``` diff --git a/src/foundation/locally-small-types.lagda.md b/src/foundation/locally-small-types.lagda.md index 305f6e9e0e..8c545e6e7b 100644 --- a/src/foundation/locally-small-types.lagda.md +++ b/src/foundation/locally-small-types.lagda.md @@ -7,12 +7,13 @@ module foundation.locally-small-types where
Imports ```agda +open import foundation.dependent-pair-types open import foundation.function-extensionality open import foundation.inhabited-subtypes open import foundation.subuniverses open import foundation.univalence +open import foundation.universe-levels -open import foundation-core.dependent-pair-types open import foundation-core.equality-dependent-pair-types open import foundation-core.equivalences open import foundation-core.homotopies @@ -22,7 +23,6 @@ open import foundation-core.small-types open import foundation-core.subtypes open import foundation-core.truncated-types open import foundation-core.truncation-levels -open import foundation-core.universe-levels ```
diff --git a/src/foundation/logical-equivalences.lagda.md b/src/foundation/logical-equivalences.lagda.md index 815593741d..45351d2ef7 100644 --- a/src/foundation/logical-equivalences.lagda.md +++ b/src/foundation/logical-equivalences.lagda.md @@ -9,12 +9,13 @@ open import foundation-core.logical-equivalences public
Imports ```agda -open import foundation-core.dependent-pair-types +open import foundation.dependent-pair-types +open import foundation.functions +open import foundation.universe-levels + open import foundation-core.equivalences -open import foundation-core.functions open import foundation-core.identity-types open import foundation-core.propositions -open import foundation-core.universe-levels ```
diff --git a/src/foundation/maybe.lagda.md b/src/foundation/maybe.lagda.md index c10ec906a9..614493a3e0 100644 --- a/src/foundation/maybe.lagda.md +++ b/src/foundation/maybe.lagda.md @@ -9,18 +9,18 @@ module foundation.maybe where ```agda open import foundation.coproduct-types open import foundation.decidable-types +open import foundation.dependent-pair-types open import foundation.equality-coproduct-types open import foundation.type-arithmetic-empty-type open import foundation.unit-type +open import foundation.universe-levels -open import foundation-core.dependent-pair-types open import foundation-core.embeddings open import foundation-core.empty-types open import foundation-core.equivalences open import foundation-core.identity-types open import foundation-core.injective-maps open import foundation-core.negation -open import foundation-core.universe-levels ``` diff --git a/src/foundation/mere-embeddings.lagda.md b/src/foundation/mere-embeddings.lagda.md index 2667effaaa..9340a3f645 100644 --- a/src/foundation/mere-embeddings.lagda.md +++ b/src/foundation/mere-embeddings.lagda.md @@ -12,9 +12,9 @@ open import foundation.embeddings open import foundation.law-of-excluded-middle open import foundation.mere-equivalences open import foundation.propositional-truncations +open import foundation.universe-levels open import foundation-core.propositions -open import foundation-core.universe-levels open import order-theory.large-preorders ``` diff --git a/src/foundation/mere-equality.lagda.md b/src/foundation/mere-equality.lagda.md index 38fe3fb83b..4c42fdedf5 100644 --- a/src/foundation/mere-equality.lagda.md +++ b/src/foundation/mere-equality.lagda.md @@ -7,16 +7,16 @@ module foundation.mere-equality where
Imports ```agda +open import foundation.dependent-pair-types open import foundation.functoriality-propositional-truncation open import foundation.propositional-truncations open import foundation.reflecting-maps-equivalence-relations +open import foundation.universe-levels -open import foundation-core.dependent-pair-types open import foundation-core.equivalence-relations open import foundation-core.identity-types open import foundation-core.propositions open import foundation-core.sets -open import foundation-core.universe-levels ```
diff --git a/src/foundation/mere-equivalences.lagda.md b/src/foundation/mere-equivalences.lagda.md index 2ea9ee3095..14a540665a 100644 --- a/src/foundation/mere-equivalences.lagda.md +++ b/src/foundation/mere-equivalences.lagda.md @@ -12,13 +12,13 @@ open import foundation.functoriality-propositional-truncation open import foundation.mere-equality open import foundation.propositional-truncations open import foundation.univalence +open import foundation.universe-levels open import foundation-core.equivalences open import foundation-core.propositions open import foundation-core.sets open import foundation-core.truncated-types open import foundation-core.truncation-levels -open import foundation-core.universe-levels ``` diff --git a/src/foundation/monomorphisms.lagda.md b/src/foundation/monomorphisms.lagda.md index 3842c0bed8..ab09c9eb7b 100644 --- a/src/foundation/monomorphisms.lagda.md +++ b/src/foundation/monomorphisms.lagda.md @@ -7,17 +7,17 @@ module foundation.monomorphisms where
Imports ```agda +open import foundation.dependent-pair-types open import foundation.embeddings +open import foundation.functions open import foundation.functoriality-function-types +open import foundation.universe-levels -open import foundation-core.dependent-pair-types open import foundation-core.equivalences -open import foundation-core.functions open import foundation-core.identity-types open import foundation-core.propositional-maps open import foundation-core.propositions open import foundation-core.truncation-levels -open import foundation-core.universe-levels ```
diff --git a/src/foundation/morphisms-cospans.lagda.md b/src/foundation/morphisms-cospans.lagda.md index 60886c72f4..8217c9a242 100644 --- a/src/foundation/morphisms-cospans.lagda.md +++ b/src/foundation/morphisms-cospans.lagda.md @@ -1,15 +1,144 @@ # Morphisms of cospans ```agda -module foundation.morphisms-cospans where +{-# OPTIONS --safe #-} -open import foundation-core.morphisms-cospans public +module foundation.morphisms-cospans where ```
Imports ```agda +open import foundation.dependent-pair-types +open import foundation.functions +open import foundation.universe-levels +open import foundation-core.cartesian-product-types +open import foundation-core.homotopies ```
+ +## Idea + +A morphism of cospans is a commuting diagram of the form + +```text + A -----> X <----- B + | | | + | | | + V V V + A' ----> X' <---- B' +``` + +## Definitions + +### Morphisms of cospans + +```agda +hom-cospan : + {l1 l2 l3 l1' l2' l3' : Level} + {A : UU l1} {B : UU l2} {X : UU l3} (f : A → X) (g : B → X) + {A' : UU l1'} {B' : UU l2'} {X' : UU l3'} (f' : A' → X') (g' : B' → X') → + UU (l1 ⊔ l2 ⊔ l3 ⊔ l1' ⊔ l2' ⊔ l3') +hom-cospan {A = A} {B} {X} f g {A'} {B'} {X'} f' g' = + Σ (A → A') (λ hA → + Σ (B → B') (λ hB → + Σ (X → X') (λ hX → + ((f' ∘ hA) ~ (hX ∘ f)) × ((g' ∘ hB) ~ (hX ∘ g))))) +``` + +### Identity morphisms of cospans + +```agda +id-hom-cospan : + {l1 l2 l3 l1' l2' l3' : Level} + {A : UU l1} {B : UU l2} {X : UU l3} (f : A → X) (g : B → X) → + hom-cospan f g f g +pr1 (id-hom-cospan f g) = id +pr1 (pr2 (id-hom-cospan f g)) = id +pr1 (pr2 (pr2 (id-hom-cospan f g))) = id +pr1 (pr2 (pr2 (pr2 (id-hom-cospan f g)))) = refl-htpy +pr2 (pr2 (pr2 (pr2 (id-hom-cospan f g)))) = refl-htpy +``` + +### Rotating cospans of cospans + +```agda +cospan-hom-cospan-rotate : + {l1 l2 l3 l1' l2' l3' l1'' l2'' l3'' : Level} + {A : UU l1} {B : UU l2} {X : UU l3} (f : A → X) (g : B → X) + {A' : UU l1'} {B' : UU l2'} {X' : UU l3'} (f' : A' → X') (g' : B' → X') + {A'' : UU l1''} {B'' : UU l2''} {X'' : UU l3''} + (f'' : A'' → X'') (g'' : B'' → X'') + (h : hom-cospan f' g' f g) (h' : hom-cospan f'' g'' f g) → + hom-cospan (pr1 h) (pr1 h') (pr1 (pr2 (pr2 h))) (pr1 (pr2 (pr2 h'))) +pr1 + ( cospan-hom-cospan-rotate f g f' g' f'' g'' + ( pair hA (pair hB (pair hX (pair HA HB)))) + ( pair hA' (pair hB' (pair hX' (pair HA' HB'))))) = f' +pr1 + ( pr2 + ( cospan-hom-cospan-rotate f g f' g' f'' g'' + ( pair hA (pair hB (pair hX (pair HA HB)))) + ( pair hA' (pair hB' (pair hX' (pair HA' HB')))))) = f'' +pr1 + ( pr2 + ( pr2 + ( cospan-hom-cospan-rotate f g f' g' f'' g'' + ( pair hA (pair hB (pair hX (pair HA HB)))) + ( pair hA' (pair hB' (pair hX' (pair HA' HB'))))))) = f +pr1 + ( pr2 + ( pr2 + ( pr2 + ( cospan-hom-cospan-rotate f g f' g' f'' g'' + ( pair hA (pair hB (pair hX (pair HA HB)))) + ( pair hA' (pair hB' (pair hX' (pair HA' HB')))))))) = inv-htpy HA +pr2 + ( pr2 + ( pr2 + ( pr2 + ( cospan-hom-cospan-rotate f g f' g' f'' g'' + ( pair hA (pair hB (pair hX (pair HA HB)))) + ( pair hA' (pair hB' (pair hX' (pair HA' HB')))))))) = inv-htpy HA' + +cospan-hom-cospan-rotate' : + {l1 l2 l3 l1' l2' l3' l1'' l2'' l3'' : Level} + {A : UU l1} {B : UU l2} {X : UU l3} (f : A → X) (g : B → X) + {A' : UU l1'} {B' : UU l2'} {X' : UU l3'} (f' : A' → X') (g' : B' → X') + {A'' : UU l1''} {B'' : UU l2''} {X'' : UU l3''} + (f'' : A'' → X'') (g'' : B'' → X'') + (h : hom-cospan f' g' f g) (h' : hom-cospan f'' g'' f g) → + hom-cospan + (pr1 (pr2 h)) (pr1 (pr2 h')) (pr1 (pr2 (pr2 h))) (pr1 (pr2 (pr2 h'))) +pr1 + ( cospan-hom-cospan-rotate' f g f' g' f'' g'' + ( pair hA (pair hB (pair hX (pair HA HB)))) + ( pair hA' (pair hB' (pair hX' (pair HA' HB'))))) = g' +pr1 + ( pr2 + ( cospan-hom-cospan-rotate' f g f' g' f'' g'' + ( pair hA (pair hB (pair hX (pair HA HB)))) + ( pair hA' (pair hB' (pair hX' (pair HA' HB')))))) = g'' +pr1 + ( pr2 + ( pr2 + ( cospan-hom-cospan-rotate' f g f' g' f'' g'' + ( pair hA (pair hB (pair hX (pair HA HB)))) + ( pair hA' (pair hB' (pair hX' (pair HA' HB'))))))) = g +pr1 + ( pr2 + ( pr2 + ( pr2 + ( cospan-hom-cospan-rotate' f g f' g' f'' g'' + ( pair hA (pair hB (pair hX (pair HA HB)))) + ( pair hA' (pair hB' (pair hX' (pair HA' HB')))))))) = inv-htpy HB +pr2 + ( pr2 + ( pr2 + ( pr2 + ( cospan-hom-cospan-rotate' f g f' g' f'' g'' + ( pair hA (pair hB (pair hX (pair HA HB)))) + ( pair hA' (pair hB' (pair hX' (pair HA' HB')))))))) = inv-htpy HB' +``` diff --git a/src/foundation/multisubsets.lagda.md b/src/foundation/multisubsets.lagda.md index 5f89ca107c..f48fd9fce6 100644 --- a/src/foundation/multisubsets.lagda.md +++ b/src/foundation/multisubsets.lagda.md @@ -9,14 +9,14 @@ module foundation.multisubsets where ```agda open import elementary-number-theory.natural-numbers +open import foundation.dependent-pair-types open import foundation.images +open import foundation.universe-levels -open import foundation-core.dependent-pair-types open import foundation-core.fibers-of-maps open import foundation-core.identity-types open import foundation-core.negation open import foundation-core.sets -open import foundation-core.universe-levels open import univalent-combinatorics.finite-types ``` diff --git a/src/foundation/multivariable-correspondences.lagda.md b/src/foundation/multivariable-correspondences.lagda.md index 4e6e3d3d07..6e422d49e7 100644 --- a/src/foundation/multivariable-correspondences.lagda.md +++ b/src/foundation/multivariable-correspondences.lagda.md @@ -9,7 +9,7 @@ module foundation.multivariable-correspondences where ```agda open import elementary-number-theory.natural-numbers -open import foundation-core.universe-levels +open import foundation.universe-levels open import univalent-combinatorics.standard-finite-types ``` diff --git a/src/foundation/multivariable-decidable-relations.lagda.md b/src/foundation/multivariable-decidable-relations.lagda.md index 7bb3aa1400..9a5467028e 100644 --- a/src/foundation/multivariable-decidable-relations.lagda.md +++ b/src/foundation/multivariable-decidable-relations.lagda.md @@ -12,8 +12,7 @@ open import elementary-number-theory.natural-numbers open import foundation.decidable-subtypes open import foundation.multivariable-correspondences open import foundation.multivariable-relations - -open import foundation-core.universe-levels +open import foundation.universe-levels open import univalent-combinatorics.standard-finite-types ``` diff --git a/src/foundation/multivariable-functoriality-set-quotients.lagda.md b/src/foundation/multivariable-functoriality-set-quotients.lagda.md index e439e033d7..940d7a71de 100644 --- a/src/foundation/multivariable-functoriality-set-quotients.lagda.md +++ b/src/foundation/multivariable-functoriality-set-quotients.lagda.md @@ -9,14 +9,14 @@ module foundation.multivariable-functoriality-set-quotients where ```agda open import elementary-number-theory.natural-numbers +open import foundation.functions open import foundation.functoriality-set-quotients open import foundation.set-quotients +open import foundation.universe-levels open import foundation.vectors-set-quotients open import foundation-core.equivalence-relations -open import foundation-core.functions open import foundation-core.homotopies -open import foundation-core.universe-levels open import linear-algebra.vectors diff --git a/src/foundation/multivariable-operations.lagda.md b/src/foundation/multivariable-operations.lagda.md index 7adcba0025..1882f5e44f 100644 --- a/src/foundation/multivariable-operations.lagda.md +++ b/src/foundation/multivariable-operations.lagda.md @@ -9,18 +9,18 @@ module foundation.multivariable-operations where ```agda open import elementary-number-theory.natural-numbers +open import foundation.dependent-pair-types +open import foundation.equality-cartesian-product-types +open import foundation.functions open import foundation.raising-universe-levels open import foundation.unit-type +open import foundation.universe-levels open import foundation-core.cartesian-product-types open import foundation-core.coproduct-types -open import foundation-core.dependent-pair-types -open import foundation-core.equality-cartesian-product-types open import foundation-core.equivalences -open import foundation-core.functions open import foundation-core.homotopies open import foundation-core.identity-types -open import foundation-core.universe-levels open import linear-algebra.vectors ``` diff --git a/src/foundation/multivariable-relations.lagda.md b/src/foundation/multivariable-relations.lagda.md index bb64c45e0f..630c453e97 100644 --- a/src/foundation/multivariable-relations.lagda.md +++ b/src/foundation/multivariable-relations.lagda.md @@ -10,9 +10,9 @@ module foundation.multivariable-relations where open import elementary-number-theory.natural-numbers open import foundation.multivariable-correspondences +open import foundation.universe-levels open import foundation-core.subtypes -open import foundation-core.universe-levels open import univalent-combinatorics.standard-finite-types ``` diff --git a/src/foundation/negation.lagda.md b/src/foundation/negation.lagda.md index 3edfa90ca0..f83225d7cb 100644 --- a/src/foundation/negation.lagda.md +++ b/src/foundation/negation.lagda.md @@ -9,12 +9,13 @@ open import foundation-core.negation public
Imports ```agda -open import foundation-core.dependent-pair-types +open import foundation.dependent-pair-types +open import foundation.universe-levels + open import foundation-core.empty-types open import foundation-core.equivalences open import foundation-core.logical-equivalences open import foundation-core.propositions -open import foundation-core.universe-levels ```
diff --git a/src/foundation/noncontractible-types.lagda.md b/src/foundation/noncontractible-types.lagda.md index 9770449727..77c76afa04 100644 --- a/src/foundation/noncontractible-types.lagda.md +++ b/src/foundation/noncontractible-types.lagda.md @@ -9,13 +9,14 @@ module foundation.noncontractible-types where ```agda open import elementary-number-theory.natural-numbers +open import foundation.dependent-pair-types +open import foundation.functions +open import foundation.universe-levels + open import foundation-core.contractible-types -open import foundation-core.dependent-pair-types open import foundation-core.empty-types -open import foundation-core.functions open import foundation-core.identity-types open import foundation-core.negation -open import foundation-core.universe-levels ``` diff --git a/src/foundation/pairs-of-distinct-elements.lagda.md b/src/foundation/pairs-of-distinct-elements.lagda.md index 0b2345e748..8ebaf9e523 100644 --- a/src/foundation/pairs-of-distinct-elements.lagda.md +++ b/src/foundation/pairs-of-distinct-elements.lagda.md @@ -7,19 +7,19 @@ module foundation.pairs-of-distinct-elements where
Imports ```agda +open import foundation.dependent-pair-types open import foundation.embeddings open import foundation.equivalences +open import foundation.fundamental-theorem-of-identity-types open import foundation.negation open import foundation.structure-identity-principle +open import foundation.subtype-identity-principle +open import foundation.universe-levels open import foundation-core.cartesian-product-types open import foundation-core.contractible-types -open import foundation-core.dependent-pair-types -open import foundation-core.fundamental-theorem-of-identity-types open import foundation-core.identity-types open import foundation-core.injective-maps -open import foundation-core.subtype-identity-principle -open import foundation-core.universe-levels ```
diff --git a/src/foundation/partial-elements.lagda.md b/src/foundation/partial-elements.lagda.md index 6d0906651e..b0de55f138 100644 --- a/src/foundation/partial-elements.lagda.md +++ b/src/foundation/partial-elements.lagda.md @@ -7,11 +7,11 @@ module foundation.partial-elements where
Imports ```agda +open import foundation.dependent-pair-types open import foundation.unit-type +open import foundation.universe-levels -open import foundation-core.dependent-pair-types open import foundation-core.propositions -open import foundation-core.universe-levels ```
diff --git a/src/foundation/partitions.lagda.md b/src/foundation/partitions.lagda.md index ee599d3dca..74e5d8f81e 100644 --- a/src/foundation/partitions.lagda.md +++ b/src/foundation/partitions.lagda.md @@ -8,10 +8,13 @@ module foundation.partitions where ```agda open import foundation.contractible-types +open import foundation.dependent-pair-types open import foundation.embeddings open import foundation.equivalences open import foundation.existential-quantification open import foundation.fiber-inclusions +open import foundation.functions +open import foundation.fundamental-theorem-of-identity-types open import foundation.identity-types open import foundation.inhabited-subtypes open import foundation.inhabited-types @@ -20,22 +23,19 @@ open import foundation.logical-equivalences open import foundation.propositional-truncations open import foundation.sigma-decompositions open import foundation.small-types +open import foundation.subtype-identity-principle open import foundation.subtypes open import foundation.surjective-maps +open import foundation.type-arithmetic-dependent-pair-types +open import foundation.universe-levels open import foundation-core.cartesian-product-types -open import foundation-core.dependent-pair-types open import foundation-core.fibers-of-maps -open import foundation-core.functions open import foundation-core.functoriality-dependent-function-types open import foundation-core.functoriality-dependent-pair-types -open import foundation-core.fundamental-theorem-of-identity-types open import foundation-core.homotopies open import foundation-core.propositions open import foundation-core.sets -open import foundation-core.subtype-identity-principle -open import foundation-core.type-arithmetic-dependent-pair-types -open import foundation-core.universe-levels ``` diff --git a/src/foundation/path-algebra.lagda.md b/src/foundation/path-algebra.lagda.md index fc9fb0b8b1..3c6178e280 100644 --- a/src/foundation/path-algebra.lagda.md +++ b/src/foundation/path-algebra.lagda.md @@ -10,12 +10,12 @@ module foundation.path-algebra where open import foundation.binary-embeddings open import foundation.binary-equivalences open import foundation.commuting-squares-of-identifications +open import foundation.functions open import foundation.identity-types +open import foundation.universe-levels open import foundation-core.constant-maps -open import foundation-core.functions open import foundation-core.homotopies -open import foundation-core.universe-levels ``` diff --git a/src/foundation/path-split-maps.lagda.md b/src/foundation/path-split-maps.lagda.md index 938188a810..2c7f9c14fb 100644 --- a/src/foundation/path-split-maps.lagda.md +++ b/src/foundation/path-split-maps.lagda.md @@ -9,12 +9,12 @@ open import foundation-core.path-split-maps public
