From 8d0adf3c9740be596d510a103752545d70b9ea95 Mon Sep 17 00:00:00 2001 From: Egbert Rijke Date: Sat, 10 Jun 2023 23:53:17 +0200 Subject: [PATCH] cleaning up transport and dependent identifications files (#650) - correcting typos in the tables of files related to identity types - Moving facts about transport from `foundation.identity-types` to `foundation.transport` - Renaming dependent paths to dependent identifications - Renaming path-overs to dependent identifications - Factoring out the computations of transport to their specific files, eg transport in cartesian products to cartesian products, transport in identity types to identity types, and so on. - Moving `functions` to `function-types` to more clearly support both operations on function types as well as other lemmas about function types. - Cleaning up dependent identifications file. There is still some conceptual work to be done here. Refactoring this is highly nontrivial. --- .../functors-large-precategories.lagda.md | 2 +- .../functors-precategories.lagda.md | 2 +- src/category-theory/groupoids.lagda.md | 2 +- .../isomorphisms-precategories.lagda.md | 2 +- ...atural-isomorphisms-precategories.lagda.md | 2 +- ...ral-transformations-precategories.lagda.md | 2 +- src/category-theory/precategories.lagda.md | 2 +- .../slice-precategories.lagda.md | 2 +- ...-radical-ideals-commutative-rings.lagda.md | 2 +- ...-radical-ideals-commutative-rings.lagda.md | 2 +- .../prime-ideals-commutative-rings.lagda.md | 2 +- ...icals-of-ideals-commutative-rings.lagda.md | 2 +- .../sums-commutative-rings.lagda.md | 2 +- .../sums-commutative-semirings.lagda.md | 2 +- .../absolute-value-integers.lagda.md | 2 +- .../addition-integers.lagda.md | 2 +- .../addition-natural-numbers.lagda.md | 2 +- .../based-induction-natural-numbers.lagda.md | 2 +- ...-strong-induction-natural-numbers.lagda.md | 2 +- .../bezouts-lemma-integers.lagda.md | 2 +- ...bounded-sums-arithmetic-functions.lagda.md | 2 +- .../congruence-integers.lagda.md | 2 +- .../decidable-types.lagda.md | 2 +- .../distance-natural-numbers.lagda.md | 2 +- .../divisibility-integers.lagda.md | 2 +- .../equality-integers.lagda.md | 2 +- .../equality-natural-numbers.lagda.md | 2 +- .../finitary-natural-numbers.lagda.md | 2 +- .../finitely-cyclic-maps.lagda.md | 2 +- .../greatest-common-divisor-integers.lagda.md | 2 +- .../inequality-integers.lagda.md | 2 +- .../inequality-natural-numbers.lagda.md | 2 +- .../initial-segments-natural-numbers.lagda.md | 2 +- .../integers.lagda.md | 2 +- ...-arithmetic-standard-finite-types.lagda.md | 2 +- .../modular-arithmetic.lagda.md | 2 +- .../multiplication-integers.lagda.md | 2 +- .../natural-numbers.lagda.md | 2 +- .../parity-natural-numbers.lagda.md | 2 +- .../powers-of-two.lagda.md | 2 +- .../sieve-of-eratosthenes.lagda.md | 2 +- ...strict-inequality-natural-numbers.lagda.md | 2 +- ...-ordered-pairs-of-natural-numbers.lagda.md | 2 +- .../strong-induction-natural-numbers.lagda.md | 2 +- .../sums-of-natural-numbers.lagda.md | 2 +- .../type-arithmetic-natural-numbers.lagda.md | 2 +- ...niversal-property-natural-numbers.lagda.md | 2 +- ...g-principle-standard-finite-types.lagda.md | 2 +- ...products-commutative-finite-rings.lagda.md | 2 +- .../dependent-products-finite-rings.lagda.md | 2 +- ...misimple-commutative-finite-rings.lagda.md | 2 +- .../abstract-quaternion-group.lagda.md | 2 +- .../concrete-quaternion-group.lagda.md | 21 +- .../delooping-sign-homomorphism.lagda.md | 2 +- .../finite-groups.lagda.md | 2 +- .../finite-semigroups.lagda.md | 2 +- .../finite-type-groups.lagda.md | 2 +- .../orbits-permutations.lagda.md | 2 +- ...ermutations-standard-finite-types.lagda.md | 2 +- src/finite-group-theory/permutations.lagda.md | 2 +- .../sign-homomorphism.lagda.md | 2 +- ...mpson-delooping-sign-homomorphism.lagda.md | 2 +- ...nspositions-standard-finite-types.lagda.md | 34 +- .../transpositions.lagda.md | 2 +- src/foundation-core.lagda.md | 4 +- .../cartesian-product-types.lagda.md | 3 +- .../coherently-invertible-maps.lagda.md | 4 +- .../commuting-cubes-of-maps.lagda.md | 2 +- .../commuting-squares-of-maps.lagda.md | 2 +- .../commuting-triangles-of-maps.lagda.md | 2 +- .../contractible-maps.lagda.md | 2 +- .../decidable-propositions.lagda.md | 2 +- ....md => dependent-identifications.lagda.md} | 14 +- .../dependent-pair-types.lagda.md | 76 +++ .../diagonal-maps-of-types.lagda.md | 2 +- src/foundation-core/embeddings.lagda.md | 2 +- src/foundation-core/empty-types.lagda.md | 2 +- src/foundation-core/endomorphisms.lagda.md | 2 +- .../equality-dependent-pair-types.lagda.md | 4 +- .../equivalence-induction.lagda.md | 2 +- src/foundation-core/equivalences.lagda.md | 4 +- src/foundation-core/fibers-of-maps.lagda.md | 2 +- .../function-extensionality.lagda.md | 2 +- .../function-types.lagda.md} | 4 +- ...oriality-dependent-function-types.lagda.md | 2 +- ...unctoriality-dependent-pair-types.lagda.md | 2 +- .../functoriality-function-types.lagda.md | 2 +- src/foundation-core/homotopies.lagda.md | 25 +- src/foundation-core/identity-types.lagda.md | 10 +- src/foundation-core/injective-maps.lagda.md | 2 +- src/foundation-core/involutions.lagda.md | 2 +- .../logical-equivalences.lagda.md | 2 +- src/foundation-core/path-split-maps.lagda.md | 2 +- src/foundation-core/propositions.lagda.md | 2 +- src/foundation-core/pullbacks.lagda.md | 2 +- src/foundation-core/retractions.lagda.md | 4 +- src/foundation-core/sections.lagda.md | 4 +- .../singleton-induction.lagda.md | 2 +- src/foundation-core/small-types.lagda.md | 1 + src/foundation-core/subtypes.lagda.md | 2 +- src/foundation-core/transport.lagda.md | 38 +- src/foundation-core/truncated-maps.lagda.md | 2 +- .../universal-property-pullbacks.lagda.md | 2 +- .../universal-property-truncation.lagda.md | 2 +- src/foundation.lagda.md | 4 +- src/foundation/0-connected-types.lagda.md | 2 +- src/foundation/0-maps.lagda.md | 2 +- ...tion-on-identifications-functions.lagda.md | 14 +- ...oproduct-and-sigma-decompositions.lagda.md | 2 +- ...law-product-and-pi-decompositions.lagda.md | 2 +- src/foundation/axiom-l.lagda.md | 2 +- src/foundation/axiom-of-choice.lagda.md | 2 +- ...valences-unordered-pairs-of-types.lagda.md | 2 +- ...inary-functoriality-set-quotients.lagda.md | 2 +- src/foundation/binary-transport.lagda.md | 2 +- src/foundation/booleans.lagda.md | 2 +- .../cartesian-product-types.lagda.md | 16 +- .../cartesian-products-set-quotients.lagda.md | 2 +- src/foundation/category-of-sets.lagda.md | 2 +- .../coherently-invertible-maps.lagda.md | 2 +- .../commuting-cubes-of-maps.lagda.md | 2 +- ...muting-squares-of-identifications.lagda.md | 2 +- .../commuting-squares-of-maps.lagda.md | 2 +- ...commuting-triangles-of-homotopies.lagda.md | 2 +- src/foundation/complements-subtypes.lagda.md | 2 +- src/foundation/complements.lagda.md | 2 +- src/foundation/cones-over-cospans.lagda.md | 2 +- src/foundation/connected-maps.lagda.md | 2 +- src/foundation/connected-types.lagda.md | 2 +- src/foundation/constant-maps.lagda.md | 2 +- .../constant-type-families.lagda.md | 4 +- src/foundation/contractible-types.lagda.md | 2 +- ...roduct-decompositions-subuniverse.lagda.md | 2 +- .../coproduct-decompositions.lagda.md | 8 +- src/foundation/coproduct-types.lagda.md | 2 +- src/foundation/coslice.lagda.md | 2 +- .../decidable-dependent-pair-types.lagda.md | 2 +- src/foundation/decidable-embeddings.lagda.md | 2 +- .../decidable-equivalence-relations.lagda.md | 2 +- src/foundation/decidable-maps.lagda.md | 2 +- .../decidable-propositions.lagda.md | 2 +- src/foundation/decidable-subtypes.lagda.md | 2 +- src/foundation/decidable-types.lagda.md | 2 +- .../dependent-binomial-theorem.lagda.md | 3 +- .../dependent-identifications.lagda.md | 466 ++++++++++++++++++ src/foundation/dependent-pair-types.lagda.md | 70 +-- src/foundation/dependent-paths.lagda.md | 380 -------------- .../descent-coproduct-types.lagda.md | 2 +- .../descent-dependent-pair-types.lagda.md | 2 +- src/foundation/descent-equivalences.lagda.md | 2 +- src/foundation/diagonals-of-maps.lagda.md | 2 +- ...rete-relaxed-sigma-decompositions.lagda.md | 2 +- .../discrete-sigma-decompositions.lagda.md | 2 +- src/foundation/disjunction.lagda.md | 2 +- .../double-negation-modality.lagda.md | 2 +- src/foundation/double-negation.lagda.md | 2 +- src/foundation/embeddings.lagda.md | 2 +- src/foundation/empty-types.lagda.md | 2 +- src/foundation/endomorphisms.lagda.md | 2 +- ...s-with-respect-to-truncated-types.lagda.md | 2 +- src/foundation/epimorphisms.lagda.md | 2 +- .../equality-cartesian-product-types.lagda.md | 43 +- .../equality-coproduct-types.lagda.md | 2 +- .../equality-dependent-pair-types.lagda.md | 6 +- .../equality-fibers-of-maps.lagda.md | 4 +- .../equivalence-extensionality.lagda.md | 2 +- src/foundation/equivalence-induction.lagda.md | 2 +- src/foundation/equivalence-relations.lagda.md | 2 +- src/foundation/equivalences-maybe.lagda.md | 2 +- src/foundation/equivalences.lagda.md | 2 +- .../exponents-set-quotients.lagda.md | 2 +- src/foundation/faithful-maps.lagda.md | 2 +- src/foundation/fiber-inclusions.lagda.md | 1 + src/foundation/fibered-equivalences.lagda.md | 2 +- src/foundation/fibered-involutions.lagda.md | 2 +- src/foundation/fibered-maps.lagda.md | 2 +- src/foundation/fibers-of-maps.lagda.md | 2 +- .../function-extensionality.lagda.md | 2 +- src/foundation/function-types.lagda.md | 69 +++ ...toriality-cartesian-product-types.lagda.md | 2 +- .../functoriality-coproduct-types.lagda.md | 2 +- ...oriality-dependent-function-types.lagda.md | 4 +- ...unctoriality-dependent-pair-types.lagda.md | 2 +- .../functoriality-fibers-of-maps.lagda.md | 2 +- .../functoriality-function-types.lagda.md | 2 +- ...oriality-propositional-truncation.lagda.md | 2 +- 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src/foundation/partitions.lagda.md | 3 +- src/foundation/path-algebra.lagda.md | 2 +- src/foundation/perfect-images.lagda.md | 2 +- src/foundation/pi-decompositions.lagda.md | 2 +- src/foundation/preidempotent-maps.lagda.md | 2 +- ...products-unordered-pairs-of-types.lagda.md | 3 +- src/foundation/projective-types.lagda.md | 2 +- .../propositional-extensionality.lagda.md | 1 + .../propositional-truncations.lagda.md | 2 +- src/foundation/pullbacks.lagda.md | 2 +- .../raising-universe-levels.lagda.md | 2 +- .../relaxed-sigma-decompositions.lagda.md | 2 +- src/foundation/repetitions-of-values.lagda.md | 2 +- src/foundation/repetitions-sequences.lagda.md | 2 +- src/foundation/retractions.lagda.md | 2 +- src/foundation/sections.lagda.md | 2 +- src/foundation/sequences.lagda.md | 3 +- src/foundation/set-presented-types.lagda.md | 2 +- src/foundation/set-quotients.lagda.md | 2 +- src/foundation/set-truncations.lagda.md | 2 +- src/foundation/sets.lagda.md | 2 +- src/foundation/sigma-decompositions.lagda.md | 2 +- src/foundation/singleton-induction.lagda.md | 2 +- src/foundation/slice.lagda.md | 2 +- src/foundation/split-surjective-maps.lagda.md | 2 +- .../structured-type-duality.lagda.md | 2 +- src/foundation/subterminal-types.lagda.md | 2 +- .../subtype-identity-principle.lagda.md | 2 +- src/foundation/subtypes.lagda.md | 2 +- src/foundation/surjective-maps.lagda.md | 2 +- src/foundation/symmetric-difference.lagda.md | 2 +- .../symmetric-identity-types.lagda.md | 2 +- src/foundation/transport.lagda.md | 187 ++----- ...vial-relaxed-sigma-decompositions.lagda.md | 2 +- .../trivial-sigma-decompositions.lagda.md | 2 +- .../truncation-equivalences.lagda.md | 2 +- src/foundation/truncation-levels.lagda.md | 2 +- src/foundation/truncation-modalities.lagda.md | 2 +- src/foundation/truncations.lagda.md | 2 +- .../type-arithmetic-booleans.lagda.md | 2 +- ...rithmetic-cartesian-product-types.lagda.md | 2 +- .../type-arithmetic-coproduct-types.lagda.md | 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...roups-generated-by-subsets-groups.lagda.md | 2 +- src/group-theory/subgroups.lagda.md | 2 +- src/group-theory/submonoids.lagda.md | 2 +- src/group-theory/subsemigroups.lagda.md | 2 +- src/group-theory/symmetric-groups.lagda.md | 2 +- src/group-theory/torsors.lagda.md | 2 +- ...transitive-concrete-group-actions.lagda.md | 2 +- .../equivalences-higher-groups.lagda.md | 2 +- ...-cartesian-products-higher-groups.lagda.md | 2 +- .../subgroups-higher-groups.lagda.md | 2 +- .../functoriality-vectors.lagda.md | 2 +- src/linear-algebra/vectors.lagda.md | 2 +- src/lists/arrays.lagda.md | 2 +- src/lists/concatenation-lists.lagda.md | 2 +- src/lists/functoriality-lists.lagda.md | 2 +- src/lists/lists-discrete-types.lagda.md | 2 +- src/lists/lists.lagda.md | 2 +- src/lists/permutation-vectors.lagda.md | 2 +- src/lists/reversing-lists.lagda.md | 2 +- src/lists/sorted-vectors.lagda.md | 2 +- .../oeis.lagda.md | 2 +- .../closure-operators-large-locales.lagda.md | 2 +- .../closure-operators-large-posets.lagda.md | 2 +- .../decidable-total-preorders.lagda.md | 2 +- .../finitely-graded-posets.lagda.md | 2 +- .../homomorphisms-frames.lagda.md | 2 +- ...morphisms-large-meet-semilattices.lagda.md | 2 +- .../homomorphisms-large-suplattices.lagda.md | 2 +- .../homomorphisms-meet-sup-lattices.lagda.md | 2 +- .../homomorphisms-sup-lattices.lagda.md | 2 +- src/order-theory/large-subframes.lagda.md | 2 +- .../nuclei-large-locales.lagda.md | 2 +- .../order-preserving-maps-posets.lagda.md | 2 +- .../order-preserving-maps-preorders.lagda.md | 2 +- src/order-theory/preorders.lagda.md | 2 +- src/order-theory/subpreorders.lagda.md | 2 +- .../closed-modalities.lagda.md | 2 +- .../extensions-of-maps.lagda.md | 2 +- ...ation-operations-function-classes.lagda.md | 2 +- .../factorizations-of-maps.lagda.md | 2 +- .../function-classes.lagda.md | 2 +- .../higher-modalities.lagda.md | 2 +- .../identity-modality.lagda.md | 2 +- .../lifting-operations.lagda.md | 2 +- .../lifting-squares.lagda.md | 2 +- .../lifts-of-maps.lagda.md | 2 +- .../local-types.lagda.md | 21 +- .../modal-operators.lagda.md | 2 +- .../null-types.lagda.md | 2 +- .../pullback-hom.lagda.md | 2 +- .../raise-modalities.lagda.md | 2 +- ...ma-closed-reflective-subuniverses.lagda.md | 2 +- .../uniquely-eliminating-modalities.lagda.md | 2 +- .../wide-function-classes.lagda.md | 2 +- src/reflection/precategory-solver.lagda.md | 2 +- ...ideals-generated-by-subsets-rings.lagda.md | 2 +- src/ring-theory/isomorphisms-rings.lagda.md | 2 +- ...ideals-generated-by-subsets-rings.lagda.md | 2 +- .../poset-of-ideals-rings.lagda.md | 2 +- .../poset-of-left-ideals-rings.lagda.md | 2 +- .../poset-of-right-ideals-rings.lagda.md | 2 +- .../powers-of-elements-rings.lagda.md | 2 +- ...ideals-generated-by-subsets-rings.lagda.md | 2 +- src/ring-theory/sums-rings.lagda.md | 2 +- src/ring-theory/sums-semirings.lagda.md | 2 +- src/set-theory/countable-sets.lagda.md | 2 +- src/set-theory/cumulative-hierarchy.lagda.md | 2 +- ...-species-of-types-in-subuniverses.lagda.md | 2 +- ...-species-of-types-in-subuniverses.lagda.md | 2 +- ...-species-of-types-in-subuniverses.lagda.md | 2 +- ...-species-of-types-in-subuniverses.lagda.md | 2 +- ...-species-of-types-in-subuniverses.lagda.md | 2 +- ...-species-of-types-in-subuniverses.lagda.md | 2 +- ...-species-of-types-in-subuniverses.lagda.md | 2 +- ...-species-of-types-in-subuniverses.lagda.md | 2 +- ...y-series-of-types-in-subuniverses.lagda.md | 2 +- src/species/morphisms-finite-species.lagda.md | 2 +- .../morphisms-species-of-types.lagda.md | 2 +- ...-species-of-types-in-subuniverses.lagda.md | 2 +- ...-species-of-types-in-subuniverses.lagda.md | 2 +- ...species-of-finite-inhabited-types.lagda.md | 2 +- .../species-of-types-in-subuniverses.lagda.md | 2 +- ...-species-of-types-in-subuniverses.lagda.md | 2 +- .../coherent-h-spaces.lagda.md | 2 +- ...types-equipped-with-endomorphisms.lagda.md | 2 +- .../finite-multiplication-magmas.lagda.md | 2 +- ...types-equipped-with-endomorphisms.lagda.md | 2 +- .../morphisms-coherent-h-spaces.lagda.md | 2 +- .../pointed-cartesian-product-types.lagda.md | 2 +- .../pointed-dependent-functions.lagda.md | 2 +- .../pointed-equivalences.lagda.md | 2 +- src/structured-types/pointed-maps.lagda.md | 2 +- ...types-equipped-with-automorphisms.lagda.md | 2 +- ...types-equipped-with-endomorphisms.lagda.md | 2 +- .../26-descent.lagda.md | 2 +- .../26-id-pushout.lagda.md | 2 +- .../27-sequences.lagda.md | 2 +- .../cavallos-trick.lagda.md | 2 +- .../cocones-under-spans.lagda.md | 2 +- .../dependent-cocones-under-spans.lagda.md | 12 +- ...ndent-universal-property-pushouts.lagda.md | 4 +- .../descent-circle.lagda.md | 46 +- .../infinite-cyclic-types.lagda.md | 2 +- .../joins-of-types.lagda.md | 2 +- .../multiplication-circle.lagda.md | 20 +- .../pullback-property-pushouts.lagda.md | 2 +- .../pushouts.lagda.md | 2 +- .../suspensions-of-types.lagda.md | 2 +- .../universal-cover-circle.lagda.md | 44 +- .../universal-property-circle.lagda.md | 4 +- .../universal-property-pushouts.lagda.md | 2 +- src/trees/bases-directed-trees.lagda.md | 2 +- .../bases-enriched-directed-trees.lagda.md | 2 +- src/trees/combinator-directed-trees.lagda.md | 2 +- ...ombinator-enriched-directed-trees.lagda.md | 2 +- src/trees/directed-trees.lagda.md | 2 +- .../equivalences-directed-trees.lagda.md | 2 +- ...ivalences-enriched-directed-trees.lagda.md | 2 +- src/trees/extensional-w-types.lagda.md | 3 +- ...riality-combinator-directed-trees.lagda.md | 2 +- src/trees/functoriality-w-types.lagda.md | 2 +- src/trees/induction-w-types.lagda.md | 2 +- ...-algebras-polynomial-endofunctors.lagda.md | 2 +- src/trees/morphisms-directed-trees.lagda.md | 2 +- src/trees/multisets.lagda.md | 2 +- src/trees/planar-binary-trees.lagda.md | 2 +- src/trees/polynomial-endofunctors.lagda.md | 2 +- .../rooted-morphisms-directed-trees.lagda.md | 2 +- ...morphisms-enriched-directed-trees.lagda.md | 2 +- src/trees/small-multisets.lagda.md | 2 +- ...oalgebras-polynomial-endofunctors.lagda.md | 3 +- ...ying-trees-of-elements-of-w-types.lagda.md | 3 +- src/trees/w-type-of-natural-numbers.lagda.md | 2 +- src/trees/w-types.lagda.md | 2 +- .../dependent-type-theories.lagda.md | 2 +- .../fibered-dependent-type-theories.lagda.md | 2 +- .../simple-type-theories.lagda.md | 2 +- .../unityped-type-theories.lagda.md | 2 +- .../2-element-decidable-subtypes.lagda.md | 2 +- .../2-element-subtypes.lagda.md | 2 +- .../2-element-types.lagda.md | 2 +- .../binomial-types.lagda.md | 2 +- .../cartesian-product-types.lagda.md | 2 +- .../coproduct-types.lagda.md | 2 +- .../counting-decidable-subtypes.lagda.md | 2 +- .../counting-dependent-pair-types.lagda.md | 2 +- src/univalent-combinatorics/counting.lagda.md | 2 +- .../cyclic-types.lagda.md | 2 +- ...ecidable-dependent-function-types.lagda.md | 2 +- .../decidable-dependent-pair-types.lagda.md | 2 +- .../decidable-equivalence-relations.lagda.md | 2 +- .../decidable-subtypes.lagda.md | 2 +- .../dependent-function-types.lagda.md | 2 +- .../dependent-pair-types.lagda.md | 2 +- ...t-truncation-over-finite-products.lagda.md | 2 +- .../equivalences-cubes.lagda.md | 2 +- .../finite-choice.lagda.md | 2 +- .../finite-types.lagda.md | 2 +- .../finitely-presented-types.lagda.md | 2 +- .../inhabited-finite-types.lagda.md | 2 +- .../necklaces.lagda.md | 2 +- ...tations-complete-undirected-graph.lagda.md | 2 +- .../partitions.lagda.md | 2 +- .../pi-finite-types.lagda.md | 2 +- .../pigeonhole-principle.lagda.md | 2 +- .../sequences-finite-types.lagda.md | 2 +- .../set-quotients-of-index-two.lagda.md | 2 +- .../standard-finite-types.lagda.md | 2 +- ...al-property-standard-finite-types.lagda.md | 2 +- 478 files changed, 1421 insertions(+), 1248 deletions(-) rename src/foundation-core/{dependent-paths.lagda.md => dependent-identifications.lagda.md} (72%) create mode 100644 src/foundation-core/dependent-pair-types.lagda.md rename src/{foundation/functions.lagda.md => foundation-core/function-types.lagda.md} (96%) create mode 100644 src/foundation/dependent-identifications.lagda.md delete mode 100644 src/foundation/dependent-paths.lagda.md create mode 100644 src/foundation/function-types.lagda.md diff --git a/src/category-theory/functors-large-precategories.lagda.md b/src/category-theory/functors-large-precategories.lagda.md index e19fd4d10b..53d1d4a608 100644 --- a/src/category-theory/functors-large-precategories.lagda.md +++ b/src/category-theory/functors-large-precategories.lagda.md @@ -10,7 +10,7 @@ module category-theory.functors-large-precategories where open import category-theory.large-precategories open import foundation.action-on-identifications-functions -open import foundation.functions +open import foundation.function-types open import foundation.identity-types open import foundation.universe-levels ``` diff --git a/src/category-theory/functors-precategories.lagda.md b/src/category-theory/functors-precategories.lagda.md index c4649a275b..b6fc0b95bc 100644 --- a/src/category-theory/functors-precategories.lagda.md +++ b/src/category-theory/functors-precategories.lagda.md @@ -12,7 +12,7 @@ open import category-theory.precategories open import foundation.action-on-identifications-functions open import foundation.cartesian-product-types open import foundation.dependent-pair-types -open import foundation.functions +open import foundation.function-types open import foundation.identity-types open import foundation.propositions open import foundation.universe-levels diff --git a/src/category-theory/groupoids.lagda.md b/src/category-theory/groupoids.lagda.md index 500a461c2c..76ef73acfd 100644 --- a/src/category-theory/groupoids.lagda.md +++ b/src/category-theory/groupoids.lagda.md @@ -18,7 +18,7 @@ open import foundation.1-types open import foundation.contractible-types open import foundation.dependent-pair-types open import foundation.equivalences -open import foundation.functions +open import foundation.function-types open import foundation.functoriality-dependent-pair-types open import foundation.fundamental-theorem-of-identity-types open import foundation.identity-types diff --git a/src/category-theory/isomorphisms-precategories.lagda.md b/src/category-theory/isomorphisms-precategories.lagda.md index 4c40ca212f..32179c74cc 100644 --- a/src/category-theory/isomorphisms-precategories.lagda.md +++ b/src/category-theory/isomorphisms-precategories.lagda.md @@ -13,7 +13,7 @@ open import foundation.action-on-identifications-functions open import foundation.cartesian-product-types open import foundation.dependent-pair-types open import foundation.equivalences -open import foundation.functions +open import foundation.function-types open import foundation.homotopies open import foundation.identity-types open import foundation.propositions diff --git a/src/category-theory/natural-isomorphisms-precategories.lagda.md b/src/category-theory/natural-isomorphisms-precategories.lagda.md index b99301dfff..b89cb31654 100644 --- a/src/category-theory/natural-isomorphisms-precategories.lagda.md +++ b/src/category-theory/natural-isomorphisms-precategories.lagda.md @@ -13,7 +13,7 @@ open import category-theory.natural-transformations-precategories open import category-theory.precategories open import foundation.dependent-pair-types -open import foundation.functions +open import foundation.function-types open import foundation.propositions open import foundation.universe-levels ``` diff --git a/src/category-theory/natural-transformations-precategories.lagda.md b/src/category-theory/natural-transformations-precategories.lagda.md index 7bb415da6a..c41ae023b3 100644 --- a/src/category-theory/natural-transformations-precategories.lagda.md +++ b/src/category-theory/natural-transformations-precategories.lagda.md @@ -14,7 +14,7 @@ open import foundation.action-on-identifications-functions open import foundation.dependent-pair-types open import foundation.embeddings open import foundation.function-extensionality -open import foundation.functions +open import foundation.function-types open import foundation.identity-types open import foundation.injective-maps open import foundation.propositions diff --git a/src/category-theory/precategories.lagda.md b/src/category-theory/precategories.lagda.md index b347e439f0..d19c565aef 100644 --- a/src/category-theory/precategories.lagda.md +++ b/src/category-theory/precategories.lagda.md @@ -10,7 +10,7 @@ module category-theory.precategories where open import foundation.cartesian-product-types open import foundation.dependent-pair-types open import foundation.function-extensionality -open import foundation.functions +open import foundation.function-types open import foundation.identity-types open import foundation.propositions open import foundation.sets diff --git a/src/category-theory/slice-precategories.lagda.md b/src/category-theory/slice-precategories.lagda.md index c81125a3f9..e2e1dbbea2 100644 --- a/src/category-theory/slice-precategories.lagda.md +++ b/src/category-theory/slice-precategories.lagda.md @@ -18,7 +18,7 @@ open import foundation.contractible-types open import foundation.dependent-pair-types open import foundation.equality-dependent-pair-types open import foundation.equivalences -open import foundation.functions +open import foundation.function-types open import foundation.functoriality-dependent-pair-types open import foundation.homotopies open import foundation.identity-types diff --git a/src/commutative-algebra/joins-radical-ideals-commutative-rings.lagda.md b/src/commutative-algebra/joins-radical-ideals-commutative-rings.lagda.md index 68e6511509..3cd6e927ed 100644 --- a/src/commutative-algebra/joins-radical-ideals-commutative-rings.lagda.md +++ b/src/commutative-algebra/joins-radical-ideals-commutative-rings.lagda.md @@ -20,7 +20,7 @@ open import commutative-algebra.radicals-of-ideals-commutative-rings open import commutative-algebra.subsets-commutative-rings open import foundation.dependent-pair-types -open import foundation.functions +open import foundation.function-types open import foundation.identity-types open import foundation.universe-levels diff --git a/src/commutative-algebra/poset-of-radical-ideals-commutative-rings.lagda.md b/src/commutative-algebra/poset-of-radical-ideals-commutative-rings.lagda.md index 5bd99204c5..d5ccc9065d 100644 --- a/src/commutative-algebra/poset-of-radical-ideals-commutative-rings.lagda.md +++ b/src/commutative-algebra/poset-of-radical-ideals-commutative-rings.lagda.md @@ -11,7 +11,7 @@ open import commutative-algebra.commutative-rings open import commutative-algebra.poset-of-ideals-commutative-rings open import commutative-algebra.radical-ideals-commutative-rings -open import foundation.functions +open import foundation.function-types open import foundation.identity-types open import foundation.propositions open import foundation.subtypes diff --git a/src/commutative-algebra/prime-ideals-commutative-rings.lagda.md b/src/commutative-algebra/prime-ideals-commutative-rings.lagda.md index 3520c31c92..eb639d4f82 100644 --- a/src/commutative-algebra/prime-ideals-commutative-rings.lagda.md +++ b/src/commutative-algebra/prime-ideals-commutative-rings.lagda.md @@ -18,7 +18,7 @@ open import elementary-number-theory.natural-numbers open import foundation.dependent-pair-types open import foundation.disjunction -open import foundation.functions +open import foundation.function-types open import foundation.identity-types open import foundation.propositions open import foundation.universe-levels diff --git a/src/commutative-algebra/radicals-of-ideals-commutative-rings.lagda.md b/src/commutative-algebra/radicals-of-ideals-commutative-rings.lagda.md index 3ee78e4b83..c328c9d0af 100644 --- a/src/commutative-algebra/radicals-of-ideals-commutative-rings.lagda.md +++ b/src/commutative-algebra/radicals-of-ideals-commutative-rings.lagda.md @@ -22,7 +22,7 @@ open import elementary-number-theory.natural-numbers open import foundation.dependent-pair-types open import foundation.existential-quantification -open import foundation.functions +open import foundation.function-types open import foundation.logical-equivalences open import foundation.propositional-truncations open import foundation.propositions diff --git a/src/commutative-algebra/sums-commutative-rings.lagda.md b/src/commutative-algebra/sums-commutative-rings.lagda.md index 57ed09bdd9..204b314815 100644 --- a/src/commutative-algebra/sums-commutative-rings.lagda.md +++ b/src/commutative-algebra/sums-commutative-rings.lagda.md @@ -14,7 +14,7 @@ open import elementary-number-theory.natural-numbers open import foundation.action-on-identifications-functions open import foundation.coproduct-types -open import foundation.functions +open import foundation.function-types open import foundation.homotopies open import foundation.identity-types open import foundation.unit-type diff --git a/src/commutative-algebra/sums-commutative-semirings.lagda.md b/src/commutative-algebra/sums-commutative-semirings.lagda.md index c0d5eab5a3..9eceddd323 100644 --- a/src/commutative-algebra/sums-commutative-semirings.lagda.md +++ b/src/commutative-algebra/sums-commutative-semirings.lagda.md @@ -12,7 +12,7 @@ open import commutative-algebra.commutative-semirings open import elementary-number-theory.addition-natural-numbers open import elementary-number-theory.natural-numbers -open import foundation.functions +open import foundation.function-types open import foundation.homotopies open import foundation.identity-types open import foundation.universe-levels diff --git a/src/elementary-number-theory/absolute-value-integers.lagda.md b/src/elementary-number-theory/absolute-value-integers.lagda.md index 0ef90566a9..8d812cedac 100644 --- a/src/elementary-number-theory/absolute-value-integers.lagda.md +++ b/src/elementary-number-theory/absolute-value-integers.lagda.md @@ -17,7 +17,7 @@ open import elementary-number-theory.natural-numbers open import foundation.action-on-identifications-functions open import foundation.coproduct-types -open import foundation.functions +open import foundation.function-types open import foundation.identity-types open import foundation.unit-type ``` diff --git a/src/elementary-number-theory/addition-integers.lagda.md b/src/elementary-number-theory/addition-integers.lagda.md index d2af1c06d5..a5c3bff0ea 100644 --- a/src/elementary-number-theory/addition-integers.lagda.md +++ b/src/elementary-number-theory/addition-integers.lagda.md @@ -19,7 +19,7 @@ open import foundation.coproduct-types open import foundation.dependent-pair-types