Imports ```agda +open import foundation.dependent-pair-types open import foundation.equivalences +open import foundation.universe-levels open import foundation-core.contractible-types -open import foundation-core.dependent-pair-types open import foundation-core.propositions -open import foundation-core.universe-levels ```
diff --git a/src/foundation/perfect-images.lagda.md b/src/foundation/perfect-images.lagda.md index 12ec0f5158..606928b8f2 100644 --- a/src/foundation/perfect-images.lagda.md +++ b/src/foundation/perfect-images.lagda.md @@ -10,23 +10,23 @@ module foundation.perfect-images where open import elementary-number-theory.natural-numbers open import foundation.decidable-types +open import foundation.dependent-pair-types open import foundation.double-negation +open import foundation.functions open import foundation.iterating-functions open import foundation.law-of-excluded-middle open import foundation.negation +open import foundation.universe-levels open import foundation-core.cartesian-product-types open import foundation-core.coproduct-types -open import foundation-core.dependent-pair-types open import foundation-core.embeddings open import foundation-core.empty-types open import foundation-core.fibers-of-maps -open import foundation-core.functions open import foundation-core.identity-types open import foundation-core.injective-maps open import foundation-core.propositional-maps open import foundation-core.propositions -open import foundation-core.universe-levels ``` diff --git a/src/foundation/pi-decompositions-subuniverse.lagda.md b/src/foundation/pi-decompositions-subuniverse.lagda.md index 5684eb72d5..3244ca8045 100644 --- a/src/foundation/pi-decompositions-subuniverse.lagda.md +++ b/src/foundation/pi-decompositions-subuniverse.lagda.md @@ -7,15 +7,15 @@ module foundation.pi-decompositions-subuniverse where
Imports ```agda +open import foundation.dependent-pair-types open import foundation.pi-decompositions open import foundation.subuniverses +open import foundation.universe-levels open import foundation-core.cartesian-product-types -open import foundation-core.dependent-pair-types open import foundation-core.equivalences open import foundation-core.homotopies open import foundation-core.propositions -open import foundation-core.universe-levels ```
diff --git a/src/foundation/pi-decompositions.lagda.md b/src/foundation/pi-decompositions.lagda.md index 3495b4bf1a..1caa7c727c 100644 --- a/src/foundation/pi-decompositions.lagda.md +++ b/src/foundation/pi-decompositions.lagda.md @@ -9,24 +9,24 @@ module foundation.pi-decompositions where
Imports ```agda +open import foundation.dependent-pair-types open import foundation.equivalence-extensionality open import foundation.equivalences +open import foundation.functions open import foundation.functoriality-dependent-function-types open import foundation.functoriality-dependent-pair-types +open import foundation.fundamental-theorem-of-identity-types open import foundation.propositional-truncations open import foundation.sets open import foundation.structure-identity-principle +open import foundation.type-arithmetic-dependent-pair-types open import foundation.type-theoretic-principle-of-choice open import foundation.univalence +open import foundation.universe-levels open import foundation-core.contractible-types -open import foundation-core.dependent-pair-types -open import foundation-core.functions -open import foundation-core.fundamental-theorem-of-identity-types open import foundation-core.homotopies open import foundation-core.identity-types -open import foundation-core.type-arithmetic-dependent-pair-types -open import foundation-core.universe-levels ```
diff --git a/src/foundation/powersets.lagda.md b/src/foundation/powersets.lagda.md index 7940218129..ab990d989a 100644 --- a/src/foundation/powersets.lagda.md +++ b/src/foundation/powersets.lagda.md @@ -8,14 +8,14 @@ module foundation.powersets where ```agda open import foundation.function-extensionality +open import foundation.functions open import foundation.propositional-extensionality +open import foundation.universe-levels open import foundation-core.equivalences -open import foundation-core.functions open import foundation-core.identity-types open import foundation-core.propositions open import foundation-core.subtypes -open import foundation-core.universe-levels open import order-theory.large-posets open import order-theory.large-preorders diff --git a/src/foundation/preidempotent-maps.lagda.md b/src/foundation/preidempotent-maps.lagda.md index eebc033159..343ccb584e 100644 --- a/src/foundation/preidempotent-maps.lagda.md +++ b/src/foundation/preidempotent-maps.lagda.md @@ -7,12 +7,13 @@ module foundation.preidempotent-maps where
Imports ```agda -open import foundation-core.dependent-pair-types -open import foundation-core.functions +open import foundation.dependent-pair-types +open import foundation.functions +open import foundation.universe-levels + open import foundation-core.homotopies open import foundation-core.propositions open import foundation-core.sets -open import foundation-core.universe-levels ```
diff --git a/src/foundation/preimages-of-subtypes.lagda.md b/src/foundation/preimages-of-subtypes.lagda.md index c5288fe37e..ac1a192b66 100644 --- a/src/foundation/preimages-of-subtypes.lagda.md +++ b/src/foundation/preimages-of-subtypes.lagda.md @@ -7,8 +7,9 @@ module foundation.preimages-of-subtypes where
Imports ```agda +open import foundation.universe-levels + open import foundation-core.subtypes -open import foundation-core.universe-levels ```
diff --git a/src/foundation/principle-of-omniscience.lagda.md b/src/foundation/principle-of-omniscience.lagda.md index d9d310af26..6053158bbf 100644 --- a/src/foundation/principle-of-omniscience.lagda.md +++ b/src/foundation/principle-of-omniscience.lagda.md @@ -9,10 +9,10 @@ module foundation.principle-of-omniscience where ```agda open import foundation.decidable-subtypes open import foundation.propositional-truncations +open import foundation.universe-levels open import foundation-core.decidable-propositions open import foundation-core.propositions -open import foundation-core.universe-levels ``` diff --git a/src/foundation/product-decompositions-subuniverse.lagda.md b/src/foundation/product-decompositions-subuniverse.lagda.md index bb3076ebc4..044928907a 100644 --- a/src/foundation/product-decompositions-subuniverse.lagda.md +++ b/src/foundation/product-decompositions-subuniverse.lagda.md @@ -8,20 +8,20 @@ module foundation.product-decompositions-subuniverse where ```agda open import foundation.cartesian-product-types +open import foundation.dependent-pair-types open import foundation.equivalences open import foundation.subuniverses +open import foundation.type-arithmetic-cartesian-product-types +open import foundation.type-arithmetic-dependent-pair-types open import foundation.unit-type open import foundation.univalence +open import foundation.universe-levels open import foundation-core.contractible-types -open import foundation-core.dependent-pair-types open import foundation-core.equality-dependent-pair-types open import foundation-core.functoriality-dependent-pair-types open import foundation-core.homotopies open import foundation-core.propositions -open import foundation-core.type-arithmetic-cartesian-product-types -open import foundation-core.type-arithmetic-dependent-pair-types -open import foundation-core.universe-levels ``` diff --git a/src/foundation/product-decompositions.lagda.md b/src/foundation/product-decompositions.lagda.md index 9770ed533f..b04c539404 100644 --- a/src/foundation/product-decompositions.lagda.md +++ b/src/foundation/product-decompositions.lagda.md @@ -8,19 +8,19 @@ module foundation.product-decompositions where ```agda open import foundation.cartesian-product-types +open import foundation.dependent-pair-types open import foundation.equivalences +open import foundation.type-arithmetic-cartesian-product-types +open import foundation.type-arithmetic-dependent-pair-types open import foundation.unit-type open import foundation.univalence +open import foundation.universe-levels open import foundation-core.contractible-types -open import foundation-core.dependent-pair-types open import foundation-core.equality-dependent-pair-types open import foundation-core.functoriality-dependent-pair-types open import foundation-core.homotopies open import foundation-core.propositions -open import foundation-core.type-arithmetic-cartesian-product-types -open import foundation-core.type-arithmetic-dependent-pair-types -open import foundation-core.universe-levels ``` diff --git a/src/foundation/products-binary-relations.lagda.md b/src/foundation/products-binary-relations.lagda.md index b5cb109bdf..1cfc60265b 100644 --- a/src/foundation/products-binary-relations.lagda.md +++ b/src/foundation/products-binary-relations.lagda.md @@ -8,11 +8,11 @@ module foundation.products-binary-relations where ```agda open import foundation.binary-relations +open import foundation.dependent-pair-types +open import foundation.universe-levels open import foundation-core.cartesian-product-types -open import foundation-core.dependent-pair-types open import foundation-core.propositions -open import foundation-core.universe-levels ``` diff --git a/src/foundation/products-equivalence-relations.lagda.md b/src/foundation/products-equivalence-relations.lagda.md index b12fa579b1..1e58c1617d 100644 --- a/src/foundation/products-equivalence-relations.lagda.md +++ b/src/foundation/products-equivalence-relations.lagda.md @@ -8,12 +8,12 @@ module foundation.products-equivalence-relations where ```agda open import foundation.binary-relations +open import foundation.dependent-pair-types open import foundation.products-binary-relations +open import foundation.universe-levels open import foundation-core.cartesian-product-types -open import foundation-core.dependent-pair-types open import foundation-core.equivalence-relations -open import foundation-core.universe-levels ``` diff --git a/src/foundation/products-of-tuples-of-types.lagda.md b/src/foundation/products-of-tuples-of-types.lagda.md index c232c635e5..dd857648e4 100644 --- a/src/foundation/products-of-tuples-of-types.lagda.md +++ b/src/foundation/products-of-tuples-of-types.lagda.md @@ -10,8 +10,7 @@ module foundation.products-of-tuples-of-types where open import elementary-number-theory.natural-numbers open import foundation.tuples-of-types - -open import foundation-core.universe-levels +open import foundation.universe-levels open import univalent-combinatorics.standard-finite-types ``` diff --git a/src/foundation/products-unordered-pairs-of-types.lagda.md b/src/foundation/products-unordered-pairs-of-types.lagda.md index 32ab9d0714..c9013920c4 100644 --- a/src/foundation/products-unordered-pairs-of-types.lagda.md +++ b/src/foundation/products-unordered-pairs-of-types.lagda.md @@ -7,18 +7,18 @@ module foundation.products-unordered-pairs-of-types where
Imports ```agda +open import foundation.dependent-pair-types +open import foundation.functions open import foundation.functoriality-cartesian-product-types open import foundation.functoriality-dependent-function-types open import foundation.identity-types open import foundation.symmetric-operations +open import foundation.universe-levels open import foundation.unordered-pairs open import foundation.unordered-pairs-of-types open import foundation-core.cartesian-product-types -open import foundation-core.dependent-pair-types open import foundation-core.equivalences -open import foundation-core.functions -open import foundation-core.universe-levels open import univalent-combinatorics.2-element-types open import univalent-combinatorics.universal-property-standard-finite-types diff --git a/src/foundation/products-unordered-tuples-of-types.lagda.md b/src/foundation/products-unordered-tuples-of-types.lagda.md index dda129fcbe..9fb480d64f 100644 --- a/src/foundation/products-unordered-tuples-of-types.lagda.md +++ b/src/foundation/products-unordered-tuples-of-types.lagda.md @@ -9,15 +9,15 @@ module foundation.products-unordered-tuples-of-types where ```agda open import elementary-number-theory.natural-numbers +open import foundation.dependent-pair-types open import foundation.functoriality-dependent-function-types open import foundation.universal-property-maybe +open import foundation.universe-levels open import foundation.unordered-tuples open import foundation.unordered-tuples-of-types open import foundation-core.cartesian-product-types -open import foundation-core.dependent-pair-types open import foundation-core.equivalences -open import foundation-core.universe-levels open import univalent-combinatorics.complements-isolated-points ``` diff --git a/src/foundation/projective-types.lagda.md b/src/foundation/projective-types.lagda.md index 326b3d9389..a158fc310c 100644 --- a/src/foundation/projective-types.lagda.md +++ b/src/foundation/projective-types.lagda.md @@ -10,13 +10,13 @@ module foundation.projective-types where open import elementary-number-theory.natural-numbers open import foundation.connected-maps +open import foundation.functions open import foundation.surjective-maps open import foundation.truncation-levels +open import foundation.universe-levels -open import foundation-core.functions open import foundation-core.sets open import foundation-core.truncated-types -open import foundation-core.universe-levels ```
diff --git a/src/foundation/proper-subtypes.lagda.md b/src/foundation/proper-subtypes.lagda.md index bc6d3a4ba2..2ca5864332 100644 --- a/src/foundation/proper-subtypes.lagda.md +++ b/src/foundation/proper-subtypes.lagda.md @@ -9,10 +9,10 @@ module foundation.proper-subtypes where ```agda open import foundation.complements-subtypes open import foundation.inhabited-subtypes +open import foundation.universe-levels open import foundation-core.propositions open import foundation-core.subtypes -open import foundation-core.universe-levels ``` diff --git a/src/foundation/propositional-extensionality.lagda.md b/src/foundation/propositional-extensionality.lagda.md index 214b23a483..624a72a527 100644 --- a/src/foundation/propositional-extensionality.lagda.md +++ b/src/foundation/propositional-extensionality.lagda.md @@ -8,26 +8,26 @@ module foundation.propositional-extensionality where ```agda open import foundation.contractible-types +open import foundation.dependent-pair-types open import foundation.empty-types +open import foundation.fundamental-theorem-of-identity-types open import foundation.identity-types open import foundation.logical-equivalences open import foundation.negation open import foundation.raising-universe-levels +open import foundation.subtype-identity-principle +open import foundation.type-arithmetic-cartesian-product-types open import foundation.unit-type open import foundation.univalence open import foundation.univalent-type-families open import foundation.universal-property-empty-type +open import foundation.universe-levels -open import foundation-core.dependent-pair-types open import foundation-core.equivalences open import foundation-core.functoriality-dependent-pair-types open import foundation-core.functoriality-function-types -open import foundation-core.fundamental-theorem-of-identity-types open import foundation-core.propositions open import foundation-core.sets -open import foundation-core.subtype-identity-principle -open import foundation-core.type-arithmetic-cartesian-product-types -open import foundation-core.universe-levels ``` diff --git a/src/foundation/propositional-maps.lagda.md b/src/foundation/propositional-maps.lagda.md index 14f5d76dcb..49b4dcbc1d 100644 --- a/src/foundation/propositional-maps.lagda.md +++ b/src/foundation/propositional-maps.lagda.md @@ -9,15 +9,15 @@ open import foundation-core.propositional-maps public
Imports ```agda +open import foundation.dependent-pair-types open import foundation.embeddings open import foundation.truncated-maps +open import foundation.universe-levels -open import foundation-core.dependent-pair-types open import foundation-core.equivalences open import foundation-core.logical-equivalences open import foundation-core.propositions open import foundation-core.truncation-levels -open import foundation-core.universe-levels ```
diff --git a/src/foundation/propositional-resizing.lagda.md b/src/foundation/propositional-resizing.lagda.md index 9f521f9c0e..b3be0891b7 100644 --- a/src/foundation/propositional-resizing.lagda.md +++ b/src/foundation/propositional-resizing.lagda.md @@ -7,10 +7,11 @@ module foundation.propositional-resizing where
Imports ```agda -open import foundation-core.dependent-pair-types +open import foundation.dependent-pair-types +open import foundation.universe-levels + open import foundation-core.propositions open import foundation-core.subtypes -open import foundation-core.universe-levels ```
diff --git a/src/foundation/propositional-truncations.lagda.md b/src/foundation/propositional-truncations.lagda.md index 8418549ca8..95f176b5d3 100644 --- a/src/foundation/propositional-truncations.lagda.md +++ b/src/foundation/propositional-truncations.lagda.md @@ -7,23 +7,23 @@ module foundation.propositional-truncations where
Imports ```agda +open import foundation.dependent-pair-types +open import foundation.functions open import foundation.functoriality-cartesian-product-types open import foundation.propositions open import foundation.truncations open import foundation.universal-property-propositional-truncation +open import foundation.universe-levels open import foundation-core.cartesian-product-types open import foundation-core.contractible-types open import foundation-core.coproduct-types -open import foundation-core.dependent-pair-types open import foundation-core.equivalences -open import foundation-core.functions open import foundation-core.homotopies open import foundation-core.identity-types open import foundation-core.sets open import foundation-core.truncated-types open import foundation-core.truncation-levels -open import foundation-core.universe-levels ```
diff --git a/src/foundation/propositions.lagda.md b/src/foundation/propositions.lagda.md index 1ef3651eb0..7299c2f02c 100644 --- a/src/foundation/propositions.lagda.md +++ b/src/foundation/propositions.lagda.md @@ -10,11 +10,11 @@ open import foundation-core.propositions public ```agda open import foundation.contractible-types +open import foundation.universe-levels open import foundation-core.retractions open import foundation-core.truncated-types open import foundation-core.truncation-levels -open import foundation-core.universe-levels ``` diff --git a/src/foundation/pullback-squares.lagda.md b/src/foundation/pullback-squares.lagda.md index 0edc8ef800..59c26f5ae6 100644 --- a/src/foundation/pullback-squares.lagda.md +++ b/src/foundation/pullback-squares.lagda.md @@ -7,11 +7,12 @@ module foundation.pullback-squares where
Imports ```agda +open import foundation.cones-over-cospans +open import foundation.dependent-pair-types +open import foundation.universe-levels + open import foundation-core.commuting-squares-of-maps -open import foundation-core.cones-over-cospans -open import foundation-core.dependent-pair-types open import foundation-core.universal-property-pullbacks -open import foundation-core.universe-levels ```
diff --git a/src/foundation/pullbacks.lagda.md b/src/foundation/pullbacks.lagda.md index 28e916a214..c94132f6cf 100644 --- a/src/foundation/pullbacks.lagda.md +++ b/src/foundation/pullbacks.lagda.md @@ -10,26 +10,26 @@ open import foundation-core.pullbacks public ```agda open import foundation.commuting-cubes-of-maps +open import foundation.cones-over-cospans +open import foundation.dependent-pair-types open import foundation.descent-equivalences open import foundation.equivalences open import foundation.function-extensionality +open import foundation.functions open import foundation.functoriality-dependent-pair-types +open import foundation.fundamental-theorem-of-identity-types open import foundation.homotopies open import foundation.identity-types open import foundation.unit-type +open import foundation.universe-levels open import foundation-core.cartesian-product-types -open import foundation-core.cones-over-cospans open import foundation-core.constant-maps open import foundation-core.contractible-types -open import foundation-core.dependent-pair-types open import foundation-core.diagonal-maps-of-types open import foundation-core.equality-dependent-pair-types -open import foundation-core.functions open import foundation-core.functoriality-dependent-function-types -open import foundation-core.fundamental-theorem-of-identity-types open import foundation-core.propositions -open import foundation-core.universe-levels ``` diff --git a/src/foundation/raising-universe-levels.lagda.md b/src/foundation/raising-universe-levels.lagda.md index 7d673a48d6..8846e95682 100644 --- a/src/foundation/raising-universe-levels.lagda.md +++ b/src/foundation/raising-universe-levels.lagda.md @@ -7,14 +7,15 @@ module foundation.raising-universe-levels where
Imports ```agda -open import foundation-core.dependent-pair-types +open import foundation.dependent-pair-types +open import foundation.functions +open import foundation.universe-levels + open import foundation-core.equivalences -open import foundation-core.functions open import foundation-core.homotopies open import foundation-core.identity-types open import foundation-core.propositions open import foundation-core.sets -open import foundation-core.universe-levels ```
diff --git a/src/foundation/reflecting-maps-equivalence-relations.lagda.md b/src/foundation/reflecting-maps-equivalence-relations.lagda.md index fca2f661e4..25be74f009 100644 --- a/src/foundation/reflecting-maps-equivalence-relations.lagda.md +++ b/src/foundation/reflecting-maps-equivalence-relations.lagda.md @@ -7,19 +7,19 @@ module foundation.reflecting-maps-equivalence-relations where
Imports ```agda +open import foundation.dependent-pair-types open import foundation.effective-maps-equivalence-relations +open import foundation.fundamental-theorem-of-identity-types open import foundation.homotopies +open import foundation.subtype-identity-principle +open import foundation.universe-levels open import foundation-core.contractible-types -open import foundation-core.dependent-pair-types open import foundation-core.equivalence-relations open import foundation-core.equivalences -open import foundation-core.fundamental-theorem-of-identity-types open import foundation-core.identity-types open import foundation-core.propositions open import foundation-core.sets -open import foundation-core.subtype-identity-principle -open import foundation-core.universe-levels ```
diff --git a/src/foundation/reflexive-relations.lagda.md b/src/foundation/reflexive-relations.lagda.md index be7dd6ad8b..dd5ce46398 100644 --- a/src/foundation/reflexive-relations.lagda.md +++ b/src/foundation/reflexive-relations.lagda.md @@ -8,9 +8,8 @@ module foundation.reflexive-relations where ```agda open import foundation.binary-relations - -open import foundation-core.dependent-pair-types -open import foundation-core.universe-levels +open import foundation.dependent-pair-types +open import foundation.universe-levels ``` diff --git a/src/foundation/relaxed-sigma-decompositions.lagda.md b/src/foundation/relaxed-sigma-decompositions.lagda.md index acbb13a825..d845f25bd9 100644 --- a/src/foundation/relaxed-sigma-decompositions.lagda.md +++ b/src/foundation/relaxed-sigma-decompositions.lagda.md @@ -9,21 +9,21 @@ module foundation.relaxed-sigma-decompositions where
Imports ```agda +open import foundation.dependent-pair-types open import foundation.equivalence-extensionality open import foundation.equivalences +open import foundation.functions +open import foundation.fundamental-theorem-of-identity-types open import foundation.structure-identity-principle +open import foundation.type-arithmetic-dependent-pair-types open import foundation.type-theoretic-principle-of-choice open import foundation.univalence +open import foundation.universe-levels open import foundation-core.contractible-types -open import foundation-core.dependent-pair-types -open import foundation-core.functions open import foundation-core.functoriality-dependent-pair-types -open import foundation-core.fundamental-theorem-of-identity-types open import foundation-core.homotopies open import foundation-core.identity-types -open import foundation-core.type-arithmetic-dependent-pair-types -open import foundation-core.universe-levels ```
diff --git a/src/foundation/repetitions-of-values.lagda.md b/src/foundation/repetitions-of-values.lagda.md index 244e305a1d..12daefca7a 100644 --- a/src/foundation/repetitions-of-values.lagda.md +++ b/src/foundation/repetitions-of-values.lagda.md @@ -7,19 +7,19 @@ module foundation.repetitions-of-values where
Imports ```agda +open import foundation.dependent-pair-types open import foundation.embeddings open import foundation.equivalences +open import foundation.functions open import foundation.identity-types open import foundation.pairs-of-distinct-elements +open import foundation.universe-levels open import foundation-core.commuting-squares-of-maps -open import foundation-core.dependent-pair-types -open import foundation-core.functions open import foundation-core.functoriality-dependent-pair-types open import foundation-core.homotopies open import foundation-core.injective-maps open import foundation-core.negation -open import foundation-core.universe-levels ```
diff --git a/src/foundation/repetitions-sequences.lagda.md b/src/foundation/repetitions-sequences.lagda.md index 153f1b0caa..6367d1ec8c 100644 --- a/src/foundation/repetitions-sequences.lagda.md +++ b/src/foundation/repetitions-sequences.lagda.md @@ -10,20 +10,20 @@ module foundation.repetitions-sequences where open import elementary-number-theory.natural-numbers open import elementary-number-theory.strictly-ordered-pairs-of-natural-numbers +open import foundation.dependent-pair-types +open import foundation.functions open import foundation.pairs-of-distinct-elements open import foundation.repetitions-of-values open import foundation.sequences open import foundation.unit-type +open import foundation.universe-levels -open import foundation-core.dependent-pair-types open import foundation-core.embeddings open import foundation-core.empty-types -open import foundation-core.functions open import foundation-core.functoriality-dependent-pair-types open import foundation-core.identity-types open import foundation-core.injective-maps open import foundation-core.negation -open import foundation-core.universe-levels ``` diff --git a/src/foundation/replacement.lagda.md b/src/foundation/replacement.lagda.md index 746a48b7b1..03a5b5719b 100644 --- a/src/foundation/replacement.lagda.md +++ b/src/foundation/replacement.lagda.md @@ -9,9 +9,9 @@ module foundation.replacement where ```agda open import foundation.images open import foundation.locally-small-types +open import foundation.universe-levels open import foundation-core.small-types -open import foundation-core.universe-levels ``` diff --git a/src/foundation/retractions.lagda.md b/src/foundation/retractions.lagda.md index 0400123a4f..bc0c3103fe 100644 --- a/src/foundation/retractions.lagda.md +++ b/src/foundation/retractions.lagda.md @@ -10,14 +10,14 @@ open import foundation-core.retractions public ```agda open import foundation.coslice +open import foundation.dependent-pair-types +open import foundation.functions +open import foundation.universe-levels -open import foundation-core.dependent-pair-types open import foundation-core.equivalences -open import foundation-core.functions open import foundation-core.homotopies open import foundation-core.identity-types open import foundation-core.injective-maps -open import foundation-core.universe-levels ``` diff --git a/src/foundation/russells-paradox.lagda.md b/src/foundation/russells-paradox.lagda.md index 239ee7b015..28d47163ac 100644 --- a/src/foundation/russells-paradox.lagda.md +++ b/src/foundation/russells-paradox.lagda.md @@ -9,6 +9,7 @@ module foundation.russells-paradox where
Imports ```agda +open import foundation.dependent-pair-types open import foundation.equivalences open import foundation.functoriality-cartesian-product-types open import foundation.identity-types @@ -17,14 +18,13 @@ open import foundation.negation open import foundation.small-types open import foundation.small-universes open import foundation.surjective-maps +open import foundation.type-arithmetic-cartesian-product-types +open import foundation.type-arithmetic-dependent-pair-types +open import foundation.universe-levels open import foundation-core.contractible-types -open import foundation-core.dependent-pair-types open import foundation-core.empty-types open import foundation-core.functoriality-dependent-pair-types -open import foundation-core.type-arithmetic-cartesian-product-types -open import foundation-core.type-arithmetic-dependent-pair-types -open import foundation-core.universe-levels open import trees.multisets open import trees.small-multisets diff --git a/src/foundation/sections.lagda.md b/src/foundation/sections.lagda.md index 626eeb7e96..5eb42f3eed 100644 --- a/src/foundation/sections.lagda.md +++ b/src/foundation/sections.lagda.md @@ -9,22 +9,22 @@ open import foundation-core.sections public
Imports ```agda +open import foundation.dependent-pair-types open import foundation.function-extensionality +open import foundation.functions open import foundation.structure-identity-principle +open import foundation.type-arithmetic-dependent-pair-types open import foundation.type-theoretic-principle-of-choice +open import foundation.universe-levels open import foundation-core.contractible-types -open import foundation-core.dependent-pair-types open import foundation-core.equivalences open import foundation-core.fibers-of-maps -open import foundation-core.functions open import foundation-core.functoriality-dependent-pair-types open import foundation-core.homotopies open import foundation-core.identity-types open import foundation-core.injective-maps open import foundation-core.retractions -open import foundation-core.type-arithmetic-dependent-pair-types -open import foundation-core.universe-levels ```
diff --git a/src/foundation/sequences.lagda.md b/src/foundation/sequences.lagda.md index 3ad94cad9e..7cf6e83bff 100644 --- a/src/foundation/sequences.lagda.md +++ b/src/foundation/sequences.lagda.md @@ -9,8 +9,8 @@ module foundation.sequences where ```agda open import elementary-number-theory.natural-numbers -open import foundation-core.functions -open import foundation-core.universe-levels +open import foundation.functions +open import foundation.universe-levels ```
diff --git a/src/foundation/set-presented-types.lagda.md b/src/foundation/set-presented-types.lagda.md index 624954077e..fc03b8d896 100644 --- a/src/foundation/set-presented-types.lagda.md +++ b/src/foundation/set-presented-types.lagda.md @@ -8,13 +8,13 @@ module foundation.set-presented-types where ```agda open import foundation.existential-quantification +open import foundation.functions open import foundation.set-truncations +open import foundation.universe-levels open import foundation-core.equivalences -open import foundation-core.functions open import foundation-core.propositions open import foundation-core.sets -open import foundation-core.universe-levels ``` diff --git a/src/foundation/set-quotients.lagda.md b/src/foundation/set-quotients.lagda.md index f88524eb4a..80ddc75860 100644 --- a/src/foundation/set-quotients.lagda.md +++ b/src/foundation/set-quotients.lagda.md @@ -7,10 +7,12 @@ module foundation.set-quotients where
Imports ```agda +open import foundation.dependent-pair-types open import foundation.effective-maps-equivalence-relations open import foundation.embeddings open import foundation.equivalence-classes open import foundation.equivalences +open import foundation.functions open import foundation.identity-types open import foundation.inhabited-subtypes open import foundation.reflecting-maps-equivalence-relations @@ -20,17 +22,15 @@ open import foundation.surjective-maps open import foundation.uniqueness-set-quotients open import foundation.universal-property-image open import foundation.universal-property-set-quotients +open import foundation.universe-levels -open import foundation-core.dependent-pair-types open import foundation-core.equivalence-relations open import foundation-core.function-extensionality -open import foundation-core.functions open import foundation-core.functoriality-dependent-function-types open import foundation-core.homotopies open import foundation-core.propositions open import foundation-core.small-types open import foundation-core.subtypes -open import foundation-core.universe-levels ```
diff --git a/src/foundation/set-truncations.lagda.md b/src/foundation/set-truncations.lagda.md index d732534a97..f01a9b6bf9 100644 --- a/src/foundation/set-truncations.lagda.md +++ b/src/foundation/set-truncations.lagda.md @@ -9,8 +9,10 @@ module foundation.set-truncations where
Imports ```agda +open import foundation.dependent-pair-types open import foundation.effective-maps-equivalence-relations open import foundation.equality-coproduct-types +open import foundation.functions open import foundation.functoriality-cartesian-product-types open import foundation.functoriality-coproduct-types open import foundation.mere-equality @@ -26,15 +28,14 @@ open import foundation.universal-property-dependent-pair-types open import foundation.universal-property-image open import foundation.universal-property-set-quotients open import foundation.universal-property-set-truncation +open import foundation.universe-levels open import foundation-core.cartesian-product-types open import foundation-core.contractible-types open import foundation-core.coproduct-types -open import foundation-core.dependent-pair-types open import foundation-core.embeddings open import foundation-core.empty-types open import foundation-core.equivalences -open import foundation-core.functions open import foundation-core.functoriality-dependent-function-types open import foundation-core.functoriality-dependent-pair-types open import foundation-core.functoriality-function-types @@ -42,7 +43,6 @@ open import foundation-core.homotopies open import foundation-core.identity-types open import foundation-core.propositions open import foundation-core.truncation-levels -open import foundation-core.universe-levels ```
diff --git a/src/foundation/sets.lagda.md b/src/foundation/sets.lagda.md index 5bcf4cc377..25a85e39eb 100644 --- a/src/foundation/sets.lagda.md +++ b/src/foundation/sets.lagda.md @@ -10,20 +10,20 @@ open import foundation-core.sets public ```agda open import foundation.contractible-types +open import foundation.dependent-pair-types +open import foundation.functions open import foundation.subuniverses open import foundation.truncated-types +open import foundation.universe-levels open import foundation-core.1-types open import foundation-core.cartesian-product-types -open import foundation-core.dependent-pair-types open import foundation-core.embeddings open import foundation-core.equivalences -open import foundation-core.functions open import foundation-core.identity-types open import foundation-core.propositional-maps open import foundation-core.propositions open import foundation-core.truncation-levels -open import foundation-core.universe-levels ``` diff --git a/src/foundation/shifting-sequences.lagda.md b/src/foundation/shifting-sequences.lagda.md index 27337c4ebd..5686c321c4 100644 --- a/src/foundation/shifting-sequences.lagda.md +++ b/src/foundation/shifting-sequences.lagda.md @@ -9,7 +9,7 @@ module foundation.shifting-sequences where ```agda open import elementary-number-theory.natural-numbers -open import foundation-core.universe-levels +open import foundation.universe-levels ``` diff --git a/src/foundation/sigma-decomposition-subuniverse.lagda.md b/src/foundation/sigma-decomposition-subuniverse.lagda.md index d1527b0936..6ec0af7de5 100644 --- a/src/foundation/sigma-decomposition-subuniverse.lagda.md +++ b/src/foundation/sigma-decomposition-subuniverse.lagda.md @@ -7,15 +7,15 @@ module foundation.sigma-decomposition-subuniverse where
Imports ```agda +open import foundation.dependent-pair-types open import foundation.relaxed-sigma-decompositions open import foundation.subuniverses +open import foundation.universe-levels open import foundation-core.cartesian-product-types -open import foundation-core.dependent-pair-types open import foundation-core.equivalences open import foundation-core.homotopies open import foundation-core.propositions -open import foundation-core.universe-levels ```
diff --git a/src/foundation/sigma-decompositions.lagda.md b/src/foundation/sigma-decompositions.lagda.md index 39bac092b7..4bf9180da1 100644 --- a/src/foundation/sigma-decompositions.lagda.md +++ b/src/foundation/sigma-decompositions.lagda.md @@ -9,24 +9,24 @@ module foundation.sigma-decompositions where
Imports ```agda +open import foundation.dependent-pair-types open import foundation.equivalence-extensionality open import foundation.equivalences +open import foundation.functions +open import foundation.fundamental-theorem-of-identity-types open import foundation.inhabited-types open import foundation.propositional-truncations open import foundation.sets open import foundation.structure-identity-principle +open import foundation.type-arithmetic-dependent-pair-types open import foundation.type-theoretic-principle-of-choice open import foundation.univalence +open import foundation.universe-levels open import foundation-core.contractible-types -open import foundation-core.dependent-pair-types -open import foundation-core.functions open import foundation-core.functoriality-dependent-pair-types -open import foundation-core.fundamental-theorem-of-identity-types open import foundation-core.homotopies open import foundation-core.identity-types -open import foundation-core.type-arithmetic-dependent-pair-types -open import foundation-core.universe-levels ```
diff --git a/src/foundation/singleton-induction.lagda.md b/src/foundation/singleton-induction.lagda.md index 9c910da4db..6b811d09d7 100644 --- a/src/foundation/singleton-induction.lagda.md +++ b/src/foundation/singleton-induction.lagda.md @@ -7,13 +7,14 @@ module foundation.singleton-induction where
Imports ```agda +open import foundation.dependent-pair-types +open import foundation.functions +open import foundation.universe-levels + open import foundation-core.contractible-types -open import foundation-core.dependent-pair-types -open import foundation-core.functions open import foundation-core.homotopies open import foundation-core.identity-types open import foundation-core.sections -open import foundation-core.universe-levels ```
diff --git a/src/foundation/singleton-subtypes.lagda.md b/src/foundation/singleton-subtypes.lagda.md index 82b986d685..ef7c9917ac 100644 --- a/src/foundation/singleton-subtypes.lagda.md +++ b/src/foundation/singleton-subtypes.lagda.md @@ -8,15 +8,15 @@ module foundation.singleton-subtypes where ```agda open import foundation.contractible-types +open import foundation.dependent-pair-types open import foundation.inhabited-subtypes open import foundation.propositional-truncations +open import foundation.universe-levels -open import foundation-core.dependent-pair-types open import foundation-core.identity-types open import foundation-core.propositions open import foundation-core.sets open import foundation-core.subtypes -open import foundation-core.universe-levels ``` diff --git a/src/foundation/slice.lagda.md b/src/foundation/slice.lagda.md index 78e3c232f9..4e1b4d9adb 100644 --- a/src/foundation/slice.lagda.md +++ b/src/foundation/slice.lagda.md @@ -7,27 +7,27 @@ module foundation.slice where