open import foundation.embeddings open import foundation.equivalences -open import foundation.functions +open import foundation.function-types open import foundation.identity-types open import foundation.injective-maps open import foundation.interchange-law diff --git a/src/elementary-number-theory/addition-natural-numbers.lagda.md b/src/elementary-number-theory/addition-natural-numbers.lagda.md index 4fdaefeea3..9edeac9343 100644 --- a/src/elementary-number-theory/addition-natural-numbers.lagda.md +++ b/src/elementary-number-theory/addition-natural-numbers.lagda.md @@ -15,7 +15,7 @@ open import foundation.cartesian-product-types open import foundation.dependent-pair-types open import foundation.embeddings open import foundation.empty-types -open import foundation.functions +open import foundation.function-types open import foundation.identity-types open import foundation.injective-maps open import foundation.interchange-law diff --git a/src/elementary-number-theory/based-induction-natural-numbers.lagda.md b/src/elementary-number-theory/based-induction-natural-numbers.lagda.md index 6ca59ad1cb..1911c83026 100644 --- a/src/elementary-number-theory/based-induction-natural-numbers.lagda.md +++ b/src/elementary-number-theory/based-induction-natural-numbers.lagda.md @@ -10,7 +10,7 @@ module elementary-number-theory.based-induction-natural-numbers where open import elementary-number-theory.inequality-natural-numbers open import elementary-number-theory.natural-numbers -open import foundation.functions +open import foundation.function-types open import foundation.identity-types open import foundation.universe-levels ``` diff --git a/src/elementary-number-theory/based-strong-induction-natural-numbers.lagda.md b/src/elementary-number-theory/based-strong-induction-natural-numbers.lagda.md index 6b36072018..2d927e209e 100644 --- a/src/elementary-number-theory/based-strong-induction-natural-numbers.lagda.md +++ b/src/elementary-number-theory/based-strong-induction-natural-numbers.lagda.md @@ -17,7 +17,7 @@ open import foundation.contractible-types open import foundation.coproduct-types open import foundation.empty-types open import foundation.function-extensionality -open import foundation.functions +open import foundation.function-types open import foundation.identity-types open import foundation.propositions open import foundation.universe-levels diff --git a/src/elementary-number-theory/bezouts-lemma-integers.lagda.md b/src/elementary-number-theory/bezouts-lemma-integers.lagda.md index 151aeb71fb..536b969aa3 100644 --- a/src/elementary-number-theory/bezouts-lemma-integers.lagda.md +++ b/src/elementary-number-theory/bezouts-lemma-integers.lagda.md @@ -26,7 +26,7 @@ open import elementary-number-theory.natural-numbers open import foundation.action-on-identifications-functions open import foundation.coproduct-types open import foundation.dependent-pair-types -open import foundation.functions +open import foundation.function-types open import foundation.identity-types open import foundation.transport open import foundation.unit-type diff --git a/src/elementary-number-theory/bounded-sums-arithmetic-functions.lagda.md b/src/elementary-number-theory/bounded-sums-arithmetic-functions.lagda.md index b9ea37e23f..7e513fd97a 100644 --- a/src/elementary-number-theory/bounded-sums-arithmetic-functions.lagda.md +++ b/src/elementary-number-theory/bounded-sums-arithmetic-functions.lagda.md @@ -14,7 +14,7 @@ open import elementary-number-theory.nonzero-natural-numbers open import foundation.coproduct-types open import foundation.decidable-propositions open import foundation.decidable-types -open import foundation.functions +open import foundation.function-types open import foundation.universe-levels open import ring-theory.rings diff --git a/src/elementary-number-theory/congruence-integers.lagda.md b/src/elementary-number-theory/congruence-integers.lagda.md index 3d8b342cd7..b7caeed9a1 100644 --- a/src/elementary-number-theory/congruence-integers.lagda.md +++ b/src/elementary-number-theory/congruence-integers.lagda.md @@ -18,7 +18,7 @@ open import elementary-number-theory.natural-numbers open import foundation.action-on-identifications-functions open import foundation.dependent-pair-types -open import foundation.functions +open import foundation.function-types open import foundation.identity-types open import foundation.transport open import foundation.universe-levels diff --git a/src/elementary-number-theory/decidable-types.lagda.md b/src/elementary-number-theory/decidable-types.lagda.md index 42f5c1a455..64e6422e08 100644 --- a/src/elementary-number-theory/decidable-types.lagda.md +++ b/src/elementary-number-theory/decidable-types.lagda.md @@ -17,7 +17,7 @@ open import foundation.coproduct-types open import foundation.decidable-types open import foundation.dependent-pair-types open import foundation.empty-types -open import foundation.functions +open import foundation.function-types open import foundation.unit-type open import foundation.universe-levels ``` diff --git a/src/elementary-number-theory/distance-natural-numbers.lagda.md b/src/elementary-number-theory/distance-natural-numbers.lagda.md index 6dd8fca3a8..4aa61b2b9e 100644 --- a/src/elementary-number-theory/distance-natural-numbers.lagda.md +++ b/src/elementary-number-theory/distance-natural-numbers.lagda.md @@ -17,7 +17,7 @@ open import foundation.action-on-identifications-binary-functions open import foundation.action-on-identifications-functions open import foundation.coproduct-types open import foundation.dependent-pair-types -open import foundation.functions +open import foundation.function-types open import foundation.identity-types open import foundation.transport open import foundation.unit-type diff --git a/src/elementary-number-theory/divisibility-integers.lagda.md b/src/elementary-number-theory/divisibility-integers.lagda.md index e1e4a80a23..0e618a6590 100644 --- a/src/elementary-number-theory/divisibility-integers.lagda.md +++ b/src/elementary-number-theory/divisibility-integers.lagda.md @@ -21,7 +21,7 @@ open import foundation.coproduct-types open import foundation.decidable-types open import foundation.dependent-pair-types open import foundation.empty-types -open import foundation.functions +open import foundation.function-types open import foundation.identity-types open import foundation.negation open import foundation.propositional-maps diff --git a/src/elementary-number-theory/equality-integers.lagda.md b/src/elementary-number-theory/equality-integers.lagda.md index cdf065cd64..8a00b0818e 100644 --- a/src/elementary-number-theory/equality-integers.lagda.md +++ b/src/elementary-number-theory/equality-integers.lagda.md @@ -21,7 +21,7 @@ open import foundation.discrete-types open import foundation.empty-types open import foundation.equality-dependent-pair-types open import foundation.equivalences -open import foundation.functions +open import foundation.function-types open import foundation.fundamental-theorem-of-identity-types open import foundation.identity-types open import foundation.propositions diff --git a/src/elementary-number-theory/equality-natural-numbers.lagda.md b/src/elementary-number-theory/equality-natural-numbers.lagda.md index 90cb4232bb..84547071d8 100644 --- a/src/elementary-number-theory/equality-natural-numbers.lagda.md +++ b/src/elementary-number-theory/equality-natural-numbers.lagda.md @@ -16,7 +16,7 @@ open import foundation.decidable-equality open import foundation.decidable-types open import foundation.dependent-pair-types open import foundation.equivalences -open import foundation.functions +open import foundation.function-types open import foundation.fundamental-theorem-of-identity-types open import foundation.identity-types open import foundation.set-truncations diff --git a/src/elementary-number-theory/finitary-natural-numbers.lagda.md b/src/elementary-number-theory/finitary-natural-numbers.lagda.md index 4cbf87f258..03193fcb2c 100644 --- a/src/elementary-number-theory/finitary-natural-numbers.lagda.md +++ b/src/elementary-number-theory/finitary-natural-numbers.lagda.md @@ -18,7 +18,7 @@ open import foundation.action-on-identifications-functions open import foundation.coproduct-types open import foundation.dependent-pair-types open import foundation.empty-types -open import foundation.functions +open import foundation.function-types open import foundation.identity-types open import foundation.injective-maps open import foundation.universe-levels diff --git a/src/elementary-number-theory/finitely-cyclic-maps.lagda.md b/src/elementary-number-theory/finitely-cyclic-maps.lagda.md index ee16f4d378..54d0492ace 100644 --- a/src/elementary-number-theory/finitely-cyclic-maps.lagda.md +++ b/src/elementary-number-theory/finitely-cyclic-maps.lagda.md @@ -13,7 +13,7 @@ open import elementary-number-theory.natural-numbers open import foundation.action-on-identifications-functions open import foundation.dependent-pair-types open import foundation.equivalences -open import foundation.functions +open import foundation.function-types open import foundation.homotopies open import foundation.identity-types open import foundation.iterating-functions diff --git a/src/elementary-number-theory/greatest-common-divisor-integers.lagda.md b/src/elementary-number-theory/greatest-common-divisor-integers.lagda.md index f48103040e..6c1acca9a6 100644 --- a/src/elementary-number-theory/greatest-common-divisor-integers.lagda.md +++ b/src/elementary-number-theory/greatest-common-divisor-integers.lagda.md @@ -20,7 +20,7 @@ open import foundation.cartesian-product-types open import foundation.coproduct-types open import foundation.dependent-pair-types open import foundation.empty-types -open import foundation.functions +open import foundation.function-types open import foundation.functoriality-cartesian-product-types open import foundation.identity-types open import foundation.logical-equivalences diff --git a/src/elementary-number-theory/inequality-integers.lagda.md b/src/elementary-number-theory/inequality-integers.lagda.md index f30cd55c72..2769bf6bef 