Imports ```agda +open import foundation.dependent-pair-types open import foundation.equivalences open import foundation.function-extensionality +open import foundation.functions +open import foundation.fundamental-theorem-of-identity-types open import foundation.homotopies open import foundation.structure-identity-principle +open import foundation.type-arithmetic-dependent-pair-types open import foundation.type-theoretic-principle-of-choice open import foundation.univalence +open import foundation.universe-levels open import foundation-core.contractible-types -open import foundation-core.dependent-pair-types open import foundation-core.embeddings open import foundation-core.equality-dependent-pair-types open import foundation-core.fibers-of-maps -open import foundation-core.functions open import foundation-core.functoriality-dependent-pair-types -open import foundation-core.fundamental-theorem-of-identity-types open import foundation-core.identity-types open import foundation-core.injective-maps open import foundation-core.propositional-maps open import foundation-core.propositions -open import foundation-core.type-arithmetic-dependent-pair-types -open import foundation-core.universe-levels open import trees.polynomial-endofunctors ``` diff --git a/src/foundation/small-maps.lagda.md b/src/foundation/small-maps.lagda.md index d94bcfc456..d4d5f9ca13 100644 --- a/src/foundation/small-maps.lagda.md +++ b/src/foundation/small-maps.lagda.md @@ -7,13 +7,13 @@ module foundation.small-maps where
Imports ```agda +open import foundation.dependent-pair-types open import foundation.locally-small-types +open import foundation.universe-levels -open import foundation-core.dependent-pair-types open import foundation-core.fibers-of-maps open import foundation-core.propositions open import foundation-core.small-types -open import foundation-core.universe-levels ```
diff --git a/src/foundation/small-types.lagda.md b/src/foundation/small-types.lagda.md index 28ae839601..dced6bbeaa 100644 --- a/src/foundation/small-types.lagda.md +++ b/src/foundation/small-types.lagda.md @@ -9,17 +9,17 @@ open import foundation-core.small-types public
Imports ```agda +open import foundation.dependent-pair-types open import foundation.images open import foundation.locally-small-types open import foundation.replacement open import foundation.surjective-maps open import foundation.uniqueness-image open import foundation.universal-property-image +open import foundation.universe-levels -open import foundation-core.dependent-pair-types open import foundation-core.embeddings open import foundation-core.homotopies -open import foundation-core.universe-levels ```
diff --git a/src/foundation/small-universes.lagda.md b/src/foundation/small-universes.lagda.md index 896648cda4..49a5bb6048 100644 --- a/src/foundation/small-universes.lagda.md +++ b/src/foundation/small-universes.lagda.md @@ -7,9 +7,10 @@ module foundation.small-universes where
Imports ```agda +open import foundation.universe-levels + open import foundation-core.cartesian-product-types open import foundation-core.small-types -open import foundation-core.universe-levels ```
diff --git a/src/foundation/split-surjective-maps.lagda.md b/src/foundation/split-surjective-maps.lagda.md index 746fb81ca8..abf38b8ce1 100644 --- a/src/foundation/split-surjective-maps.lagda.md +++ b/src/foundation/split-surjective-maps.lagda.md @@ -7,17 +7,17 @@ module foundation.split-surjective-maps where
Imports ```agda +open import foundation.dependent-pair-types +open import foundation.functions open import foundation.type-theoretic-principle-of-choice +open import foundation.universe-levels open import foundation-core.cartesian-product-types -open import foundation-core.dependent-pair-types open import foundation-core.equivalences open import foundation-core.fibers-of-maps -open import foundation-core.functions open import foundation-core.injective-maps open import foundation-core.retractions open import foundation-core.sections -open import foundation-core.universe-levels ```
diff --git a/src/foundation/standard-apartness-relations.lagda.md b/src/foundation/standard-apartness-relations.lagda.md index b219ae2e40..eeb57dd90c 100644 --- a/src/foundation/standard-apartness-relations.lagda.md +++ b/src/foundation/standard-apartness-relations.lagda.md @@ -9,14 +9,14 @@ module foundation.standard-apartness-relations where ```agda open import foundation.apartness-relations open import foundation.decidable-types +open import foundation.dependent-pair-types open import foundation.law-of-excluded-middle open import foundation.tight-apartness-relations +open import foundation.universe-levels -open import foundation-core.dependent-pair-types open import foundation-core.identity-types open import foundation-core.logical-equivalences open import foundation-core.negation -open import foundation-core.universe-levels ```
diff --git a/src/foundation/strongly-extensional-maps.lagda.md b/src/foundation/strongly-extensional-maps.lagda.md index 8d3745442e..4db8992245 100644 --- a/src/foundation/strongly-extensional-maps.lagda.md +++ b/src/foundation/strongly-extensional-maps.lagda.md @@ -8,8 +8,7 @@ module foundation.strongly-extensional-maps where ```agda open import foundation.apartness-relations - -open import foundation-core.universe-levels +open import foundation.universe-levels ``` diff --git a/src/foundation/structure-identity-principle.lagda.md b/src/foundation/structure-identity-principle.lagda.md index 229e0b6b9f..d2b5e07e54 100644 --- a/src/foundation/structure-identity-principle.lagda.md +++ b/src/foundation/structure-identity-principle.lagda.md @@ -7,13 +7,14 @@ module foundation.structure-identity-principle where
Imports ```agda +open import foundation.dependent-pair-types +open import foundation.fundamental-theorem-of-identity-types +open import foundation.type-arithmetic-dependent-pair-types +open import foundation.universe-levels + open import foundation-core.contractible-types -open import foundation-core.dependent-pair-types open import foundation-core.equivalences -open import foundation-core.fundamental-theorem-of-identity-types open import foundation-core.identity-types -open import foundation-core.type-arithmetic-dependent-pair-types -open import foundation-core.universe-levels ```
diff --git a/src/foundation/structure.lagda.md b/src/foundation/structure.lagda.md index 81a7a7996a..d411448fd3 100644 --- a/src/foundation/structure.lagda.md +++ b/src/foundation/structure.lagda.md @@ -7,13 +7,13 @@ module foundation.structure where
Imports ```agda +open import foundation.dependent-pair-types open import foundation.univalence +open import foundation.universe-levels -open import foundation-core.dependent-pair-types open import foundation-core.equivalences open import foundation-core.fibers-of-maps open import foundation-core.identity-types -open import foundation-core.universe-levels ```
diff --git a/src/foundation/structured-type-duality.lagda.md b/src/foundation/structured-type-duality.lagda.md index 48457baf99..895d88c9a9 100644 --- a/src/foundation/structured-type-duality.lagda.md +++ b/src/foundation/structured-type-duality.lagda.md @@ -7,19 +7,19 @@ module foundation.structured-type-duality where
Imports ```agda +open import foundation.dependent-pair-types open import foundation.equivalences +open import foundation.functions open import foundation.structure +open import foundation.type-arithmetic-dependent-pair-types open import foundation.type-duality open import foundation.type-theoretic-principle-of-choice open import foundation.univalence +open import foundation.universe-levels -open import foundation-core.dependent-pair-types open import foundation-core.fibers-of-maps -open import foundation-core.functions open import foundation-core.functoriality-dependent-function-types open import foundation-core.functoriality-dependent-pair-types -open import foundation-core.type-arithmetic-dependent-pair-types -open import foundation-core.universe-levels ```
diff --git a/src/foundation/subterminal-types.lagda.md b/src/foundation/subterminal-types.lagda.md index ece912f1ea..66a8c79943 100644 --- a/src/foundation/subterminal-types.lagda.md +++ b/src/foundation/subterminal-types.lagda.md @@ -7,15 +7,15 @@ module foundation.subterminal-types where
Imports ```agda +open import foundation.functions open import foundation.unit-type +open import foundation.universe-levels open import foundation-core.contractible-types open import foundation-core.embeddings open import foundation-core.equivalences -open import foundation-core.functions open import foundation-core.identity-types open import foundation-core.propositions -open import foundation-core.universe-levels ```
diff --git a/src/foundation/subtype-duality.lagda.md b/src/foundation/subtype-duality.lagda.md index 447154efa5..3d8bde41d4 100644 --- a/src/foundation/subtype-duality.lagda.md +++ b/src/foundation/subtype-duality.lagda.md @@ -7,17 +7,17 @@ module foundation.subtype-duality where
Imports ```agda +open import foundation.dependent-pair-types open import foundation.inhabited-types open import foundation.propositional-maps open import foundation.structured-type-duality open import foundation.surjective-maps +open import foundation.universe-levels -open import foundation-core.dependent-pair-types open import foundation-core.embeddings open import foundation-core.equivalences open import foundation-core.functoriality-dependent-pair-types open import foundation-core.propositions -open import foundation-core.universe-levels ```
diff --git a/src/foundation/subtype-identity-principle.lagda.md b/src/foundation/subtype-identity-principle.lagda.md index 32d5866be0..0c3f0c7e63 100644 --- a/src/foundation/subtype-identity-principle.lagda.md +++ b/src/foundation/subtype-identity-principle.lagda.md @@ -2,14 +2,108 @@ ```agda module foundation.subtype-identity-principle where - -open import foundation-core.subtype-identity-principle public ```
Imports ```agda +open import foundation.dependent-pair-types +open import foundation.functions +open import foundation.fundamental-theorem-of-identity-types +open import foundation.type-arithmetic-dependent-pair-types +open import foundation.universe-levels +open import foundation-core.contractible-types +open import foundation-core.equivalences +open import foundation-core.identity-types +open import foundation-core.propositions ```
+ +## Idea + +The subtype identity principle allows us to efficiently characterize the +identity type of a subtype, using a characterization of the identity type of the +base type. + +## Lemma + +The following is a general construction that will help us show that the identity +type of a subtype agrees with the identity type of the original type. We already +know that the first projection of a family of propositions is an embedding, but +the following lemma still has its uses. + +```agda +module _ + {l1 l2 : Level} {A : UU l1} {B : A → UU l2} + where + + abstract + is-contr-total-Eq-subtype : + {l3 : Level} {P : A → UU l3} → + is-contr (Σ A B) → ((x : A) → is-prop (P x)) → + (a : A) (b : B a) (p : P a) → + is-contr (Σ (Σ A P) (B ∘ pr1)) + is-contr-total-Eq-subtype {l3} {P} + is-contr-AB is-subtype-P a b p = + is-contr-equiv + ( Σ (Σ A B) (P ∘ pr1)) + ( equiv-right-swap-Σ) + ( is-contr-equiv + ( P a) + ( left-unit-law-Σ-is-contr + ( is-contr-AB) + ( pair a b)) + ( is-proof-irrelevant-is-prop (is-subtype-P a) p)) +``` + +## Theorem + +### The subtype identity principle + +```agda +module _ + {l1 l2 l3 : Level} {A : UU l1} {P : A → UU l2} + (is-prop-P : (x : A) → is-prop (P x)) {Eq-A : A → UU l3} + {a : A} (p : P a) (refl-A : Eq-A a) + where + + abstract + subtype-identity-principle : + {f : (x : A) → a ＝ x → Eq-A x} + (h : (z : (Σ A P)) → (pair a p) ＝ z → Eq-A (pr1 z)) → + ((x : A) → is-equiv (f x)) → (z : Σ A P) → is-equiv (h z) + subtype-identity-principle {f} h H = + fundamental-theorem-id + ( is-contr-total-Eq-subtype + ( fundamental-theorem-id' f H) + ( is-prop-P) + ( a) + ( refl-A) + ( p)) + ( h) + +module _ + {l1 l2 l3 : Level} {A : UU l1} (P : A → Prop l2) {Eq-A : A → UU l3} + {a : A} (p : type-Prop (P a)) (refl-A : Eq-A a) + where + + map-extensionality-type-subtype : + (f : (x : A) → (a ＝ x) ≃ Eq-A x) → + (z : Σ A (type-Prop ∘ P)) → (pair a p) ＝ z → Eq-A (pr1 z) + map-extensionality-type-subtype f .(pair a p) refl = refl-A + + extensionality-type-subtype : + (f : (x : A) → (a ＝ x) ≃ Eq-A x) → + (z : Σ A (type-Prop ∘ P)) → (pair a p ＝ z) ≃ Eq-A (pr1 z) + pr1 (extensionality-type-subtype f z) = map-extensionality-type-subtype f z + pr2 (extensionality-type-subtype f z) = + subtype-identity-principle + ( is-prop-type-Prop ∘ P) + ( p) + ( refl-A) + ( map-extensionality-type-subtype f) + ( is-equiv-map-equiv ∘ f) + ( z) +``` diff --git a/src/foundation/subtypes.lagda.md b/src/foundation/subtypes.lagda.md index 9b68351d80..fcfbcba326 100644 --- a/src/foundation/subtypes.lagda.md +++ b/src/foundation/subtypes.lagda.md @@ -9,21 +9,21 @@ open import foundation-core.subtypes public
Imports ```agda +open import foundation.dependent-pair-types open import foundation.equality-dependent-function-types +open import foundation.functions open import foundation.propositional-extensionality +open import foundation.universe-levels open import foundation-core.contractible-types -open import foundation-core.dependent-pair-types open import foundation-core.embeddings open import foundation-core.equivalences -open import foundation-core.functions open import foundation-core.homotopies open import foundation-core.identity-types open import foundation-core.injective-maps open import foundation-core.logical-equivalences open import foundation-core.propositions open import foundation-core.sets -open import foundation-core.universe-levels ```
diff --git a/src/foundation/subuniverses.lagda.md b/src/foundation/subuniverses.lagda.md index 9ed244b96c..02be574f64 100644 --- a/src/foundation/subuniverses.lagda.md +++ b/src/foundation/subuniverses.lagda.md @@ -7,19 +7,19 @@ module foundation.subuniverses where
Imports ```agda +open import foundation.dependent-pair-types open import foundation.equality-dependent-function-types +open import foundation.fundamental-theorem-of-identity-types +open import foundation.subtype-identity-principle open import foundation.univalence +open import foundation.universe-levels open import foundation-core.contractible-types -open import foundation-core.dependent-pair-types open import foundation-core.embeddings open import foundation-core.equivalences -open import foundation-core.fundamental-theorem-of-identity-types open import foundation-core.identity-types open import foundation-core.propositions -open import foundation-core.subtype-identity-principle open import foundation-core.subtypes -open import foundation-core.universe-levels ```
diff --git a/src/foundation/surjective-maps.lagda.md b/src/foundation/surjective-maps.lagda.md index 8eae5d03e2..b7047a93f4 100644 --- a/src/foundation/surjective-maps.lagda.md +++ b/src/foundation/surjective-maps.lagda.md @@ -9,33 +9,33 @@ module foundation.surjective-maps where ```agda open import foundation.connected-maps open import foundation.contractible-types +open import foundation.dependent-pair-types open import foundation.embeddings +open import foundation.equality-cartesian-product-types +open import foundation.functions open import foundation.functoriality-cartesian-product-types +open import foundation.fundamental-theorem-of-identity-types open import foundation.homotopies open import foundation.identity-types open import foundation.propositional-truncations open import foundation.structure-identity-principle +open import foundation.subtype-identity-principle open import foundation.truncated-types open import foundation.univalence open import foundation.universal-property-propositional-truncation +open import foundation.universe-levels open import foundation-core.constant-maps open import foundation-core.contractible-maps -open import foundation-core.dependent-pair-types -open import foundation-core.equality-cartesian-product-types open import foundation-core.equivalences open import foundation-core.fibers-of-maps -open import foundation-core.functions open import foundation-core.functoriality-dependent-function-types -open import foundation-core.fundamental-theorem-of-identity-types open import foundation-core.propositional-maps open import foundation-core.propositions open import foundation-core.sections open import foundation-core.sets -open import foundation-core.subtype-identity-principle open import foundation-core.truncated-maps open import foundation-core.truncation-levels -open import foundation-core.universe-levels open import orthogonal-factorization-systems.extensions-of-maps ``` diff --git a/src/foundation/symmetric-difference.lagda.md b/src/foundation/symmetric-difference.lagda.md index 02b0b595ee..5a12db14b3 100644 --- a/src/foundation/symmetric-difference.lagda.md +++ b/src/foundation/symmetric-difference.lagda.md @@ -9,18 +9,18 @@ module foundation.symmetric-difference where ```agda open import foundation.decidable-subtypes open import foundation.decidable-types +open import foundation.dependent-pair-types open import foundation.exclusive-disjunction +open import foundation.functions open import foundation.identity-types hiding (inv) open import foundation.intersections-subtypes +open import foundation.universe-levels open import foundation-core.coproduct-types open import foundation-core.decidable-propositions -open import foundation-core.dependent-pair-types open import foundation-core.equivalences -open import foundation-core.functions open import foundation-core.propositions open import foundation-core.subtypes -open import foundation-core.universe-levels ``` diff --git a/src/foundation/symmetric-identity-types.lagda.md b/src/foundation/symmetric-identity-types.lagda.md index 4e93a1e3f6..64becb9868 100644 --- a/src/foundation/symmetric-identity-types.lagda.md +++ b/src/foundation/symmetric-identity-types.lagda.md @@ -7,22 +7,22 @@ module foundation.symmetric-identity-types where
Imports ```agda +open import foundation.dependent-pair-types open import foundation.function-extensionality +open import foundation.functions open import foundation.functoriality-dependent-function-types +open import foundation.fundamental-theorem-of-identity-types open import foundation.homotopies open import foundation.identity-types open import foundation.structure-identity-principle +open import foundation.universe-levels open import foundation.unordered-pairs open import foundation-core.contractible-types open import foundation-core.coproduct-types -open import foundation-core.dependent-pair-types open import foundation-core.equality-dependent-pair-types open import foundation-core.equivalences -open import foundation-core.functions open import foundation-core.functoriality-dependent-pair-types -open import foundation-core.fundamental-theorem-of-identity-types -open import foundation-core.universe-levels open import univalent-combinatorics.standard-finite-types ``` diff --git a/src/foundation/symmetric-operations.lagda.md b/src/foundation/symmetric-operations.lagda.md index b58863b504..bc994a6c53 100644 --- a/src/foundation/symmetric-operations.lagda.md +++ b/src/foundation/symmetric-operations.lagda.md @@ -7,17 +7,17 @@ module foundation.symmetric-operations where
Imports ```agda +open import foundation.dependent-pair-types open import foundation.equivalence-extensionality open import foundation.functoriality-coproduct-types open import foundation.universal-property-propositional-truncation-into-sets +open import foundation.universe-levels open import foundation.unordered-pairs open import foundation-core.coproduct-types -open import foundation-core.dependent-pair-types open import foundation-core.equivalences open import foundation-core.identity-types open import foundation-core.sets -open import foundation-core.universe-levels open import univalent-combinatorics.2-element-types open import univalent-combinatorics.finite-types diff --git a/src/foundation/tight-apartness-relations.lagda.md b/src/foundation/tight-apartness-relations.lagda.md index 2536456a34..4874b072f0 100644 --- a/src/foundation/tight-apartness-relations.lagda.md +++ b/src/foundation/tight-apartness-relations.lagda.md @@ -9,15 +9,15 @@ module foundation.tight-apartness-relations where ```agda open import foundation.apartness-relations open import foundation.binary-relations +open import foundation.dependent-pair-types open import foundation.function-extensionality open import foundation.propositional-truncations +open import foundation.universe-levels open import foundation-core.cartesian-product-types -open import foundation-core.dependent-pair-types open import foundation-core.identity-types open import foundation-core.negation open import foundation-core.propositions -open import foundation-core.universe-levels ```
diff --git a/src/foundation/transport.lagda.md b/src/foundation/transport.lagda.md index 17274c8fc4..b4711c8f6f 100644 --- a/src/foundation/transport.lagda.md +++ b/src/foundation/transport.lagda.md @@ -7,17 +7,17 @@ module foundation.transport where
Imports ```agda +open import foundation.dependent-pair-types +open import foundation.equality-cartesian-product-types open import foundation.function-extensionality +open import foundation.functions +open import foundation.universe-levels open import foundation-core.cartesian-product-types -open import foundation-core.dependent-pair-types -open import foundation-core.equality-cartesian-product-types open import foundation-core.equality-dependent-pair-types open import foundation-core.equivalences -open import foundation-core.functions open import foundation-core.homotopies open import foundation-core.identity-types -open import foundation-core.universe-levels ```
diff --git a/src/foundation/trivial-relaxed-sigma-decompositions.lagda.md b/src/foundation/trivial-relaxed-sigma-decompositions.lagda.md index 3952e87c5b..2cba62d7e5 100644 --- a/src/foundation/trivial-relaxed-sigma-decompositions.lagda.md +++ b/src/foundation/trivial-relaxed-sigma-decompositions.lagda.md @@ -8,18 +8,18 @@ module foundation.trivial-relaxed-sigma-decompositions where ```agda open import foundation.contractible-types +open import foundation.dependent-pair-types open import foundation.equivalences +open import foundation.functions open import foundation.relaxed-sigma-decompositions +open import foundation.type-arithmetic-dependent-pair-types open import foundation.unit-type +open import foundation.universe-levels -open import foundation-core.dependent-pair-types open import foundation-core.equality-dependent-pair-types -open import foundation-core.functions open import foundation-core.identity-types open import foundation-core.propositions open import foundation-core.subtypes -open import foundation-core.type-arithmetic-dependent-pair-types -open import foundation-core.universe-levels ```
diff --git a/src/foundation/trivial-sigma-decompositions.lagda.md b/src/foundation/trivial-sigma-decompositions.lagda.md index 4bba03444d..3a0d8ec4a0 100644 --- a/src/foundation/trivial-sigma-decompositions.lagda.md +++ b/src/foundation/trivial-sigma-decompositions.lagda.md @@ -8,22 +8,22 @@ module foundation.trivial-sigma-decompositions where ```agda open import foundation.contractible-types +open import foundation.dependent-pair-types open import foundation.equivalences +open import foundation.functions open import foundation.functoriality-propositional-truncation open import foundation.inhabited-types open import foundation.sigma-decompositions +open import foundation.type-arithmetic-dependent-pair-types open import foundation.type-arithmetic-empty-type open import foundation.unit-type +open import foundation.universe-levels -open import foundation-core.dependent-pair-types open import foundation-core.empty-types open import foundation-core.equality-dependent-pair-types -open import foundation-core.functions open import foundation-core.identity-types open import foundation-core.propositions open import foundation-core.subtypes -open import foundation-core.type-arithmetic-dependent-pair-types -open import foundation-core.universe-levels ``` diff --git a/src/foundation/truncated-equality.lagda.md b/src/foundation/truncated-equality.lagda.md index 1e2526a119..c88cab61f7 100644 --- a/src/foundation/truncated-equality.lagda.md +++ b/src/foundation/truncated-equality.lagda.md @@ -8,11 +8,11 @@ module foundation.truncated-equality where ```agda open import foundation.truncations +open import foundation.universe-levels open import foundation-core.identity-types open import foundation-core.truncated-types open import foundation-core.truncation-levels -open import foundation-core.universe-levels ``` diff --git a/src/foundation/truncated-maps.lagda.md b/src/foundation/truncated-maps.lagda.md index 91bc3f74aa..e5a6f037e2 100644 --- a/src/foundation/truncated-maps.lagda.md +++ b/src/foundation/truncated-maps.lagda.md @@ -9,15 +9,16 @@ open import foundation-core.truncated-maps public
Imports ```agda -open import foundation-core.cones-over-cospans -open import foundation-core.dependent-pair-types +open import foundation.cones-over-cospans +open import foundation.dependent-pair-types +open import foundation.functoriality-fibers-of-maps +open import foundation.universe-levels + open import foundation-core.fibers-of-maps -open import foundation-core.functoriality-fibers-of-maps open import foundation-core.propositions open import foundation-core.pullbacks open import foundation-core.truncated-types open import foundation-core.truncation-levels -open import foundation-core.universe-levels ```
diff --git a/src/foundation/truncated-types.lagda.md b/src/foundation/truncated-types.lagda.md index f269d51001..e6588ad2a4 100644 --- a/src/foundation/truncated-types.lagda.md +++ b/src/foundation/truncated-types.lagda.md @@ -9,17 +9,17 @@ open import foundation-core.truncated-types public
Imports ```agda +open import foundation.dependent-pair-types +open import foundation.subtype-identity-principle open import foundation.univalence +open import foundation.universe-levels open import foundation-core.contractible-types -open import foundation-core.dependent-pair-types open import foundation-core.embeddings open import foundation-core.equivalences open import foundation-core.identity-types -open import foundation-core.subtype-identity-principle open import foundation-core.subtypes open import foundation-core.truncation-levels -open import foundation-core.universe-levels ```
diff --git a/src/foundation/truncation-equivalences.lagda.md b/src/foundation/truncation-equivalences.lagda.md index f36846d884..b994fd2464 100644 --- a/src/foundation/truncation-equivalences.lagda.md +++ b/src/foundation/truncation-equivalences.lagda.md @@ -8,16 +8,16 @@ module foundation.truncation-equivalences where ```agda open import foundation.commuting-squares-of-maps +open import foundation.dependent-pair-types open import foundation.equivalences +open import foundation.functions open import foundation.functoriality-truncation open import foundation.truncations +open import foundation.universe-levels -open import foundation-core.dependent-pair-types -open import foundation-core.functions open import foundation-core.homotopies open import foundation-core.truncated-types open import foundation-core.truncation-levels -open import foundation-core.universe-levels ``` diff --git a/src/foundation/truncation-images-of-maps.lagda.md b/src/foundation/truncation-images-of-maps.lagda.md index 2639f19068..b3a569f4d6 100644 --- a/src/foundation/truncation-images-of-maps.lagda.md +++ b/src/foundation/truncation-images-of-maps.lagda.md @@ -7,13 +7,13 @@ module foundation.truncation-images-of-maps where
Imports ```agda +open import foundation.dependent-pair-types open import foundation.truncations +open import foundation.universe-levels -open import foundation-core.dependent-pair-types open import foundation-core.fibers-of-maps open import foundation-core.identity-types open import foundation-core.truncation-levels -open import foundation-core.universe-levels ```
diff --git a/src/foundation/truncation-levels.lagda.md b/src/foundation/truncation-levels.lagda.md index 182cf425cc..9fe6d62edf 100644 --- a/src/foundation/truncation-levels.lagda.md +++ b/src/foundation/truncation-levels.lagda.md @@ -11,7 +11,8 @@ open import foundation-core.truncation-levels public ```agda open import elementary-number-theory.natural-numbers -open import foundation-core.functions +open import foundation.functions + open import foundation-core.identity-types ``` diff --git a/src/foundation/truncations.lagda.md b/src/foundation/truncations.lagda.md index 62c937a08b..a341d939e2 100644 --- a/src/foundation/truncations.lagda.md +++ b/src/foundation/truncations.lagda.md @@ -7,26 +7,26 @@ module foundation.truncations where
Imports ```agda +open import foundation.dependent-pair-types open import foundation.function-extensionality +open import foundation.functions open import foundation.functoriality-dependent-function-types +open import foundation.fundamental-theorem-of-identity-types open import foundation.identity-types open import foundation.truncated-types open import foundation.universal-property-dependent-pair-types +open import foundation.universe-levels open import foundation-core.contractible-maps open import foundation-core.contractible-types -open import foundation-core.dependent-pair-types open import foundation-core.equality-dependent-pair-types open import foundation-core.equivalences open import foundation-core.fibers-of-maps -open import foundation-core.functions open import foundation-core.functoriality-dependent-pair-types -open import foundation-core.fundamental-theorem-of-identity-types open import foundation-core.homotopies open import foundation-core.propositions open import foundation-core.truncation-levels open import foundation-core.universal-property-truncation -open import foundation-core.universe-levels ```
diff --git a/src/foundation/tuples-of-types.lagda.md b/src/foundation/tuples-of-types.lagda.md index ed4c05ba66..49b8e03fc5 100644 --- a/src/foundation/tuples-of-types.lagda.md +++ b/src/foundation/tuples-of-types.lagda.md @@ -9,7 +9,7 @@ module foundation.tuples-of-types where ```agda open import elementary-number-theory.natural-numbers -open import foundation-core.universe-levels +open import foundation.universe-levels open import univalent-combinatorics.standard-finite-types ``` diff --git a/src/foundation/type-arithmetic-booleans.lagda.md b/src/foundation/type-arithmetic-booleans.lagda.md index 8d6339344c..86598e064b 100644 --- a/src/foundation/type-arithmetic-booleans.lagda.md +++ b/src/foundation/type-arithmetic-booleans.lagda.md @@ -8,14 +8,14 @@ module foundation.type-arithmetic-booleans where ```agda open import foundation.booleans +open import foundation.dependent-pair-types +open import foundation.functions +open import foundation.universe-levels open import foundation-core.coproduct-types -open import foundation-core.dependent-pair-types open import foundation-core.equivalences -open import foundation-core.functions open import foundation-core.homotopies open import foundation-core.identity-types -open import foundation-core.universe-levels ``` diff --git a/src/foundation/type-arithmetic-cartesian-product-types.lagda.md b/src/foundation/type-arithmetic-cartesian-product-types.lagda.md index bfe4956efc..505154ed7e 100644 --- a/src/foundation/type-arithmetic-cartesian-product-types.lagda.md +++ b/src/foundation/type-arithmetic-cartesian-product-types.lagda.md @@ -2,14 +2,161 @@ ```agda module foundation.type-arithmetic-cartesian-product-types where - -open import foundation-core.type-arithmetic-cartesian-product-types public ```
Imports ```agda +open import foundation.dependent-pair-types +open import foundation.equality-cartesian-product-types +open import foundation.functions +open import foundation.type-arithmetic-dependent-pair-types +open import foundation.universe-levels +open import foundation-core.cartesian-product-types +open import foundation-core.contractible-types +open import foundation-core.equivalences +open import foundation-core.homotopies +open import foundation-core.identity-types +open import foundation-core.propositions ```
+ +## Idea + +We prove laws for the manipulation of cartesian products with respect to +themselves and dependent pair types. + +## Laws + +### Commutativity of cartesian products + +```agda +module _ + {l1 l2 : Level} {A : UU l1} {B : UU l2} + where + + map-commutative-prod : A × B → B × A + pr1 (map-commutative-prod (pair a b)) = b + pr2 (map-commutative-prod (pair a b)) = a + + map-inv-commutative-prod : B × A → A × B + pr1 (map-inv-commutative-prod (pair b a)) = a + pr2 (map-inv-commutative-prod (pair b a)) = b + + issec-map-inv-commutative-prod : + (map-commutative-prod ∘ map-inv-commutative-prod) ~ id + issec-map-inv-commutative-prod (pair b a) = refl + + isretr-map-inv-commutative-prod : + (map-inv-commutative-prod ∘ map-commutative-prod) ~ id + isretr-map-inv-commutative-prod (pair a b) = refl + + is-equiv-map-commutative-prod : is-equiv map-commutative-prod + is-equiv-map-commutative-prod = + is-equiv-has-inverse + map-inv-commutative-prod + issec-map-inv-commutative-prod + isretr-map-inv-commutative-prod + + commutative-prod : (A × B) ≃ (B × A) + pr1 commutative-prod = map-commutative-prod + pr2 commutative-prod = is-equiv-map-commutative-prod +``` + +### Associativity of cartesian products + +```agda +module _ + {l1 l2 l3 : Level} (A : UU l1) (B : UU l2) (C : UU l3) + where + + map-associative-prod : (A × B) × C → A × (B × C) + map-associative-prod = map-associative-Σ A (λ x → B) (λ w → C) + + map-inv-associative-prod : A × (B × C) → (A × B) × C + map-inv-associative-prod = map-inv-associative-Σ A (λ x → B) (λ w → C) + + issec-map-inv-associative-prod : + (map-associative-prod ∘ map-inv-associative-prod) ~ id + issec-map-inv-associative-prod = + issec-map-inv-associative-Σ A (λ x → B) (λ w → C) + + isretr-map-inv-associative-prod : + (map-inv-associative-prod ∘ map-associative-prod) ~ id + isretr-map-inv-associative-prod = + isretr-map-inv-associative-Σ A (λ x → B) (λ w → C) + + is-equiv-map-associative-prod : is-equiv map-associative-prod + is-equiv-map-associative-prod = + is-equiv-map-associative-Σ A (λ x → B) (λ w → C) + + associative-prod : ((A × B) × C) ≃ (A × (B × C)) + associative-prod = associative-Σ A (λ x → B) (λ w → C) +``` + +### The unit laws of cartesian product types with respect to contractible types + +```agda +module _ + {l1 l2 : Level} {A : UU l1} {B : UU l2} (is-contr-B : is-contr B) + where + + right-unit-law-prod-is-contr : (A × B) ≃ A + right-unit-law-prod-is-contr = right-unit-law-Σ-is-contr (λ a → is-contr-B) + +module _ + {l1 l2 : Level} {A : UU l1} {B : UU l2} (is-contr-A : is-contr A) + where + + left-unit-law-prod-is-contr : (A × B) ≃ B + left-unit-law-prod-is-contr = + left-unit-law-Σ-is-contr is-contr-A (center is-contr-A) + + is-equiv-pr2-prod-is-contr : is-equiv (pr2 {B = λ a → B}) + is-equiv-pr2-prod-is-contr = + is-equiv-comp + ( pr1) + ( map-commutative-prod) + ( is-equiv-map-commutative-prod) + ( is-equiv-pr1-is-contr λ b → is-contr-A) + + equiv-pr2-prod-is-contr : (A × B) ≃ B + pr1 equiv-pr2-prod-is-contr = pr2 + pr2 equiv-pr2-prod-is-contr = is-equiv-pr2-prod-is-contr +``` + +### Adding redundant property + +```agda +equiv-add-redundant-prop : + {l1 l2 : Level} {A : UU l1} {B : UU l2} → + (is-prop B) → (f : A → B) → (A ≃ (A × B)) +pr1 (equiv-add-redundant-prop is-prop-B f) a = a , f a +pr2 (equiv-add-redundant-prop is-prop-B f) = + is-equiv-has-inverse + ( pr1) + ( λ p → eq-pair refl (eq-is-prop is-prop-B)) + ( λ a → refl) +``` + +## See also + +- Functorial properties of cartesian products are recorded in + [`foundation.functoriality-cartesian-product-types`](foundation.functoriality-cartesian-product-types.md). +- Equality proofs in cartesian product types are characterized in + [`foundation.equality-cartesian-product-types`](foundation.equality-cartesian-product-types.md). +- The universal property of cartesian product types is treated in + [`foundation.universal-property-cartesian-product-types`](foundation.universal-property-cartesian-product-types.md). + +- Arithmetical laws involving dependent pair types are recorded in + [`foundation.type-arithmetic-dependent-pair-types`](foundation.type-arithmetic-dependent-pair-types.md). + - Arithmetical laws involving dependent product types are recorded in + [`foundation.type-arithmetic-dependent-function-types`](foundation.type-arithmetic-dependent-function-types.md). +- Arithmetical laws involving coproduct types are recorded in + [`foundation.type-arithmetic-coproduct-types`](foundation.type-arithmetic-coproduct-types.md). +- Arithmetical laws involving the unit type are recorded in + [`foundation.type-arithmetic-unit-type`](foundation.type-arithmetic-unit-type.md). +- Arithmetical laws involving the empty type are recorded in + [`foundation.type-arithmetic-empty-type`](foundation.type-arithmetic-empty-type.md). diff --git a/src/foundation/type-arithmetic-coproduct-types.lagda.md b/src/foundation/type-arithmetic-coproduct-types.lagda.md index 41448dc83c..45cbf5b8f1 100644 --- a/src/foundation/type-arithmetic-coproduct-types.lagda.md +++ b/src/foundation/type-arithmetic-coproduct-types.lagda.md @@ -8,17 +8,17 @@ module foundation.type-arithmetic-coproduct-types where ```agda open import foundation.coproduct-types +open import foundation.dependent-pair-types open import foundation.equality-coproduct-types +open import foundation.functions +open import foundation.universe-levels open import foundation-core.cartesian-product-types open import foundation-core.contractible-types -open import foundation-core.dependent-pair-types open import foundation-core.empty-types open import foundation-core.equivalences -open import foundation-core.functions open import foundation-core.homotopies open import foundation-core.identity-types -open import foundation-core.universe-levels ``` diff --git a/src/foundation/type-arithmetic-dependent-function-types.lagda.md b/src/foundation/type-arithmetic-dependent-function-types.lagda.md index e382dc320c..4d468e1846 100644 --- a/src/foundation/type-arithmetic-dependent-function-types.lagda.md +++ b/src/foundation/type-arithmetic-dependent-function-types.lagda.md @@ -7,17 +7,17 @@ module foundation.type-arithmetic-dependent-function-types where
Imports ```agda +open import foundation.dependent-pair-types open import foundation.functoriality-dependent-function-types open import foundation.type-arithmetic-unit-type open import foundation.unit-type +open import foundation.universe-levels open import foundation-core.contractible-types -open import foundation-core.dependent-pair-types open import foundation-core.equivalences open import foundation-core.homotopies open import foundation-core.identity-types open import foundation-core.univalence -open import foundation-core.universe-levels ```
diff --git a/src/foundation/type-arithmetic-dependent-pair-types.lagda.md b/src/foundation/type-arithmetic-dependent-pair-types.lagda.md index ffdcfffc76..483a6c596d 100644 --- a/src/foundation/type-arithmetic-dependent-pair-types.lagda.md +++ b/src/foundation/type-arithmetic-dependent-pair-types.lagda.md @@ -2,14 +2,24 @@ ```agda module foundation.type-arithmetic-dependent-pair-types where - -open import foundation-core.type-arithmetic-dependent-pair-types public ```
Imports ```agda +open import foundation.dependent-pair-types +open import foundation.functions +open import foundation.universe-levels +open import foundation-core.cartesian-product-types +open import foundation-core.contractible-maps +open import foundation-core.contractible-types +open import foundation-core.equality-dependent-pair-types +open import foundation-core.equivalences +open import foundation-core.fibers-of-maps +open import foundation-core.homotopies +open import foundation-core.identity-types +open import foundation-core.singleton-induction ```