100644 --- a/src/elementary-number-theory/inequality-integers.lagda.md +++ b/src/elementary-number-theory/inequality-integers.lagda.md @@ -15,7 +15,7 @@ open import elementary-number-theory.natural-numbers open import foundation.action-on-identifications-functions open import foundation.coproduct-types -open import foundation.functions +open import foundation.function-types open import foundation.functoriality-coproduct-types open import foundation.identity-types open import foundation.propositions diff --git a/src/elementary-number-theory/inequality-natural-numbers.lagda.md b/src/elementary-number-theory/inequality-natural-numbers.lagda.md index b33b117ae6..48385b34d4 100644 --- a/src/elementary-number-theory/inequality-natural-numbers.lagda.md +++ b/src/elementary-number-theory/inequality-natural-numbers.lagda.md @@ -17,7 +17,7 @@ open import foundation.coproduct-types open import foundation.decidable-types open import foundation.dependent-pair-types open import foundation.empty-types -open import foundation.functions +open import foundation.function-types open import foundation.functoriality-coproduct-types open import foundation.identity-types open import foundation.negation diff --git a/src/elementary-number-theory/initial-segments-natural-numbers.lagda.md b/src/elementary-number-theory/initial-segments-natural-numbers.lagda.md index 4de715f99d..d6d10e541f 100644 --- a/src/elementary-number-theory/initial-segments-natural-numbers.lagda.md +++ b/src/elementary-number-theory/initial-segments-natural-numbers.lagda.md @@ -11,7 +11,7 @@ open import elementary-number-theory.maximum-natural-numbers open import elementary-number-theory.natural-numbers open import foundation.dependent-pair-types -open import foundation.functions +open import foundation.function-types open import foundation.identity-types open import foundation.propositions open import foundation.subtypes diff --git a/src/elementary-number-theory/integers.lagda.md b/src/elementary-number-theory/integers.lagda.md index d1e6c878ff..f5a4d7131f 100644 --- a/src/elementary-number-theory/integers.lagda.md +++ b/src/elementary-number-theory/integers.lagda.md @@ -16,7 +16,7 @@ open import foundation.embeddings open import foundation.empty-types open import foundation.equality-coproduct-types open import foundation.equivalences -open import foundation.functions +open import foundation.function-types open import foundation.homotopies open import foundation.identity-types open import foundation.injective-maps diff --git a/src/elementary-number-theory/modular-arithmetic-standard-finite-types.lagda.md b/src/elementary-number-theory/modular-arithmetic-standard-finite-types.lagda.md index bb679cf570..90585e758a 100644 --- a/src/elementary-number-theory/modular-arithmetic-standard-finite-types.lagda.md +++ b/src/elementary-number-theory/modular-arithmetic-standard-finite-types.lagda.md @@ -23,7 +23,7 @@ open import foundation.decidable-propositions open import foundation.decidable-types open import foundation.dependent-pair-types open import foundation.equivalences -open import foundation.functions +open import foundation.function-types open import foundation.identity-types open import foundation.injective-maps open import foundation.split-surjective-maps diff --git a/src/elementary-number-theory/modular-arithmetic.lagda.md b/src/elementary-number-theory/modular-arithmetic.lagda.md index c07c16c408..2d63d8d4d4 100644 --- a/src/elementary-number-theory/modular-arithmetic.lagda.md +++ b/src/elementary-number-theory/modular-arithmetic.lagda.md @@ -29,7 +29,7 @@ open import foundation.dependent-pair-types open import foundation.discrete-types open import foundation.empty-types open import foundation.equivalences -open import foundation.functions +open import foundation.function-types open import foundation.identity-types open import foundation.injective-maps open import foundation.negation diff --git a/src/elementary-number-theory/multiplication-integers.lagda.md b/src/elementary-number-theory/multiplication-integers.lagda.md index 0552374a34..2f916303dd 100644 --- a/src/elementary-number-theory/multiplication-integers.lagda.md +++ b/src/elementary-number-theory/multiplication-integers.lagda.md @@ -21,7 +21,7 @@ open import foundation.action-on-identifications-functions open import foundation.coproduct-types open import foundation.embeddings open import foundation.empty-types -open import foundation.functions +open import foundation.function-types open import foundation.identity-types open import foundation.injective-maps open import foundation.interchange-law diff --git a/src/elementary-number-theory/natural-numbers.lagda.md b/src/elementary-number-theory/natural-numbers.lagda.md index f523435d7d..b4c2cf4a34 100644 --- a/src/elementary-number-theory/natural-numbers.lagda.md +++ b/src/elementary-number-theory/natural-numbers.lagda.md @@ -11,7 +11,7 @@ open import foundation.action-on-identifications-functions open import foundation.dependent-pair-types open import foundation.empty-types open import foundation.equivalences -open import foundation.functions +open import foundation.function-types open import foundation.homotopies open import foundation.identity-types open import foundation.injective-maps diff --git a/src/elementary-number-theory/parity-natural-numbers.lagda.md b/src/elementary-number-theory/parity-natural-numbers.lagda.md index b776266ee9..402bd6f7f4 100644 --- a/src/elementary-number-theory/parity-natural-numbers.lagda.md +++ b/src/elementary-number-theory/parity-natural-numbers.lagda.md @@ -16,7 +16,7 @@ open import foundation.action-on-identifications-functions open import foundation.decidable-types open import foundation.dependent-pair-types open import foundation.empty-types -open import foundation.functions +open import foundation.function-types open import foundation.identity-types open import foundation.negation open import foundation.universe-levels diff --git a/src/elementary-number-theory/powers-of-two.lagda.md b/src/elementary-number-theory/powers-of-two.lagda.md index 16687e4502..6c41d84e12 100644 --- a/src/elementary-number-theory/powers-of-two.lagda.md +++ b/src/elementary-number-theory/powers-of-two.lagda.md @@ -19,7 +19,7 @@ open import elementary-number-theory.strong-induction-natural-numbers open import foundation.action-on-identifications-functions open import foundation.dependent-pair-types open import foundation.equality-cartesian-product-types -open import foundation.functions +open import foundation.function-types open import foundation.split-surjective-maps open import foundation.transport open import foundation.universe-levels diff --git a/src/elementary-number-theory/sieve-of-eratosthenes.lagda.md b/src/elementary-number-theory/sieve-of-eratosthenes.lagda.md index 616d607102..f76a992745 100644 --- a/src/elementary-number-theory/sieve-of-eratosthenes.lagda.md +++ b/src/elementary-number-theory/sieve-of-eratosthenes.lagda.md @@ -22,7 +22,7 @@ open import foundation.coproduct-types open import foundation.decidable-types open import foundation.dependent-pair-types open import foundation.empty-types -open import foundation.functions +open import foundation.function-types open import foundation.identity-types open import foundation.unit-type open import foundation.universe-levels diff --git a/src/elementary-number-theory/strict-inequality-natural-numbers.lagda.md b/src/elementary-number-theory/strict-inequality-natural-numbers.lagda.md index 855396d96a..b128457342 100644 --- a/src/elementary-number-theory/strict-inequality-natural-numbers.lagda.md +++ b/src/elementary-number-theory/strict-inequality-natural-numbers.lagda.md @@ -17,7 +17,7 @@ open import foundation.coproduct-types open import foundation.decidable-types open import foundation.dependent-pair-types open import foundation.empty-types -open import foundation.functions +open import foundation.function-types open import foundation.functoriality-coproduct-types open import foundation.identity-types open import foundation.negation diff --git a/src/elementary-number-theory/strictly-ordered-pairs-of-natural-numbers.lagda.md b/src/elementary-number-theory/strictly-ordered-pairs-of-natural-numbers.lagda.md index 804eb96353..92bb7aae2f 100644 --- a/src/elementary-number-theory/strictly-ordered-pairs-of-natural-numbers.lagda.md +++ b/src/elementary-number-theory/strictly-ordered-pairs-of-natural-numbers.lagda.md @@ -13,7 +13,7 @@ open import elementary-number-theory.strict-inequality-natural-numbers open import foundation.action-on-identifications-functions open import foundation.dependent-pair-types open import foundation.empty-types -open import foundation.functions +open import foundation.function-types open import foundation.functoriality-dependent-pair-types open import foundation.identity-types open import foundation.negation diff --git a/src/elementary-number-theory/strong-induction-natural-numbers.lagda.md b/src/elementary-number-theory/strong-induction-natural-numbers.lagda.md index cfe3876b71..d255688989 100644 --- a/src/elementary-number-theory/strong-induction-natural-numbers.lagda.md +++ b/src/elementary-number-theory/strong-induction-natural-numbers.lagda.md @@ -16,7 +16,7 @@ open import foundation.coproduct-types open import foundation.dependent-pair-types open import foundation.empty-types open import foundation.function-extensionality -open import foundation.functions +open import foundation.function-types open import foundation.identity-types open import foundation.propositions open import foundation.universe-levels diff --git a/src/elementary-number-theory/sums-of-natural-numbers.lagda.md b/src/elementary-number-theory/sums-of-natural-numbers.lagda.md index 4a1adbfa6b..12503cc6a1 100644 --- a/src/elementary-number-theory/sums-of-natural-numbers.lagda.md +++ b/src/elementary-number-theory/sums-of-natural-numbers.lagda.md @@ -17,7 +17,7 @@ open import foundation.constant-maps open import foundation.coproduct-types open import foundation.dependent-pair-types open import foundation.equivalences -open import foundation.functions +open import foundation.function-types