@@ -17,7 +27,379 @@ open import foundation-core.type-arithmetic-dependent-pair-types public ## Idea We prove laws for the manipulation of dependent pair types with respect to -themselves and contractible types. +themselves and arithmetical laws with respect to contractible types. + +## Properties + +### The left unit law for Σ using a contractible base type + +```agda +module _ + {l1 l2 : Level} {A : UU l1} {B : A → UU l2} (C : is-contr A) (a : A) + where + + map-inv-left-unit-law-Σ-is-contr : B a → Σ A B + map-inv-left-unit-law-Σ-is-contr b = pair a b + + map-left-unit-law-Σ-is-contr : Σ A B → B a + map-left-unit-law-Σ-is-contr = + ind-Σ + ( ind-singleton-is-contr a C + ( λ x → B x → B a) + ( id)) + + issec-map-inv-left-unit-law-Σ-is-contr : + ( map-left-unit-law-Σ-is-contr ∘ map-inv-left-unit-law-Σ-is-contr) ~ id + issec-map-inv-left-unit-law-Σ-is-contr b = + ap + ( λ (f : B a → B a) → f b) + ( compute-ind-singleton-is-contr a C (λ x → B x → B a) id) + + isretr-map-inv-left-unit-law-Σ-is-contr : + ( map-inv-left-unit-law-Σ-is-contr ∘ map-left-unit-law-Σ-is-contr) ~ id + isretr-map-inv-left-unit-law-Σ-is-contr = + ind-Σ + ( ind-singleton-is-contr a C + ( λ x → + ( y : B x) → + Id + ( ( map-inv-left-unit-law-Σ-is-contr ∘ + map-left-unit-law-Σ-is-contr) + ( pair x y)) + ( pair x y)) + ( λ y → ap + ( map-inv-left-unit-law-Σ-is-contr) + ( ap + ( λ f → f y) + ( compute-ind-singleton-is-contr a C (λ x → B x → B a) id)))) + + is-equiv-map-left-unit-law-Σ-is-contr : + is-equiv map-left-unit-law-Σ-is-contr + is-equiv-map-left-unit-law-Σ-is-contr = + is-equiv-has-inverse + map-inv-left-unit-law-Σ-is-contr + issec-map-inv-left-unit-law-Σ-is-contr + isretr-map-inv-left-unit-law-Σ-is-contr + + left-unit-law-Σ-is-contr : Σ A B ≃ B a + pr1 left-unit-law-Σ-is-contr = map-left-unit-law-Σ-is-contr + pr2 left-unit-law-Σ-is-contr = is-equiv-map-left-unit-law-Σ-is-contr + + abstract + is-equiv-map-inv-left-unit-law-Σ-is-contr : + is-equiv map-inv-left-unit-law-Σ-is-contr + is-equiv-map-inv-left-unit-law-Σ-is-contr = + is-equiv-has-inverse + map-left-unit-law-Σ-is-contr + isretr-map-inv-left-unit-law-Σ-is-contr + issec-map-inv-left-unit-law-Σ-is-contr + + inv-left-unit-law-Σ-is-contr : B a ≃ Σ A B + pr1 inv-left-unit-law-Σ-is-contr = map-inv-left-unit-law-Σ-is-contr + pr2 inv-left-unit-law-Σ-is-contr = is-equiv-map-inv-left-unit-law-Σ-is-contr +``` + +### Right unit law for dependent pair types + +```agda +module _ + {l1 l2 : Level} {A : UU l1} {B : A → UU l2} + where + + abstract + is-equiv-pr1-is-contr : ((a : A) → is-contr (B a)) → is-equiv (pr1 {B = B}) + is-equiv-pr1-is-contr is-contr-B = + is-equiv-is-contr-map + ( λ x → is-contr-equiv + ( B x) + ( equiv-fib-pr1 B x) + ( is-contr-B x)) + + equiv-pr1 : ((a : A) → is-contr (B a)) → (Σ A B) ≃ A + pr1 (equiv-pr1 is-contr-B) = pr1 + pr2 (equiv-pr1 is-contr-B) = is-equiv-pr1-is-contr is-contr-B + + right-unit-law-Σ-is-contr : ((a : A) → is-contr (B a)) → (Σ A B) ≃ A + right-unit-law-Σ-is-contr = equiv-pr1 + + abstract + is-contr-is-equiv-pr1 : is-equiv (pr1 {B = B}) → ((a : A) → is-contr (B a)) + is-contr-is-equiv-pr1 is-equiv-pr1-B a = + is-contr-equiv' + ( fib pr1 a) + ( equiv-fib-pr1 B a) + ( is-contr-map-is-equiv is-equiv-pr1-B a) + + map-inv-right-unit-law-Σ-is-contr : + ((a : A) → is-contr (B a)) → A → Σ A B + map-inv-right-unit-law-Σ-is-contr H a = (a , center (H a)) + + issec-map-inv-right-unit-law-Σ-is-contr : + (H : (a : A) → is-contr (B a)) → + ( pr1 ∘ map-inv-right-unit-law-Σ-is-contr H) ~ id + issec-map-inv-right-unit-law-Σ-is-contr H = refl-htpy + + isretr-map-inv-right-unit-law-Σ-is-contr : + (H : (a : A) → is-contr (B a)) → + ( map-inv-right-unit-law-Σ-is-contr H ∘ pr1) ~ id + isretr-map-inv-right-unit-law-Σ-is-contr H (a , b) = + eq-pair-Σ refl (eq-is-contr (H a)) + + is-equiv-map-inv-right-unit-law-Σ-is-contr : + (H : (a : A) → is-contr (B a)) → + is-equiv (map-inv-right-unit-law-Σ-is-contr H) + is-equiv-map-inv-right-unit-law-Σ-is-contr H = + is-equiv-has-inverse + ( pr1) + ( isretr-map-inv-right-unit-law-Σ-is-contr H) + ( issec-map-inv-right-unit-law-Σ-is-contr H) + + inv-right-unit-law-Σ-is-contr : + (H : (a : A) → is-contr (B a)) → A ≃ Σ A B + pr1 (inv-right-unit-law-Σ-is-contr H) = map-inv-right-unit-law-Σ-is-contr H + pr2 (inv-right-unit-law-Σ-is-contr H) = + is-equiv-map-inv-right-unit-law-Σ-is-contr H +``` + +### Associativity of dependent pair types + +There are two ways to express associativity for dependent pair types. We +formalize both ways. + +```agda +module _ + {l1 l2 l3 : Level} (A : UU l1) (B : A → UU l2) (C : Σ A B → UU l3) + where + + map-associative-Σ : Σ (Σ A B) C → Σ A (λ x → Σ (B x) (λ y → C (pair x y))) + pr1 (map-associative-Σ ((x , y) , z)) = x + pr1 (pr2 (map-associative-Σ ((x , y) , z))) = y + pr2 (pr2 (map-associative-Σ ((x , y) , z))) = z + + map-inv-associative-Σ : Σ A (λ x → Σ (B x) (λ y → C (pair x y))) → Σ (Σ A B) C + pr1 (pr1 (map-inv-associative-Σ (x , y , z))) = x + pr2 (pr1 (map-inv-associative-Σ (x , y , z))) = y + pr2 (map-inv-associative-Σ (x , y , z)) = z + + isretr-map-inv-associative-Σ : + (map-inv-associative-Σ ∘ map-associative-Σ) ~ id + isretr-map-inv-associative-Σ (pair (pair x y) z) = refl + + issec-map-inv-associative-Σ : (map-associative-Σ ∘ map-inv-associative-Σ) ~ id + issec-map-inv-associative-Σ (pair x (pair y z)) = refl + + abstract + is-equiv-map-associative-Σ : is-equiv map-associative-Σ + is-equiv-map-associative-Σ = + is-equiv-has-inverse + map-inv-associative-Σ + issec-map-inv-associative-Σ + isretr-map-inv-associative-Σ + + associative-Σ : Σ (Σ A B) C ≃ Σ A (λ x → Σ (B x) (λ y → C (pair x y))) + pr1 associative-Σ = map-associative-Σ + pr2 associative-Σ = is-equiv-map-associative-Σ + + inv-associative-Σ : Σ A (λ x → Σ (B x) (λ y → C (pair x y))) ≃ Σ (Σ A B) C + pr1 inv-associative-Σ = map-inv-associative-Σ + pr2 inv-associative-Σ = + is-equiv-has-inverse + map-associative-Σ + isretr-map-inv-associative-Σ + issec-map-inv-associative-Σ +``` + +### Associativity, second formulation + +```agda +module _ + {l1 l2 l3 : Level} (A : UU l1) (B : A → UU l2) (C : (x : A) → B x → UU l3) + where + + map-associative-Σ' : + Σ (Σ A B) (λ w → C (pr1 w) (pr2 w)) → Σ A (λ x → Σ (B x) (C x)) + pr1 (map-associative-Σ' ((x , y) , z)) = x + pr1 (pr2 (map-associative-Σ' ((x , y) , z))) = y + pr2 (pr2 (map-associative-Σ' ((x , y) , z))) = z + + map-inv-associative-Σ' : + Σ A (λ x → Σ (B x) (C x)) → Σ (Σ A B) (λ w → C (pr1 w) (pr2 w)) + pr1 (pr1 (map-inv-associative-Σ' (x , y , z))) = x + pr2 (pr1 (map-inv-associative-Σ' (x , y , z))) = y + pr2 (map-inv-associative-Σ' (x , y , z)) = z + + issec-map-inv-associative-Σ' : + (map-associative-Σ' ∘ map-inv-associative-Σ') ~ id + issec-map-inv-associative-Σ' (pair x (pair y z)) = refl + + isretr-map-inv-associative-Σ' : + ( map-inv-associative-Σ' ∘ map-associative-Σ') ~ id + isretr-map-inv-associative-Σ' (pair (pair x y) z) = refl + + is-equiv-map-associative-Σ' : is-equiv map-associative-Σ' + is-equiv-map-associative-Σ' = + is-equiv-has-inverse + map-inv-associative-Σ' + issec-map-inv-associative-Σ' + isretr-map-inv-associative-Σ' + + associative-Σ' : + Σ (Σ A B) (λ w → C (pr1 w) (pr2 w)) ≃ Σ A (λ x → Σ (B x) (C x)) + pr1 associative-Σ' = map-associative-Σ' + pr2 associative-Σ' = is-equiv-map-associative-Σ' + + inv-associative-Σ' : + Σ A (λ x → Σ (B x) (C x)) ≃ Σ (Σ A B) (λ w → C (pr1 w) (pr2 w)) + pr1 inv-associative-Σ' = map-inv-associative-Σ' + pr2 inv-associative-Σ' = + is-equiv-has-inverse + map-associative-Σ' + isretr-map-inv-associative-Σ' + issec-map-inv-associative-Σ' +``` + +### The interchange law + +```agda +module _ + { l1 l2 l3 l4 : Level} { A : UU l1} {B : A → UU l2} {C : A → UU l3} + ( D : (x : A) → B x → C x → UU l4) + where + + map-interchange-Σ-Σ : + Σ (Σ A B) (λ t → Σ (C (pr1 t)) (D (pr1 t) (pr2 t))) → + Σ (Σ A C) (λ t → Σ (B (pr1 t)) (λ y → D (pr1 t) y (pr2 t))) + pr1 (pr1 (map-interchange-Σ-Σ t)) = pr1 (pr1 t) + pr2 (pr1 (map-interchange-Σ-Σ t)) = pr1 (pr2 t) + pr1 (pr2 (map-interchange-Σ-Σ t)) = pr2 (pr1 t) + pr2 (pr2 (map-interchange-Σ-Σ t)) = pr2 (pr2 t) + + map-inv-interchange-Σ-Σ : + Σ (Σ A C) (λ t → Σ (B (pr1 t)) (λ y → D (pr1 t) y (pr2 t))) → + Σ (Σ A B) (λ t → Σ (C (pr1 t)) (D (pr1 t) (pr2 t))) + pr1 (pr1 (map-inv-interchange-Σ-Σ t)) = pr1 (pr1 t) + pr2 (pr1 (map-inv-interchange-Σ-Σ t)) = pr1 (pr2 t) + pr1 (pr2 (map-inv-interchange-Σ-Σ t)) = pr2 (pr1 t) + pr2 (pr2 (map-inv-interchange-Σ-Σ t)) = pr2 (pr2 t) + + issec-map-inv-interchange-Σ-Σ : + ( map-interchange-Σ-Σ ∘ map-inv-interchange-Σ-Σ) ~ id + issec-map-inv-interchange-Σ-Σ (pair (pair a c) (pair b d)) = refl + + isretr-map-inv-interchange-Σ-Σ : + ( map-inv-interchange-Σ-Σ ∘ map-interchange-Σ-Σ) ~ id + isretr-map-inv-interchange-Σ-Σ (pair (pair a b) (pair c d)) = refl + + abstract + is-equiv-map-interchange-Σ-Σ : is-equiv map-interchange-Σ-Σ + is-equiv-map-interchange-Σ-Σ = + is-equiv-has-inverse + map-inv-interchange-Σ-Σ + issec-map-inv-interchange-Σ-Σ + isretr-map-inv-interchange-Σ-Σ + + interchange-Σ-Σ : + Σ (Σ A B) (λ t → Σ (C (pr1 t)) (D (pr1 t) (pr2 t))) ≃ + Σ (Σ A C) (λ t → Σ (B (pr1 t)) (λ y → D (pr1 t) y (pr2 t))) + pr1 interchange-Σ-Σ = map-interchange-Σ-Σ + pr2 interchange-Σ-Σ = is-equiv-map-interchange-Σ-Σ + + eq-interchange-Σ-Σ-is-contr : + {a : A} {b : B a} → is-contr (Σ (C a) (D a b)) → + {x y : Σ (C a) (D a b)} → + map-equiv interchange-Σ-Σ ((a , b) , x) ＝ + map-equiv interchange-Σ-Σ ((a , b) , y) + eq-interchange-Σ-Σ-is-contr H = + ap (map-equiv interchange-Σ-Σ) (ap (pair _) (eq-is-contr H)) +``` + +### Swapping the order of quantification in a Σ-type, on the left + +```agda +module _ + {l1 l2 l3 : Level} {A : UU l1} {B : UU l2} {C : A → B → UU l3} + where + + map-left-swap-Σ : Σ A (λ x → Σ B (C x)) → Σ B (λ y → Σ A (λ x → C x y)) + pr1 (map-left-swap-Σ (a , b , c)) = b + pr1 (pr2 (map-left-swap-Σ (a , b , c))) = a + pr2 (pr2 (map-left-swap-Σ (a , b , c))) = c + + map-inv-left-swap-Σ : + Σ B (λ y → Σ A (λ x → C x y)) → Σ A (λ x → Σ B (C x)) + pr1 (map-inv-left-swap-Σ (b , a , c)) = a + pr1 (pr2 (map-inv-left-swap-Σ (b , a , c))) = b + pr2 (pr2 (map-inv-left-swap-Σ (b , a , c))) = c + + isretr-map-inv-left-swap-Σ : (map-inv-left-swap-Σ ∘ map-left-swap-Σ) ~ id + isretr-map-inv-left-swap-Σ (pair a (pair b c)) = refl + + issec-map-inv-left-swap-Σ : (map-left-swap-Σ ∘ map-inv-left-swap-Σ) ~ id + issec-map-inv-left-swap-Σ (pair b (pair a c)) = refl + + abstract + is-equiv-map-left-swap-Σ : is-equiv map-left-swap-Σ + is-equiv-map-left-swap-Σ = + is-equiv-has-inverse + map-inv-left-swap-Σ + issec-map-inv-left-swap-Σ + isretr-map-inv-left-swap-Σ + + equiv-left-swap-Σ : Σ A (λ a → Σ B (C a)) ≃ Σ B (λ b → Σ A (λ a → C a b)) + pr1 equiv-left-swap-Σ = map-left-swap-Σ + pr2 equiv-left-swap-Σ = is-equiv-map-left-swap-Σ +``` + +### Swapping the order of quantification in a Σ-type, on the right + +```agda +module _ + {l1 l2 l3 : Level} {A : UU l1} {B : A → UU l2} {C : A → UU l3} + where + + map-right-swap-Σ : Σ (Σ A B) (C ∘ pr1) → Σ (Σ A C) (B ∘ pr1) + pr1 (pr1 (map-right-swap-Σ ((a , b) , c))) = a + pr2 (pr1 (map-right-swap-Σ ((a , b) , c))) = c + pr2 (map-right-swap-Σ ((a , b) , c)) = b + + map-inv-right-swap-Σ : Σ (Σ A C) (B ∘ pr1) → Σ (Σ A B) (C ∘ pr1) + pr1 (pr1 (map-inv-right-swap-Σ ((a , c) , b))) = a + pr2 (pr1 (map-inv-right-swap-Σ ((a , c) , b))) = b + pr2 (map-inv-right-swap-Σ ((a , c) , b)) = c + + issec-map-inv-right-swap-Σ : (map-right-swap-Σ ∘ map-inv-right-swap-Σ) ~ id + issec-map-inv-right-swap-Σ (pair (pair x y) z) = refl + + isretr-map-inv-right-swap-Σ : (map-inv-right-swap-Σ ∘ map-right-swap-Σ) ~ id + isretr-map-inv-right-swap-Σ (pair (pair x z) y) = refl + + is-equiv-map-right-swap-Σ : is-equiv map-right-swap-Σ + is-equiv-map-right-swap-Σ = + is-equiv-has-inverse + map-inv-right-swap-Σ + issec-map-inv-right-swap-Σ + isretr-map-inv-right-swap-Σ + + equiv-right-swap-Σ : Σ (Σ A B) (C ∘ pr1) ≃ Σ (Σ A C) (B ∘ pr1) + pr1 equiv-right-swap-Σ = map-right-swap-Σ + pr2 equiv-right-swap-Σ = is-equiv-map-right-swap-Σ +``` + +### Distributive laws of cartesian products over Σ + +```agda +left-distributive-prod-Σ : + {l1 l2 l3 : Level} {A : UU l1} {B : UU l2} {C : B → UU l3} → + (A × (Σ B C)) ≃ Σ B (λ b → A × (C b)) +left-distributive-prod-Σ = + equiv-left-swap-Σ + +right-distributive-prod-Σ : + {l1 l2 l3 : Level} {A : UU l1} {B : A → UU l2} {C : UU l3} → + ((Σ A B) × C) ≃ Σ A (λ a → B a × C) +right-distributive-prod-Σ {A} = + associative-Σ _ _ _ +``` ## See also diff --git a/src/foundation/type-arithmetic-empty-type.lagda.md b/src/foundation/type-arithmetic-empty-type.lagda.md index c0a7e7e496..ef42003019 100644 --- a/src/foundation/type-arithmetic-empty-type.lagda.md +++ b/src/foundation/type-arithmetic-empty-type.lagda.md @@ -8,17 +8,17 @@ module foundation.type-arithmetic-empty-type where ```agda open import foundation.coproduct-types +open import foundation.dependent-pair-types +open import foundation.functions +open import foundation.universe-levels open import foundation-core.cartesian-product-types open import foundation-core.contractible-maps open import foundation-core.contractible-types -open import foundation-core.dependent-pair-types open import foundation-core.empty-types open import foundation-core.equivalences -open import foundation-core.functions open import foundation-core.homotopies open import foundation-core.identity-types -open import foundation-core.universe-levels ``` diff --git a/src/foundation/type-arithmetic-unit-type.lagda.md b/src/foundation/type-arithmetic-unit-type.lagda.md index 23fa4a341e..c46342ab4c 100644 --- a/src/foundation/type-arithmetic-unit-type.lagda.md +++ b/src/foundation/type-arithmetic-unit-type.lagda.md @@ -7,16 +7,16 @@ module foundation.type-arithmetic-unit-type where
Imports ```agda +open import foundation.dependent-pair-types open import foundation.function-extensionality +open import foundation.functions open import foundation.unit-type +open import foundation.universe-levels open import foundation-core.cartesian-product-types -open import foundation-core.dependent-pair-types open import foundation-core.equivalences -open import foundation-core.functions open import foundation-core.homotopies open import foundation-core.identity-types -open import foundation-core.universe-levels ```
diff --git a/src/foundation/type-duality.lagda.md b/src/foundation/type-duality.lagda.md index bb945e3b60..abac66d498 100644 --- a/src/foundation/type-duality.lagda.md +++ b/src/foundation/type-duality.lagda.md @@ -7,28 +7,28 @@ module foundation.type-duality where
Imports ```agda +open import foundation.dependent-pair-types open import foundation.equivalences open import foundation.function-extensionality +open import foundation.functions +open import foundation.fundamental-theorem-of-identity-types open import foundation.locally-small-types open import foundation.slice +open import foundation.type-arithmetic-dependent-pair-types open import foundation.unit-type open import foundation.univalence +open import foundation.universe-levels open import foundation-core.contractible-maps open import foundation-core.contractible-types -open import foundation-core.dependent-pair-types open import foundation-core.embeddings open import foundation-core.fibers-of-maps -open import foundation-core.functions open import foundation-core.functoriality-dependent-function-types open import foundation-core.functoriality-dependent-pair-types -open import foundation-core.fundamental-theorem-of-identity-types open import foundation-core.homotopies open import foundation-core.identity-types open import foundation-core.injective-maps open import foundation-core.small-types -open import foundation-core.type-arithmetic-dependent-pair-types -open import foundation-core.universe-levels open import trees.polynomial-endofunctors ``` diff --git a/src/foundation/type-theoretic-principle-of-choice.lagda.md b/src/foundation/type-theoretic-principle-of-choice.lagda.md index 137a50e760..65d9cc883f 100644 --- a/src/foundation/type-theoretic-principle-of-choice.lagda.md +++ b/src/foundation/type-theoretic-principle-of-choice.lagda.md @@ -7,16 +7,16 @@ module foundation.type-theoretic-principle-of-choice where
Imports ```agda +open import foundation.dependent-pair-types open import foundation.function-extensionality +open import foundation.functions open import foundation.structure-identity-principle +open import foundation.universe-levels -open import foundation-core.dependent-pair-types open import foundation-core.equality-dependent-pair-types open import foundation-core.equivalences -open import foundation-core.functions open import foundation-core.homotopies open import foundation-core.identity-types -open import foundation-core.universe-levels ```
diff --git a/src/foundation/unions-subtypes.lagda.md b/src/foundation/unions-subtypes.lagda.md index 87823bacfb..ec82d12d43 100644 --- a/src/foundation/unions-subtypes.lagda.md +++ b/src/foundation/unions-subtypes.lagda.md @@ -10,9 +10,9 @@ module foundation.unions-subtypes where open import foundation.decidable-subtypes open import foundation.disjunction open import foundation.large-locale-of-subtypes +open import foundation.universe-levels open import foundation-core.subtypes -open import foundation-core.universe-levels ```
diff --git a/src/foundation/unique-existence.lagda.md b/src/foundation/unique-existence.lagda.md index b66dc60c21..f668a70013 100644 --- a/src/foundation/unique-existence.lagda.md +++ b/src/foundation/unique-existence.lagda.md @@ -7,9 +7,10 @@ module foundation.unique-existence where
Imports ```agda +open import foundation.dependent-pair-types +open import foundation.universe-levels + open import foundation-core.contractible-types -open import foundation-core.dependent-pair-types -open import foundation-core.universe-levels ```
diff --git a/src/foundation/uniqueness-image.lagda.md b/src/foundation/uniqueness-image.lagda.md index bc5c6510f3..aac1422bbb 100644 --- a/src/foundation/uniqueness-image.lagda.md +++ b/src/foundation/uniqueness-image.lagda.md @@ -7,21 +7,21 @@ module foundation.uniqueness-image where
Imports ```agda +open import foundation.dependent-pair-types open import foundation.equivalences +open import foundation.functions open import foundation.images open import foundation.slice +open import foundation.type-arithmetic-dependent-pair-types open import foundation.universal-property-image +open import foundation.universe-levels open import foundation-core.contractible-types -open import foundation-core.dependent-pair-types open import foundation-core.embeddings -open import foundation-core.functions open import foundation-core.functoriality-dependent-pair-types open import foundation-core.homotopies open import foundation-core.propositions open import foundation-core.sections -open import foundation-core.type-arithmetic-dependent-pair-types -open import foundation-core.universe-levels ```
diff --git a/src/foundation/uniqueness-set-quotients.lagda.md b/src/foundation/uniqueness-set-quotients.lagda.md index f5cfeb21ef..8468ef1800 100644 --- a/src/foundation/uniqueness-set-quotients.lagda.md +++ b/src/foundation/uniqueness-set-quotients.lagda.md @@ -7,21 +7,21 @@ module foundation.uniqueness-set-quotients where
Imports ```agda +open import foundation.dependent-pair-types open import foundation.equivalences +open import foundation.functions open import foundation.reflecting-maps-equivalence-relations open import foundation.sets +open import foundation.subtype-identity-principle open import foundation.universal-property-set-quotients +open import foundation.universe-levels open import foundation-core.contractible-types -open import foundation-core.dependent-pair-types open import foundation-core.equivalence-relations open import foundation-core.function-extensionality -open import foundation-core.functions open import foundation-core.homotopies open import foundation-core.identity-types open import foundation-core.injective-maps -open import foundation-core.subtype-identity-principle -open import foundation-core.universe-levels ```
diff --git a/src/foundation/uniqueness-set-truncations.lagda.md b/src/foundation/uniqueness-set-truncations.lagda.md index e19e545c15..78b16d8588 100644 --- a/src/foundation/uniqueness-set-truncations.lagda.md +++ b/src/foundation/uniqueness-set-truncations.lagda.md @@ -7,17 +7,17 @@ module foundation.uniqueness-set-truncations where
Imports ```agda +open import foundation.dependent-pair-types +open import foundation.functions open import foundation.mere-equality open import foundation.sets open import foundation.uniqueness-set-quotients open import foundation.universal-property-set-truncation +open import foundation.universe-levels open import foundation-core.contractible-types -open import foundation-core.dependent-pair-types open import foundation-core.equivalences -open import foundation-core.functions open import foundation-core.homotopies -open import foundation-core.universe-levels ```
diff --git a/src/foundation/uniqueness-truncation.lagda.md b/src/foundation/uniqueness-truncation.lagda.md index 3b8a63eb08..78d89d7df5 100644 --- a/src/foundation/uniqueness-truncation.lagda.md +++ b/src/foundation/uniqueness-truncation.lagda.md @@ -7,11 +7,12 @@ module foundation.uniqueness-truncation where
Imports ```agda -open import foundation-core.functions +open import foundation.functions +open import foundation.universe-levels + open import foundation-core.homotopies open import foundation-core.truncated-types open import foundation-core.truncation-levels -open import foundation-core.universe-levels ```
diff --git a/src/foundation/unit-type.lagda.md b/src/foundation/unit-type.lagda.md index 14f337f645..0e9bdd076b 100644 --- a/src/foundation/unit-type.lagda.md +++ b/src/foundation/unit-type.lagda.md @@ -7,18 +7,18 @@ module foundation.unit-type where
Imports ```agda +open import foundation.dependent-pair-types open import foundation.raising-universe-levels +open import foundation.universe-levels open import foundation-core.constant-maps open import foundation-core.contractible-types -open import foundation-core.dependent-pair-types open import foundation-core.equivalences open import foundation-core.identity-types open import foundation-core.propositions open import foundation-core.sets open import foundation-core.truncated-types open import foundation-core.truncation-levels -open import foundation-core.universe-levels ```
diff --git a/src/foundation/unital-binary-operations.lagda.md b/src/foundation/unital-binary-operations.lagda.md index ff157ad725..bcb9a54e2b 100644 --- a/src/foundation/unital-binary-operations.lagda.md +++ b/src/foundation/unital-binary-operations.lagda.md @@ -7,11 +7,12 @@ module foundation.unital-binary-operations where
Imports ```agda +open import foundation.dependent-pair-types +open import foundation.universe-levels + open import foundation-core.cartesian-product-types -open import foundation-core.dependent-pair-types open import foundation-core.homotopies open import foundation-core.identity-types -open import foundation-core.universe-levels ```
diff --git a/src/foundation/univalence-action-on-equivalences.lagda.md b/src/foundation/univalence-action-on-equivalences.lagda.md index 036127687b..05067d1a1d 100644 --- a/src/foundation/univalence-action-on-equivalences.lagda.md +++ b/src/foundation/univalence-action-on-equivalences.lagda.md @@ -7,20 +7,20 @@ module foundation.univalence-action-on-equivalences where
Imports ```agda +open import foundation.dependent-pair-types open import foundation.function-extensionality open import foundation.identity-types open import foundation.sets open import foundation.subuniverses open import foundation.univalence +open import foundation.universe-levels open import foundation-core.contractible-types -open import foundation-core.dependent-pair-types open import foundation-core.equality-dependent-pair-types open import foundation-core.equivalences open import foundation-core.injective-maps open import foundation-core.propositions open import foundation-core.subtypes -open import foundation-core.universe-levels ```
diff --git a/src/foundation/univalence-implies-function-extensionality.lagda.md b/src/foundation/univalence-implies-function-extensionality.lagda.md index a8341aaadd..a1aedfb757 100644 --- a/src/foundation/univalence-implies-function-extensionality.lagda.md +++ b/src/foundation/univalence-implies-function-extensionality.lagda.md @@ -7,19 +7,19 @@ module foundation.univalence-implies-function-extensionality where
Imports ```agda +open import foundation.dependent-pair-types open import foundation.equivalence-induction +open import foundation.functions +open import foundation.type-arithmetic-dependent-pair-types +open import foundation.universe-levels open import foundation.weak-function-extensionality open import foundation-core.contractible-maps open import foundation-core.contractible-types -open import foundation-core.dependent-pair-types open import foundation-core.fibers-of-maps open import foundation-core.function-extensionality -open import foundation-core.functions open import foundation-core.homotopies open import foundation-core.identity-types -open import foundation-core.type-arithmetic-dependent-pair-types -open import foundation-core.universe-levels ```
diff --git a/src/foundation/univalence.lagda.md b/src/foundation/univalence.lagda.md index 1dde88411a..63cbdea5ed 100644 --- a/src/foundation/univalence.lagda.md +++ b/src/foundation/univalence.lagda.md @@ -9,18 +9,18 @@ open import foundation-core.univalence public
Imports ```agda +open import foundation.dependent-pair-types open import foundation.equality-dependent-function-types open import foundation.equivalences +open import foundation.functions +open import foundation.fundamental-theorem-of-identity-types +open import foundation.universe-levels open import foundation-core.contractible-types -open import foundation-core.dependent-pair-types -open import foundation-core.functions open import foundation-core.functoriality-dependent-pair-types -open import foundation-core.fundamental-theorem-of-identity-types open import foundation-core.homotopies open import foundation-core.identity-types open import foundation-core.injective-maps -open import foundation-core.universe-levels ```
diff --git a/src/foundation/univalent-type-families.lagda.md b/src/foundation/univalent-type-families.lagda.md index 095e1740dc..f2da3797cb 100644 --- a/src/foundation/univalent-type-families.lagda.md +++ b/src/foundation/univalent-type-families.lagda.md @@ -8,9 +8,9 @@ module foundation.univalent-type-families where ```agda open import foundation.identity-types +open import foundation.universe-levels open import foundation-core.equivalences -open import foundation-core.universe-levels ``` diff --git a/src/foundation/universal-property-booleans.lagda.md b/src/foundation/universal-property-booleans.lagda.md index fa65cd3ad6..267b3dedce 100644 --- a/src/foundation/universal-property-booleans.lagda.md +++ b/src/foundation/universal-property-booleans.lagda.md @@ -8,16 +8,16 @@ module foundation.universal-property-booleans where ```agda open import foundation.booleans +open import foundation.dependent-pair-types +open import foundation.equality-cartesian-product-types open import foundation.function-extensionality +open import foundation.functions +open import foundation.universe-levels open import foundation-core.cartesian-product-types -open import foundation-core.dependent-pair-types -open import foundation-core.equality-cartesian-product-types open import foundation-core.equivalences -open import foundation-core.functions open import foundation-core.homotopies open import foundation-core.identity-types -open import foundation-core.universe-levels ``` diff --git a/src/foundation/universal-property-cartesian-product-types.lagda.md b/src/foundation/universal-property-cartesian-product-types.lagda.md index c57311cd74..d4ffa19392 100644 --- a/src/foundation/universal-property-cartesian-product-types.lagda.md +++ b/src/foundation/universal-property-cartesian-product-types.lagda.md @@ -7,21 +7,21 @@ module foundation.universal-property-cartesian-product-types where
Imports ```agda +open import foundation.cones-over-cospans +open import foundation.dependent-pair-types +open import foundation.functions open import foundation.unit-type +open import foundation.universe-levels open import foundation-core.cartesian-product-types -open import foundation-core.cones-over-cospans open import foundation-core.constant-maps open import foundation-core.contractible-types -open import foundation-core.dependent-pair-types open import foundation-core.equality-dependent-pair-types open import foundation-core.equivalences -open import foundation-core.functions open import foundation-core.homotopies open import foundation-core.identity-types open import foundation-core.pullbacks open import foundation-core.universal-property-pullbacks -open import foundation-core.universe-levels ```
diff --git a/src/foundation/universal-property-coproduct-types.lagda.md b/src/foundation/universal-property-coproduct-types.lagda.md index 6c03900ec4..bf1d98547b 100644 --- a/src/foundation/universal-property-coproduct-types.lagda.md +++ b/src/foundation/universal-property-coproduct-types.lagda.md @@ -7,16 +7,16 @@ module foundation.universal-property-coproduct-types where
Imports ```agda +open import foundation.dependent-pair-types +open import foundation.equality-cartesian-product-types open import foundation.equivalences open import foundation.function-extensionality +open import foundation.functions +open import foundation.universe-levels open import foundation-core.cartesian-product-types open import foundation-core.coproduct-types -open import foundation-core.dependent-pair-types -open import foundation-core.equality-cartesian-product-types -open import foundation-core.functions open import foundation-core.identity-types -open import foundation-core.universe-levels ```
diff --git a/src/foundation/universal-property-dependent-pair-types.lagda.md b/src/foundation/universal-property-dependent-pair-types.lagda.md index 7be23ce643..29487b71cb 100644 --- a/src/foundation/universal-property-dependent-pair-types.lagda.md +++ b/src/foundation/universal-property-dependent-pair-types.lagda.md @@ -7,13 +7,13 @@ module foundation.universal-property-dependent-pair-types where
Imports ```agda +open import foundation.dependent-pair-types open import foundation.function-extensionality +open import foundation.universe-levels -open import foundation-core.dependent-pair-types open import foundation-core.equivalences open import foundation-core.homotopies open import foundation-core.identity-types -open import foundation-core.universe-levels ```
diff --git a/src/foundation/universal-property-empty-type.lagda.md b/src/foundation/universal-property-empty-type.lagda.md index 70ea5c4014..d178d004f8 100644 --- a/src/foundation/universal-property-empty-type.lagda.md +++ b/src/foundation/universal-property-empty-type.lagda.md @@ -7,14 +7,14 @@ module foundation.universal-property-empty-type where
Imports ```agda +open import foundation.dependent-pair-types open import foundation.equivalences open import foundation.function-extensionality +open import foundation.functions +open import foundation.universe-levels open import foundation-core.contractible-types -open import foundation-core.dependent-pair-types open import foundation-core.empty-types -open import foundation-core.functions -open import foundation-core.universe-levels ```
diff --git a/src/foundation/universal-property-fiber-products.lagda.md b/src/foundation/universal-property-fiber-products.lagda.md index 000f5038e7..fff71c9a53 100644 --- a/src/foundation/universal-property-fiber-products.lagda.md +++ b/src/foundation/universal-property-fiber-products.lagda.md @@ -7,20 +7,21 @@ module foundation.universal-property-fiber-products where
Imports ```agda +open import foundation.cones-over-cospans +open import foundation.dependent-pair-types +open import foundation.equality-cartesian-product-types +open import foundation.functions +open import foundation.universe-levels + open import foundation-core.cartesian-product-types -open import foundation-core.cones-over-cospans -open import foundation-core.dependent-pair-types -open import foundation-core.equality-cartesian-product-types open import foundation-core.equality-dependent-pair-types open import foundation-core.equivalences open import foundation-core.fibers-of-maps -open import foundation-core.functions open import foundation-core.functoriality-dependent-pair-types open import foundation-core.homotopies open import foundation-core.identity-types open import foundation-core.pullbacks open import foundation-core.universal-property-pullbacks -open import foundation-core.universe-levels ```
diff --git a/src/foundation/universal-property-identity-types.lagda.md b/src/foundation/universal-property-identity-types.lagda.md index 1e45ef5d25..4a067d11c3 100644 --- a/src/foundation/universal-property-identity-types.lagda.md +++ b/src/foundation/universal-property-identity-types.lagda.md @@ -7,13 +7,13 @@ module foundation.universal-property-identity-types where
Imports ```agda +open import foundation.dependent-pair-types open import foundation.function-extensionality open import foundation.identity-types +open import foundation.universe-levels -open import foundation-core.dependent-pair-types open import foundation-core.equivalences open import foundation-core.functoriality-dependent-function-types -open import foundation-core.universe-levels ```
diff --git a/src/foundation/universal-property-image.lagda.md b/src/foundation/universal-property-image.lagda.md index d3ce937f3c..99f2c23a36 100644 --- a/src/foundation/universal-property-image.lagda.md +++ b/src/foundation/universal-property-image.lagda.md @@ -7,19 +7,20 @@ module foundation.universal-property-image where
Imports ```agda +open import foundation.dependent-pair-types open import foundation.embeddings +open import foundation.functions open import foundation.identity-types open import foundation.images open import foundation.propositional-truncations open import foundation.slice open import foundation.surjective-maps +open import foundation.universe-levels open import foundation-core.contractible-maps open import foundation-core.contractible-types -open import foundation-core.dependent-pair-types open import foundation-core.equivalences open import foundation-core.fibers-of-maps -open import foundation-core.functions open import foundation-core.functoriality-dependent-function-types open import foundation-core.functoriality-dependent-pair-types open import foundation-core.homotopies @@ -28,7 +29,6 @@ open import foundation-core.propositional-maps open import foundation-core.propositions open import foundation-core.sections open import foundation-core.subtypes -open import foundation-core.universe-levels ```
diff --git a/src/foundation/universal-property-maybe.lagda.md b/src/foundation/universal-property-maybe.lagda.md index e9cd5677d1..6644e2b569 100644 --- a/src/foundation/universal-property-maybe.lagda.md +++ b/src/foundation/universal-property-maybe.lagda.md @@ -7,17 +7,17 @@ module foundation.universal-property-maybe where
Imports ```agda +open import foundation.dependent-pair-types open import foundation.function-extensionality +open import foundation.functions open import foundation.maybe +open import foundation.universe-levels open import foundation-core.cartesian-product-types open import foundation-core.coproduct-types -open import foundation-core.dependent-pair-types open import foundation-core.equivalences -open import foundation-core.functions open import foundation-core.homotopies open import foundation-core.identity-types -open import foundation-core.universe-levels ```
diff --git a/src/foundation/universal-property-propositional-truncation-into-sets.lagda.md b/src/foundation/universal-property-propositional-truncation-into-sets.lagda.md index eaa1e4ba72..bccc688bff 100644 --- a/src/foundation/universal-property-propositional-truncation-into-sets.lagda.md +++ b/src/foundation/universal-property-propositional-truncation-into-sets.lagda.md @@ -7,20 +7,20 @@ module foundation.universal-property-propositional-truncation-into-sets where
Imports ```agda +open import foundation.dependent-pair-types open import foundation.function-extensionality +open import foundation.functions open import foundation.propositional-truncations +open import foundation.universe-levels open import foundation.weakly-constant-maps -open import foundation-core.dependent-pair-types open import foundation-core.equivalences open import foundation-core.fibers-of-maps -open import foundation-core.functions open import foundation-core.homotopies open import foundation-core.identity-types open import foundation-core.propositions open import foundation-core.sets open import foundation-core.subtypes -open import foundation-core.universe-levels ```
diff --git a/src/foundation/universal-property-propositional-truncation.lagda.md b/src/foundation/universal-property-propositional-truncation.lagda.md index 7b3fbcdb6f..209b1c3cd2 100644 --- a/src/foundation/universal-property-propositional-truncation.lagda.md +++ b/src/foundation/universal-property-propositional-truncation.lagda.md @@ -7,24 +7,24 @@ module foundation.universal-property-propositional-truncation where
Imports ```agda +open import foundation.dependent-pair-types open import foundation.equivalences open import foundation.function-extensionality +open import foundation.functions open import foundation.functoriality-cartesian-product-types +open import foundation.subtype-identity-principle open import foundation.type-theoretic-principle-of-choice open import foundation.unit-type open import foundation.universal-property-dependent-pair-types +open import foundation.universe-levels open import foundation-core.contractible-maps open import foundation-core.contractible-types -open import foundation-core.dependent-pair-types -open import foundation-core.functions open import foundation-core.functoriality-dependent-function-types open import foundation-core.functoriality-dependent-pair-types open import foundation-core.homotopies open import foundation-core.identity-types open import foundation-core.propositions -open import foundation-core.subtype-identity-principle -open import foundation-core.universe-levels ```
diff --git a/src/foundation/universal-property-pullbacks.lagda.md b/src/foundation/universal-property-pullbacks.lagda.md index 2d4dccc1a5..08c1058685 100644 --- a/src/foundation/universal-property-pullbacks.lagda.md +++ b/src/foundation/universal-property-pullbacks.lagda.md @@ -9,15 +9,15 @@ open import foundation-core.universal-property-pullbacks public
Imports ```agda +open import foundation.cones-over-cospans +open import foundation.dependent-pair-types open import foundation.equivalences +open import foundation.subtype-identity-principle +open import foundation.universe-levels -open import foundation-core.cones-over-cospans open import foundation-core.contractible-types -open import foundation-core.dependent-pair-types open import foundation-core.identity-types open import foundation-core.propositions -open import foundation-core.subtype-identity-principle -open import foundation-core.universe-levels ```
diff --git a/src/foundation/universal-property-set-quotients.lagda.md b/src/foundation/universal-property-set-quotients.lagda.md index 481c8b4bac..588f600d00 100644 --- a/src/foundation/universal-property-set-quotients.lagda.md +++ b/src/foundation/universal-property-set-quotients.lagda.md @@ -9,11 +9,14 @@ module foundation.universal-property-set-quotients where