open import foundation.homotopies open import foundation.identity-types open import foundation.unit-type 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 f0e807e405..5a8b2345ea 100644 --- a/src/elementary-number-theory/type-arithmetic-natural-numbers.lagda.md +++ b/src/elementary-number-theory/type-arithmetic-natural-numbers.lagda.md @@ -17,7 +17,7 @@ open import elementary-number-theory.powers-of-two open import foundation.action-on-identifications-functions open import foundation.dependent-pair-types -open import foundation.functions +open import foundation.function-types open import foundation.functoriality-cartesian-product-types open import foundation.functoriality-coproduct-types open import foundation.iterating-functions diff --git a/src/elementary-number-theory/universal-property-natural-numbers.lagda.md b/src/elementary-number-theory/universal-property-natural-numbers.lagda.md index 6f3c84b2cb..821544c4ed 100644 --- a/src/elementary-number-theory/universal-property-natural-numbers.lagda.md +++ b/src/elementary-number-theory/universal-property-natural-numbers.lagda.md @@ -14,7 +14,7 @@ open import foundation.cartesian-product-types open import foundation.contractible-types open import foundation.dependent-pair-types open import foundation.equivalences -open import foundation.functions +open import foundation.function-types open import foundation.fundamental-theorem-of-identity-types open import foundation.homotopies open import foundation.identity-types diff --git a/src/elementary-number-theory/well-ordering-principle-standard-finite-types.lagda.md b/src/elementary-number-theory/well-ordering-principle-standard-finite-types.lagda.md index 94bce95e55..631db0439b 100644 --- a/src/elementary-number-theory/well-ordering-principle-standard-finite-types.lagda.md +++ b/src/elementary-number-theory/well-ordering-principle-standard-finite-types.lagda.md @@ -21,7 +21,7 @@ open import foundation.empty-types open import foundation.equivalence-extensionality open import foundation.equivalences open import foundation.existential-quantification -open import foundation.functions +open import foundation.function-types open import foundation.functoriality-coproduct-types open import foundation.functoriality-dependent-pair-types open import foundation.functoriality-propositional-truncation diff --git a/src/finite-algebra/dependent-products-commutative-finite-rings.lagda.md b/src/finite-algebra/dependent-products-commutative-finite-rings.lagda.md index 8c9a9c3be7..9d54f7d636 100644 --- a/src/finite-algebra/dependent-products-commutative-finite-rings.lagda.md +++ b/src/finite-algebra/dependent-products-commutative-finite-rings.lagda.md @@ -15,7 +15,7 @@ open import finite-algebra.dependent-products-finite-rings open import finite-algebra.finite-rings open import foundation.dependent-pair-types -open import foundation.functions +open import foundation.function-types open import foundation.identity-types open import foundation.sets open import foundation.universe-levels diff --git a/src/finite-algebra/dependent-products-finite-rings.lagda.md b/src/finite-algebra/dependent-products-finite-rings.lagda.md index 6b14814fac..18b15b2f33 100644 --- a/src/finite-algebra/dependent-products-finite-rings.lagda.md +++ b/src/finite-algebra/dependent-products-finite-rings.lagda.md @@ -10,7 +10,7 @@ module finite-algebra.dependent-products-finite-rings where open import finite-algebra.finite-rings open import foundation.dependent-pair-types -open import foundation.functions +open import foundation.function-types open import foundation.identity-types open import foundation.sets open import foundation.universe-levels diff --git a/src/finite-algebra/semisimple-commutative-finite-rings.lagda.md b/src/finite-algebra/semisimple-commutative-finite-rings.lagda.md index 4ea419b3bb..a52e06300b 100644 --- a/src/finite-algebra/semisimple-commutative-finite-rings.lagda.md +++ b/src/finite-algebra/semisimple-commutative-finite-rings.lagda.md @@ -16,7 +16,7 @@ open import finite-algebra.homomorphisms-commutative-finite-rings open import foundation.dependent-pair-types open import foundation.existential-quantification -open import foundation.functions +open import foundation.function-types open import foundation.propositional-truncations open import foundation.universe-levels diff --git a/src/finite-group-theory/abstract-quaternion-group.lagda.md b/src/finite-group-theory/abstract-quaternion-group.lagda.md index 1561fb5510..fa8ff62a4c 100644 --- a/src/finite-group-theory/abstract-quaternion-group.lagda.md +++ b/src/finite-group-theory/abstract-quaternion-group.lagda.md @@ -13,7 +13,7 @@ open import foundation.decidable-types open import foundation.dependent-pair-types open import foundation.empty-types open import foundation.equivalences -open import foundation.functions +open import foundation.function-types open import foundation.homotopies open import foundation.identity-types open import foundation.negation diff --git a/src/finite-group-theory/concrete-quaternion-group.lagda.md b/src/finite-group-theory/concrete-quaternion-group.lagda.md index e6839fd7e9..0c66d78790 100644 --- a/src/finite-group-theory/concrete-quaternion-group.lagda.md +++ b/src/finite-group-theory/concrete-quaternion-group.lagda.md @@ -13,6 +13,7 @@ open import foundation.dependent-pair-types open import foundation.equivalences open import foundation.identity-types open import foundation.isolated-points +open import foundation.transport open import foundation.universe-levels open import univalent-combinatorics.complements-isolated-points @@ -26,18 +27,6 @@ open import univalent-combinatorics.equivalences-cubes ## Definition ```agda -{- -equiv-face-standard-cube : - {k : ℕ} (d : dim-standard-cube (succ-ℕ k)) - (a : axis-standard-cube (succ-ℕ k) d) → - equiv-cube (face-cube (standard-cube (succ-ℕ k)) d a) (standard-cube k) -equiv-face-standard-cube {zero-ℕ} d a = - pair - {! is-equiv-is-empty!} - {!!} -equiv-face-standard-cube {succ-ℕ k} d a = {!!} --} - equiv-face-cube : (k : ℕ) (X Y : cube (succ-ℕ k)) (e : equiv-cube (succ-ℕ k) X Y) (d : dim-cube (succ-ℕ k) X) (a : axis-cube (succ-ℕ k) X d) → @@ -101,14 +90,6 @@ labelling-faces-cube-with-labeled-faces : labelling-cube k (face-cube k (cube-cube-with-labeled-faces k X) d a) labelling-faces-cube-with-labeled-faces k X = pr2 X -{- -standard-cube-with-labeled-faces : - (k : ℕ) → cube-with-labeled-faces k -standard-cube-with-labeled-faces k = - pair ( standard-cube (succ-ℕ k)) - ( λ d a → {!!}) --} - equiv-cube-with-labeled-faces : {k : ℕ} (X Y : cube-with-labeled-faces k) → UU lzero equiv-cube-with-labeled-faces {k} X Y = diff --git a/src/finite-group-theory/delooping-sign-homomorphism.lagda.md b/src/finite-group-theory/delooping-sign-homomorphism.lagda.md index 2f7b14e0f8..ee6ca4441d 100644 --- a/src/finite-group-theory/delooping-sign-homomorphism.lagda.md +++ b/src/finite-group-theory/delooping-sign-homomorphism.lagda.md @@ -33,7 +33,7 @@ open import foundation.equivalence-extensionality open import foundation.equivalence-relations open import foundation.equivalences open import foundation.function-extensionality -open import foundation.functions +open import foundation.function-types open import foundation.functoriality-propositional-truncation open import foundation.functoriality-set-quotients open import foundation.identity-types diff --git a/src/finite-group-theory/finite-groups.lagda.md b/src/finite-group-theory/finite-groups.lagda.md index a190291a02..f1a7d9d569 100644 --- a/src/finite-group-theory/finite-groups.lagda.md +++ b/src/finite-group-theory/finite-groups.lagda.md @@ -18,7 +18,7 @@ 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.function-types open import foundation.homotopies open import foundation.identity-types open import foundation.injective-maps diff --git a/src/finite-group-theory/finite-semigroups.lagda.md b/src/finite-group-theory/finite-semigroups.lagda.md index 21357cfbdb..41e1c80077 100644 --- a/src/finite-group-theory/finite-semigroups.lagda.md +++ b/src/finite-group-theory/finite-semigroups.lagda.md @@ -10,7 +10,7 @@ module finite-group-theory.finite-semigroups where open import elementary-number-theory.natural-numbers open import foundation.equivalences -open import foundation.functions +open import foundation.function-types open import foundation.functoriality-dependent-pair-types open import foundation.identity-types open import foundation.mere-equivalences diff --git a/src/finite-group-theory/finite-type-groups.lagda.md b/src/finite-group-theory/finite-type-groups.lagda.md index 7516b56988..13648a21e2 100644 --- a/src/finite-group-theory/finite-type-groups.lagda.md +++ b/src/finite-group-theory/finite-type-groups.lagda.md @@ -15,7 +15,7 @@ open import foundation.action-on-identifications-functions open import foundation.dependent-pair-types open import foundation.equality-dependent-pair-types open import foundation.function-extensionality -open import foundation.functions +open import foundation.function-types open import foundation.identity-types open import foundation.propositional-truncations open import foundation.propositions diff --git a/src/finite-group-theory/orbits-permutations.lagda.md b/src/finite-group-theory/orbits-permutations.lagda.md index f169696fb2..87f20fd6eb 100644 --- a/src/finite-group-theory/orbits-permutations.lagda.md +++ b/src/finite-group-theory/orbits-permutations.lagda.md @@ -38,7 +38,7 @@ open import foundation.equivalence-classes open import foundation.equivalence-extensionality open import foundation.equivalence-relations open import foundation.equivalences -open import foundation.functions +open import foundation.function-types open import foundation.identity-types open import foundation.injective-maps open import foundation.iterating-functions diff --git a/src/finite-group-theory/permutations-standard-finite-types.lagda.md b/src/finite-group-theory/permutations-standard-finite-types.lagda.md index 7fcedb019d..16994ecdee 100644 --- a/src/finite-group-theory/permutations-standard-finite-types.lagda.md +++ b/src/finite-group-theory/permutations-standard-finite-types.lagda.md @@ -24,7 +24,7 @@ open import foundation.equality-dependent-pair-types open import foundation.equivalence-extensionality open import foundation.equivalences open import foundation.equivalences-maybe -open import foundation.functions +open import foundation.function-types open import foundation.homotopies open import foundation.identity-types open import foundation.injective-maps diff --git a/src/finite-group-theory/permutations.lagda.md b/src/finite-group-theory/permutations.lagda.md index d3768ff691..735f0bf701 100644 --- a/src/finite-group-theory/permutations.lagda.md +++ b/src/finite-group-theory/permutations.lagda.md @@ -26,7 +26,7 @@ open import foundation.empty-types open import foundation.equality-dependent-pair-types open import foundation.equivalence-extensionality open import foundation.equivalences -open import foundation.functions +open import foundation.function-types open import foundation.identity-types open import foundation.iterating-functions open import foundation.iterating-involutions diff --git a/src/finite-group-theory/sign-homomorphism.lagda.md b/src/finite-group-theory/sign-homomorphism.lagda.md index 240959a05a..c291cbb7d1 100644 --- a/src/finite-group-theory/sign-homomorphism.lagda.md +++ b/src/finite-group-theory/sign-homomorphism.lagda.md @@ -21,7 +21,7 @@ open import foundation.decidable-propositions open import foundation.dependent-pair-types open import foundation.equivalence-extensionality open import foundation.equivalences -open import foundation.functions +open import foundation.function-types open import foundation.identity-types open import foundation.propositional-truncations open import foundation.sets diff --git a/src/finite-group-theory/simpson-delooping-sign-homomorphism.lagda.md b/src/finite-group-theory/simpson-delooping-sign-homomorphism.lagda.md index 15046d7b5b..26273191cd 100644 --- a/src/finite-group-theory/simpson-delooping-sign-homomorphism.lagda.md +++ b/src/finite-group-theory/simpson-delooping-sign-homomorphism.lagda.md @@ -32,7 +32,7 @@ open import foundation.equivalence-classes open import foundation.equivalence-extensionality open import foundation.equivalence-relations open import foundation.equivalences -open import foundation.functions +open import foundation.function-types open import foundation.identity-types open import foundation.involutions open import foundation.logical-equivalences diff --git a/src/finite-group-theory/transpositions-standard-finite-types.lagda.md b/src/finite-group-theory/transpositions-standard-finite-types.lagda.md index 8bf7464985..b04b3d2cd6 100644 --- a/src/finite-group-theory/transpositions-standard-finite-types.lagda.md +++ b/src/finite-group-theory/transpositions-standard-finite-types.lagda.md @@ -9,36 +9,38 @@ module finite-group-theory.transpositions-standard-finite-types where
Imports ```agda -open import univalent-combinatorics.standard-finite-types -open import univalent-combinatorics.equality-standard-finite-types - -open import elementary-number-theory.natural-numbers open import elementary-number-theory.inequality-natural-numbers +open import elementary-number-theory.natural-numbers + +open import finite-group-theory.permutations-standard-finite-types +open import finite-group-theory.transpositions open import foundation.action-on-identifications-functions open import foundation.automorphisms -open import foundation.universe-levels -open import foundation.dependent-pair-types open import foundation.coproduct-types -open import foundation.unit-type +open import foundation.dependent-pair-types +open import foundation.empty-types +open import foundation.equivalence-extensionality open import foundation.equivalences +open import foundation.function-types +open import foundation.functoriality-coproduct-types +open import foundation.homotopies open import foundation.identity-types open import foundation.negation -open import foundation.homotopies -open import foundation.functions -open import foundation.empty-types -open import foundation.functoriality-coproduct-types -open import foundation.equivalence-extensionality open import foundation.propositions +open import foundation.unit-type +open import foundation.universe-levels -open import lists.lists -open import lists.functoriality-lists open import lists.flattening-lists +open import lists.functoriality-lists +open import lists.lists -open import finite-group-theory.permutations-standard-finite-types -open import finite-group-theory.transpositions +open import univalent-combinatorics.equality-standard-finite-types +open import univalent-combinatorics.standard-finite-types ``` +
+ ## Idea Given `i` and `j` in `Fin n`, the transposition associated to `i` and `j` swap diff --git a/src/finite-group-theory/transpositions.lagda.md b/src/finite-group-theory/transpositions.lagda.md index a2a0ec8cd3..377a9425d3 100644 --- a/src/finite-group-theory/transpositions.lagda.md +++ b/src/finite-group-theory/transpositions.lagda.md @@ -27,7 +27,7 @@ open import foundation.equivalences open import foundation.equivalences-maybe open import foundation.fibers-of-maps open import foundation.function-extensionality -open import foundation.functions +open import foundation.function-types open import foundation.functoriality-coproduct-types open import foundation.homotopies open import foundation.identity-types diff --git a/src/foundation-core.lagda.md b/src/foundation-core.lagda.md index bce6596818..ad4d0da68a 100644 --- a/src/foundation-core.lagda.md +++ b/src/foundation-core.lagda.md @@ -16,7 +16,8 @@ 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.decidable-propositions public -open import foundation-core.dependent-paths public +open import foundation-core.dependent-identifications 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 @@ -28,6 +29,7 @@ open import foundation-core.equivalence-relations public open import foundation-core.equivalences public open import foundation-core.fibers-of-maps public open import foundation-core.function-extensionality public +open import foundation-core.function-types public open import foundation-core.functoriality-dependent-function-types public open import foundation-core.functoriality-dependent-pair-types public open import foundation-core.functoriality-function-types public diff --git a/src/foundation-core/cartesian-product-types.lagda.md b/src/foundation-core/cartesian-product-types.lagda.md index 4b58d81047..7833a040a4 100644 --- a/src/foundation-core/cartesian-product-types.lagda.md +++ b/src/foundation-core/cartesian-product-types.lagda.md @@ -7,8 +7,9 @@ module foundation-core.cartesian-product-types where
Imports ```agda -open import foundation.dependent-pair-types open import foundation.universe-levels + +open import foundation-core.dependent-pair-types ```
diff --git a/src/foundation-core/coherently-invertible-maps.lagda.md b/src/foundation-core/coherently-invertible-maps.lagda.md index c1013c4525..b96fae19f5 100644 --- a/src/foundation-core/coherently-invertible-maps.lagda.md +++ b/src/foundation-core/coherently-invertible-maps.lagda.md @@ -9,11 +9,11 @@ module foundation-core.coherently-invertible-maps where ```agda open import foundation.action-on-identifications-functions 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.function-types open import foundation-core.homotopies open import foundation-core.identity-types ``` diff --git a/src/foundation-core/commuting-cubes-of-maps.lagda.md b/src/foundation-core/commuting-cubes-of-maps.lagda.md index 2844188673..862584b3e7 100644 --- a/src/foundation-core/commuting-cubes-of-maps.lagda.md +++ b/src/foundation-core/commuting-cubes-of-maps.lagda.md @@ -10,10 +10,10 @@ module foundation-core.commuting-cubes-of-maps where open import foundation.action-on-identifications-functions 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.function-types open import foundation-core.homotopies open import foundation-core.identity-types ``` diff --git a/src/foundation-core/commuting-squares-of-maps.lagda.md b/src/foundation-core/commuting-squares-of-maps.lagda.md index ff33b0b6ab..8b1c450787 100644 --- a/src/foundation-core/commuting-squares-of-maps.lagda.md +++ b/src/foundation-core/commuting-squares-of-maps.lagda.md @@ -7,10 +7,10 @@ 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.function-types open import foundation-core.homotopies ``` diff --git a/src/foundation-core/commuting-triangles-of-maps.lagda.md b/src/foundation-core/commuting-triangles-of-maps.lagda.md index 9d7e5820b9..96a7707b2f 100644 --- a/src/foundation-core/commuting-triangles-of-maps.lagda.md +++ b/src/foundation-core/commuting-triangles-of-maps.lagda.md @@ -7,9 +7,9 @@ module foundation-core.commuting-triangles-of-maps where
Imports ```agda -open import foundation.functions open import foundation.universe-levels +open import foundation-core.function-types open import foundation-core.homotopies ``` diff --git a/src/foundation-core/contractible-maps.lagda.md b/src/foundation-core/contractible-maps.lagda.md index 014b68e4a5..7b94d30881 100644 --- a/src/foundation-core/contractible-maps.lagda.md +++ b/src/foundation-core/contractible-maps.lagda.md @@ -9,13 +9,13 @@ module foundation-core.contractible-maps where ```agda open import foundation.action-on-identifications-functions 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.equivalences open import foundation-core.fibers-of-maps +open import foundation-core.function-types open import foundation-core.homotopies open import foundation-core.identity-types ``` diff --git a/src/foundation-core/decidable-propositions.lagda.md b/src/foundation-core/decidable-propositions.lagda.md index 6353e975dc..af46769d4a 100644 --- a/src/foundation-core/decidable-propositions.lagda.md +++ b/src/foundation-core/decidable-propositions.lagda.md @@ -11,7 +11,6 @@ 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 @@ -19,6 +18,7 @@ open import foundation.universe-levels open import foundation-core.cartesian-product-types open import foundation-core.empty-types +open import foundation-core.function-types open import foundation-core.functoriality-dependent-pair-types open import foundation-core.propositions open import foundation-core.subtypes diff --git a/src/foundation-core/dependent-paths.lagda.md b/src/foundation-core/dependent-identifications.lagda.md similarity index 72% rename from src/foundation-core/dependent-paths.lagda.md rename to src/foundation-core/dependent-identifications.lagda.md index 53653c6651..cf3edc31b8 100644 --- a/src/foundation-core/dependent-paths.lagda.md +++ b/src/foundation-core/dependent-identifications.lagda.md @@ -1,7 +1,7 @@ -# Dependent paths +# Dependent identifications ```agda -module foundation-core.dependent-paths where +module foundation-core.dependent-identifications where ```