Imports ```agda +open import foundation.dependent-pair-types open import foundation.effective-maps-equivalence-relations open import foundation.epimorphisms-with-respect-to-sets open import foundation.equivalence-classes open import foundation.existential-quantification open import foundation.function-extensionality +open import foundation.functions +open import foundation.fundamental-theorem-of-identity-types open import foundation.identity-types open import foundation.images open import foundation.locally-small-types @@ -23,18 +26,16 @@ open import foundation.reflecting-maps-equivalence-relations open import foundation.sets open import foundation.surjective-maps open import foundation.universal-property-image +open import foundation.universe-levels open import foundation-core.cartesian-product-types open import foundation-core.contractible-maps open import foundation-core.contractible-types -open import foundation-core.dependent-pair-types open import foundation-core.embeddings open import foundation-core.equivalence-relations open import foundation-core.equivalences open import foundation-core.fibers-of-maps -open import foundation-core.functions open import foundation-core.functoriality-dependent-pair-types -open import foundation-core.fundamental-theorem-of-identity-types open import foundation-core.homotopies open import foundation-core.injective-maps open import foundation-core.propositional-maps @@ -42,7 +43,6 @@ open import foundation-core.propositions open import foundation-core.small-types open import foundation-core.subtypes open import foundation-core.univalence -open import foundation-core.universe-levels ```
diff --git a/src/foundation/universal-property-set-truncation.lagda.md b/src/foundation/universal-property-set-truncation.lagda.md index ca715ea065..d76ce9015f 100644 --- a/src/foundation/universal-property-set-truncation.lagda.md +++ b/src/foundation/universal-property-set-truncation.lagda.md @@ -7,24 +7,24 @@ module foundation.universal-property-set-truncation where
Imports ```agda +open import foundation.dependent-pair-types open import foundation.equivalences open import foundation.function-extensionality +open import foundation.functions open import foundation.mere-equality open import foundation.reflecting-maps-equivalence-relations open import foundation.sets +open import foundation.type-arithmetic-dependent-pair-types open import foundation.type-theoretic-principle-of-choice open import foundation.universal-property-set-quotients +open import foundation.universe-levels open import foundation-core.contractible-maps open import foundation-core.contractible-types -open import foundation-core.dependent-pair-types -open import foundation-core.functions open import foundation-core.functoriality-dependent-pair-types open import foundation-core.homotopies open import foundation-core.identity-types open import foundation-core.propositions -open import foundation-core.type-arithmetic-dependent-pair-types -open import foundation-core.universe-levels ```
diff --git a/src/foundation/universal-property-truncation.lagda.md b/src/foundation/universal-property-truncation.lagda.md index abb3a6ce52..63a5d1d63b 100644 --- a/src/foundation/universal-property-truncation.lagda.md +++ b/src/foundation/universal-property-truncation.lagda.md @@ -10,6 +10,7 @@ open import foundation-core.universal-property-truncation public ```agda open import foundation.contractible-types +open import foundation.dependent-pair-types open import foundation.function-extensionality open import foundation.identity-types open import foundation.propositional-truncations @@ -18,16 +19,15 @@ open import foundation.type-arithmetic-dependent-function-types open import foundation.type-theoretic-principle-of-choice open import foundation.universal-property-dependent-pair-types open import foundation.universal-property-identity-types +open import foundation.universe-levels open import foundation-core.contractible-maps -open import foundation-core.dependent-pair-types open import foundation-core.equivalences open import foundation-core.fibers-of-maps open import foundation-core.functoriality-dependent-function-types open import foundation-core.functoriality-dependent-pair-types open import foundation-core.truncated-types open import foundation-core.truncation-levels -open import foundation-core.universe-levels ``` diff --git a/src/foundation/universal-property-unit-type.lagda.md b/src/foundation/universal-property-unit-type.lagda.md index c1d3847aa7..f736576930 100644 --- a/src/foundation/universal-property-unit-type.lagda.md +++ b/src/foundation/universal-property-unit-type.lagda.md @@ -8,14 +8,14 @@ module foundation.universal-property-unit-type where ```agda open import foundation.contractible-types +open import foundation.dependent-pair-types open import foundation.equivalences +open import foundation.functions open import foundation.unit-type +open import foundation.universe-levels open import foundation-core.constant-maps -open import foundation-core.dependent-pair-types -open import foundation-core.functions open import foundation-core.homotopies -open import foundation-core.universe-levels ``` diff --git a/src/foundation/universe-levels.lagda.md b/src/foundation/universe-levels.lagda.md index 785a6b7e65..f3f99c1253 100644 --- a/src/foundation/universe-levels.lagda.md +++ b/src/foundation/universe-levels.lagda.md @@ -5,13 +5,14 @@ module foundation.universe-levels where -open import foundation-core.universe-levels public +open import Agda.Primitive + using (Level ; lzero ; lsuc ; _⊔_) + renaming (Set to UU ; Setω to UUω) + public ``` -
Imports +## Idea -```agda - -``` - -
+We import Agda's built in mechanism of universe levels. The universes are called +`UU`, which stands for _univalent universe_, although we will not immediately +assume that universes are univalent. diff --git a/src/foundation/unordered-pairs-of-types.lagda.md b/src/foundation/unordered-pairs-of-types.lagda.md index d655b9035c..6819cbe635 100644 --- a/src/foundation/unordered-pairs-of-types.lagda.md +++ b/src/foundation/unordered-pairs-of-types.lagda.md @@ -7,16 +7,16 @@ module foundation.unordered-pairs-of-types where
Imports ```agda +open import foundation.dependent-pair-types +open import foundation.fundamental-theorem-of-identity-types open import foundation.structure-identity-principle open import foundation.univalence +open import foundation.universe-levels open import foundation.unordered-pairs open import foundation-core.contractible-types -open import foundation-core.dependent-pair-types open import foundation-core.equivalences -open import foundation-core.fundamental-theorem-of-identity-types open import foundation-core.identity-types -open import foundation-core.universe-levels open import univalent-combinatorics.2-element-types ``` diff --git a/src/foundation/unordered-pairs.lagda.md b/src/foundation/unordered-pairs.lagda.md index e4094152e8..e7273187e8 100644 --- a/src/foundation/unordered-pairs.lagda.md +++ b/src/foundation/unordered-pairs.lagda.md @@ -8,26 +8,26 @@ module foundation.unordered-pairs where ```agda open import foundation.decidable-equality +open import foundation.dependent-pair-types open import foundation.existential-quantification open import foundation.function-extensionality +open import foundation.functions +open import foundation.fundamental-theorem-of-identity-types open import foundation.homotopies open import foundation.mere-equivalences open import foundation.propositional-truncations open import foundation.structure-identity-principle +open import foundation.universe-levels open import foundation-core.contractible-types open import foundation-core.coproduct-types -open import foundation-core.dependent-pair-types open import foundation-core.embeddings open import foundation-core.equivalences -open import foundation-core.functions open import foundation-core.functoriality-dependent-pair-types open import foundation-core.functoriality-function-types -open import foundation-core.fundamental-theorem-of-identity-types open import foundation-core.identity-types open import foundation-core.propositions open import foundation-core.sets -open import foundation-core.universe-levels open import univalent-combinatorics.2-element-types open import univalent-combinatorics.equality-standard-finite-types diff --git a/src/foundation/unordered-tuples-of-types.lagda.md b/src/foundation/unordered-tuples-of-types.lagda.md index 10e1c2bf53..ded645fd43 100644 --- a/src/foundation/unordered-tuples-of-types.lagda.md +++ b/src/foundation/unordered-tuples-of-types.lagda.md @@ -9,16 +9,16 @@ module foundation.unordered-tuples-of-types where ```agda open import elementary-number-theory.natural-numbers +open import foundation.dependent-pair-types +open import foundation.fundamental-theorem-of-identity-types open import foundation.structure-identity-principle open import foundation.univalence +open import foundation.universe-levels open import foundation.unordered-tuples open import foundation-core.contractible-types -open import foundation-core.dependent-pair-types open import foundation-core.equivalences -open import foundation-core.fundamental-theorem-of-identity-types open import foundation-core.identity-types -open import foundation-core.universe-levels open import univalent-combinatorics.finite-types ``` diff --git a/src/foundation/unordered-tuples.lagda.md b/src/foundation/unordered-tuples.lagda.md index 416ca2e554..da9ffc8820 100644 --- a/src/foundation/unordered-tuples.lagda.md +++ b/src/foundation/unordered-tuples.lagda.md @@ -10,19 +10,19 @@ module foundation.unordered-tuples where open import elementary-number-theory.natural-numbers open import foundation.decidable-equality +open import foundation.dependent-pair-types +open import foundation.functions +open import foundation.fundamental-theorem-of-identity-types open import foundation.homotopies open import foundation.structure-identity-principle +open import foundation.universe-levels open import foundation-core.contractible-types -open import foundation-core.dependent-pair-types open import foundation-core.equivalences -open import foundation-core.functions open import foundation-core.functoriality-dependent-pair-types open import foundation-core.functoriality-function-types -open import foundation-core.fundamental-theorem-of-identity-types open import foundation-core.identity-types open import foundation-core.sets -open import foundation-core.universe-levels open import univalent-combinatorics.complements-isolated-points open import univalent-combinatorics.equality-finite-types diff --git a/src/foundation/vectors-set-quotients.lagda.md b/src/foundation/vectors-set-quotients.lagda.md index 87d2c93357..f954662a7a 100644 --- a/src/foundation/vectors-set-quotients.lagda.md +++ b/src/foundation/vectors-set-quotients.lagda.md @@ -12,7 +12,10 @@ module foundation.vectors-set-quotients where open import elementary-number-theory.natural-numbers open import foundation.cartesian-products-set-quotients +open import foundation.dependent-pair-types +open import foundation.equality-cartesian-product-types open import foundation.function-extensionality +open import foundation.functions open import foundation.multivariable-operations open import foundation.products-equivalence-relations open import foundation.raising-universe-levels @@ -21,21 +24,18 @@ open import foundation.set-quotients open import foundation.sets open import foundation.unit-type open import foundation.universal-property-set-quotients +open import foundation.universe-levels open import foundation-core.cartesian-product-types open import foundation-core.coproduct-types -open import foundation-core.dependent-pair-types -open import foundation-core.equality-cartesian-product-types open import foundation-core.equality-dependent-pair-types open import foundation-core.equivalence-relations open import foundation-core.equivalences -open import foundation-core.functions open import foundation-core.homotopies open import foundation-core.identity-types open import foundation-core.propositions open import foundation-core.retractions open import foundation-core.sections -open import foundation-core.universe-levels open import linear-algebra.vectors diff --git a/src/foundation/weak-function-extensionality.lagda.md b/src/foundation/weak-function-extensionality.lagda.md index 500ab36d6c..2be687323b 100644 --- a/src/foundation/weak-function-extensionality.lagda.md +++ b/src/foundation/weak-function-extensionality.lagda.md @@ -9,18 +9,18 @@ module foundation.weak-function-extensionality where ```agda open import foundation.decidable-equality open import foundation.decidable-types +open import foundation.dependent-pair-types +open import foundation.fundamental-theorem-of-identity-types +open import foundation.universe-levels open import foundation-core.contractible-types open import foundation-core.coproduct-types -open import foundation-core.dependent-pair-types open import foundation-core.empty-types open import foundation-core.equality-dependent-pair-types open import foundation-core.equivalences open import foundation-core.function-extensionality -open import foundation-core.fundamental-theorem-of-identity-types open import foundation-core.identity-types open import foundation-core.propositions -open import foundation-core.universe-levels ```
diff --git a/src/foundation/weak-limited-principle-of-omniscience.lagda.md b/src/foundation/weak-limited-principle-of-omniscience.lagda.md index 93d5f267b4..8e5480c9da 100644 --- a/src/foundation/weak-limited-principle-of-omniscience.lagda.md +++ b/src/foundation/weak-limited-principle-of-omniscience.lagda.md @@ -11,10 +11,10 @@ open import elementary-number-theory.natural-numbers open import foundation.disjunction open import foundation.negation +open import foundation.universe-levels open import foundation-core.propositions open import foundation-core.sets -open import foundation-core.universe-levels open import univalent-combinatorics.standard-finite-types ``` diff --git a/src/foundation/weakly-constant-maps.lagda.md b/src/foundation/weakly-constant-maps.lagda.md index ebc7cfd45e..420bc3e44f 100644 --- a/src/foundation/weakly-constant-maps.lagda.md +++ b/src/foundation/weakly-constant-maps.lagda.md @@ -7,11 +7,12 @@ module foundation.weakly-constant-maps where
Imports ```agda -open import foundation-core.dependent-pair-types +open import foundation.dependent-pair-types +open import foundation.universe-levels + open import foundation-core.identity-types open import foundation-core.propositions open import foundation-core.sets -open import foundation-core.universe-levels ```
diff --git a/src/orthogonal-factorization-systems/local-types.lagda.md b/src/orthogonal-factorization-systems/local-types.lagda.md index bd8d3819d2..ad83346db1 100644 --- a/src/orthogonal-factorization-systems/local-types.lagda.md +++ b/src/orthogonal-factorization-systems/local-types.lagda.md @@ -7,24 +7,24 @@ module orthogonal-factorization-systems.local-types where
Imports ```agda +open import foundation.dependent-pair-types open import foundation.equivalences +open import foundation.functions open import foundation.type-arithmetic-dependent-function-types open import foundation.type-arithmetic-unit-type open import foundation.unit-type open import foundation.universal-property-empty-type +open import foundation.universe-levels open import foundation-core.contractible-maps open import foundation-core.contractible-types -open import foundation-core.dependent-pair-types open import foundation-core.empty-types open import foundation-core.function-extensionality -open import foundation-core.functions open import foundation-core.functoriality-dependent-function-types open import foundation-core.identity-types open import foundation-core.propositions open import foundation-core.retractions open import foundation-core.sections -open import foundation-core.universe-levels ```
diff --git a/src/reflection/boolean-reflection.lagda.md b/src/reflection/boolean-reflection.lagda.md index 70840f2184..6908c83cf5 100644 --- a/src/reflection/boolean-reflection.lagda.md +++ b/src/reflection/boolean-reflection.lagda.md @@ -9,11 +9,11 @@ module reflection.boolean-reflection where ```agda open import foundation.booleans open import foundation.decidable-types +open import foundation.universe-levels open import foundation-core.coproduct-types open import foundation-core.empty-types open import foundation-core.identity-types -open import foundation-core.universe-levels ``` diff --git a/src/reflection/type-checking-monad.lagda.md b/src/reflection/type-checking-monad.lagda.md index fd0b450434..4220ac6b1f 100644 --- a/src/reflection/type-checking-monad.lagda.md +++ b/src/reflection/type-checking-monad.lagda.md @@ -14,12 +14,11 @@ open import elementary-number-theory.natural-numbers open import foundation.booleans open import foundation.cartesian-product-types +open import foundation.dependent-pair-types open import foundation.identity-types open import foundation.unit-type open import foundation.universe-levels -open import foundation-core.dependent-pair-types - open import lists.lists open import primitives.strings diff --git a/src/set-theory/countable-sets.lagda.md b/src/set-theory/countable-sets.lagda.md index dbb802a3cb..f061f9d49c 100644 --- a/src/set-theory/countable-sets.lagda.md +++ b/src/set-theory/countable-sets.lagda.md @@ -15,9 +15,11 @@ open import foundation.coproduct-types open import foundation.decidable-propositions open import foundation.decidable-subtypes open import foundation.decidable-types +open import foundation.dependent-pair-types open import foundation.empty-types open import foundation.equality-coproduct-types open import foundation.existential-quantification +open import foundation.functions open import foundation.functoriality-cartesian-product-types open import foundation.functoriality-coproduct-types open import foundation.maybe @@ -32,9 +34,7 @@ open import foundation.unit-type open import foundation.universe-levels open import foundation-core.cartesian-product-types -open import foundation-core.dependent-pair-types open import foundation-core.fibers-of-maps -open import foundation-core.functions open import foundation-core.identity-types open import univalent-combinatorics.standard-finite-types diff --git a/src/structured-types/commuting-squares-of-pointed-maps.lagda.md b/src/structured-types/commuting-squares-of-pointed-maps.lagda.md index 74c2702ec7..a49e3d8c0b 100644 --- a/src/structured-types/commuting-squares-of-pointed-maps.lagda.md +++ b/src/structured-types/commuting-squares-of-pointed-maps.lagda.md @@ -7,7 +7,7 @@ module structured-types.commuting-squares-of-pointed-maps where
Imports ```agda -open import foundation-core.universe-levels +open import foundation.universe-levels open import structured-types.pointed-homotopies open import structured-types.pointed-maps diff --git a/src/structured-types/types-equipped-with-automorphisms.lagda.md b/src/structured-types/types-equipped-with-automorphisms.lagda.md index 2606d8b2c8..d32a40d763 100644 --- a/src/structured-types/types-equipped-with-automorphisms.lagda.md +++ b/src/structured-types/types-equipped-with-automorphisms.lagda.md @@ -2,17 +2,15 @@ ```agda module structured-types.types-equipped-with-automorphisms where - -open import foundation.universe-levels -open import foundation.dependent-pair-types -open import foundation.equivalences -open import foundation.automorphisms ```
Imports ```agda - +open import foundation.automorphisms +open import foundation.dependent-pair-types +open import foundation.equivalences +open import foundation.universe-levels ```
diff --git a/src/synthetic-homotopy-theory/joins-of-types.lagda.md b/src/synthetic-homotopy-theory/joins-of-types.lagda.md index 3bea88dcea..eb8458f791 100644 --- a/src/synthetic-homotopy-theory/joins-of-types.lagda.md +++ b/src/synthetic-homotopy-theory/joins-of-types.lagda.md @@ -13,6 +13,7 @@ open import foundation.dependent-pair-types open import foundation.disjunction open import foundation.empty-types open import foundation.equivalences +open import foundation.functions open import foundation.identity-types open import foundation.propositions open import foundation.type-arithmetic-cartesian-product-types @@ -22,8 +23,6 @@ open import foundation.type-arithmetic-unit-type open import foundation.unit-type open import foundation.universe-levels -open import foundation-core.functions - open import synthetic-homotopy-theory.cocones-under-spans open import synthetic-homotopy-theory.pushouts open import synthetic-homotopy-theory.universal-property-pushouts