Imports @@ -31,13 +31,13 @@ where `tr` is the [transport](foundation-core.transport.md) function. ## Definition ```agda -path-over : +dependent-identification : {l1 l2 : Level} {A : UU l1} (B : A → UU l2) {x x' : A} (p : x ＝ x') → B x → B x' → UU l2 -path-over B p u v = (tr B p u ＝ v) +dependent-identification B p u v = (tr B p u ＝ v) -refl-path-over : +refl-dependent-identification : {l1 l2 : Level} {A : UU l1} (B : A → UU l2) {x : A} {y : B x} → - path-over B refl y y -refl-path-over B = refl + dependent-identification B refl y y +refl-dependent-identification B = refl ``` diff --git a/src/foundation-core/dependent-pair-types.lagda.md b/src/foundation-core/dependent-pair-types.lagda.md new file mode 100644 index 0000000000..f080c46625 --- /dev/null +++ b/src/foundation-core/dependent-pair-types.lagda.md @@ -0,0 +1,76 @@ +# Dependent pair types + +```agda +module foundation-core.dependent-pair-types where +``` + +
Imports + +```agda +open import foundation.universe-levels +``` + +
+ +## Idea + +Consider a type family `B` over `A`. The **dependent pair type** `Σ A B` is the +type consisting of **(dependent) pairs** `(a , b)` where `a : A` and `b : B a`. +Such pairs are sometimes called dependent pairs because the type of `b` depends +on the value of the first component `a`. + +## 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 7bc33c78d6..4b38fc3bf9 100644 --- a/src/foundation-core/diagonal-maps-of-types.lagda.md +++ b/src/foundation-core/diagonal-maps-of-types.lagda.md @@ -10,12 +10,12 @@ module foundation-core.diagonal-maps-of-types where open import foundation.action-on-identifications-functions 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.equivalences open import foundation-core.fibers-of-maps +open import foundation-core.function-types open import foundation-core.homotopies open import foundation-core.identity-types open import foundation-core.propositions diff --git a/src/foundation-core/embeddings.lagda.md b/src/foundation-core/embeddings.lagda.md index 1e0f607f8b..4d230156ae 100644 --- a/src/foundation-core/embeddings.lagda.md +++ b/src/foundation-core/embeddings.lagda.md @@ -9,10 +9,10 @@ module foundation-core.embeddings where ```agda open import foundation.action-on-identifications-functions open import foundation.dependent-pair-types -open import foundation.functions open import foundation.universe-levels open import foundation-core.equivalences +open import foundation-core.function-types open import foundation-core.identity-types ``` diff --git a/src/foundation-core/empty-types.lagda.md b/src/foundation-core/empty-types.lagda.md index 5a2c12bd60..c6e97dd980 100644 --- a/src/foundation-core/empty-types.lagda.md +++ b/src/foundation-core/empty-types.lagda.md @@ -8,12 +8,12 @@ module foundation-core.empty-types where ```agda open import foundation.dependent-pair-types -open import foundation.functions open import foundation.propositions open import foundation.universe-levels open import foundation-core.embeddings open import foundation-core.equivalences +open import foundation-core.function-types open import foundation-core.sets open import foundation-core.truncated-types open import foundation-core.truncation-levels diff --git a/src/foundation-core/endomorphisms.lagda.md b/src/foundation-core/endomorphisms.lagda.md index c7cdc965b5..c4677ddbe5 100644 --- a/src/foundation-core/endomorphisms.lagda.md +++ b/src/foundation-core/endomorphisms.lagda.md @@ -8,10 +8,10 @@ module foundation-core.endomorphisms where ```agda open import foundation.dependent-pair-types -open import foundation.functions open import foundation.sets open import foundation.universe-levels +open import foundation-core.function-types open import foundation-core.truncated-types open import foundation-core.truncation-levels diff --git a/src/foundation-core/equality-dependent-pair-types.lagda.md b/src/foundation-core/equality-dependent-pair-types.lagda.md index 6378ae237d..3ec6434f47 100644 --- a/src/foundation-core/equality-dependent-pair-types.lagda.md +++ b/src/foundation-core/equality-dependent-pair-types.lagda.md @@ -7,11 +7,11 @@ module foundation-core.equality-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.dependent-pair-types open import foundation-core.equivalences +open import foundation-core.function-types open import foundation-core.homotopies open import foundation-core.identity-types open import foundation-core.transport diff --git a/src/foundation-core/equivalence-induction.lagda.md b/src/foundation-core/equivalence-induction.lagda.md index 62ed07fccd..a69a720e82 100644 --- a/src/foundation-core/equivalence-induction.lagda.md +++ b/src/foundation-core/equivalence-induction.lagda.md @@ -8,11 +8,11 @@ module foundation-core.equivalence-induction where ```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.equivalences +open import foundation-core.function-types open import foundation-core.homotopies open import foundation-core.identity-types open import foundation-core.sections diff --git a/src/foundation-core/equivalences.lagda.md b/src/foundation-core/equivalences.lagda.md index cbf2204d46..e8d6c7bc1f 100644 --- a/src/foundation-core/equivalences.lagda.md +++ b/src/foundation-core/equivalences.lagda.md @@ -9,12 +9,12 @@ module foundation-core.equivalences where ```agda open import foundation.action-on-identifications-binary-functions open import foundation.action-on-identifications-functions -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.function-types open import foundation-core.homotopies open import foundation-core.identity-types open import foundation-core.retractions diff --git a/src/foundation-core/fibers-of-maps.lagda.md b/src/foundation-core/fibers-of-maps.lagda.md index 7952d59912..c8140cbda8 100644 --- a/src/foundation-core/fibers-of-maps.lagda.md +++ b/src/foundation-core/fibers-of-maps.lagda.md @@ -10,10 +10,10 @@ module foundation-core.fibers-of-maps where open import foundation.action-on-identifications-functions open import foundation.dependent-pair-types open import foundation.function-extensionality -open import foundation.functions open import foundation.universe-levels open import foundation-core.equivalences +open import foundation-core.function-types open import foundation-core.homotopies open import foundation-core.identity-types open import foundation-core.transport diff --git a/src/foundation-core/function-extensionality.lagda.md b/src/foundation-core/function-extensionality.lagda.md index 49c2339c19..8a9830b6e0 100644 --- a/src/foundation-core/function-extensionality.lagda.md +++ b/src/foundation-core/function-extensionality.lagda.md @@ -8,10 +8,10 @@ module foundation-core.function-extensionality where ```agda open import foundation.action-on-identifications-functions -open import foundation.functions open import foundation.universe-levels open import foundation-core.equivalences +open import foundation-core.function-types open import foundation-core.homotopies open import foundation-core.identity-types ``` diff --git a/src/foundation/functions.lagda.md b/src/foundation-core/function-types.lagda.md similarity index 96% rename from src/foundation/functions.lagda.md rename to src/foundation-core/function-types.lagda.md index 91bae6329b..779fe5beea 100644 --- a/src/foundation/functions.lagda.md +++ b/src/foundation-core/function-types.lagda.md @@ -1,7 +1,7 @@ -# Functions +# Function types ```agda -module foundation.functions where +module foundation-core.function-types where ```
Imports diff --git a/src/foundation-core/functoriality-dependent-function-types.lagda.md b/src/foundation-core/functoriality-dependent-function-types.lagda.md index 3bedb20104..a843564ab3 100644 --- a/src/foundation-core/functoriality-dependent-function-types.lagda.md +++ b/src/foundation-core/functoriality-dependent-function-types.lagda.md @@ -11,7 +11,6 @@ open import foundation.action-on-identifications-dependent-functions open import foundation.action-on-identifications-functions 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 @@ -21,6 +20,7 @@ open import foundation-core.contractible-maps open import foundation-core.contractible-types open import foundation-core.equivalences open import foundation-core.fibers-of-maps +open import foundation-core.function-types open import foundation-core.functoriality-dependent-pair-types open import foundation-core.homotopies open import foundation-core.identity-types diff --git a/src/foundation-core/functoriality-dependent-pair-types.lagda.md b/src/foundation-core/functoriality-dependent-pair-types.lagda.md index 641db3e306..4647b63ae2 100644 --- a/src/foundation-core/functoriality-dependent-pair-types.lagda.md +++ b/src/foundation-core/functoriality-dependent-pair-types.lagda.md @@ -8,7 +8,6 @@ module foundation-core.functoriality-dependent-pair-types where ```agda open import foundation.dependent-pair-types -open import foundation.functions open import foundation.universe-levels open import foundation-core.contractible-maps @@ -16,6 +15,7 @@ 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.function-types open import foundation-core.homotopies open import foundation-core.identity-types open import foundation-core.transport diff --git a/src/foundation-core/functoriality-function-types.lagda.md b/src/foundation-core/functoriality-function-types.lagda.md index a2a2830043..a2491192ee 100644 --- a/src/foundation-core/functoriality-function-types.lagda.md +++ b/src/foundation-core/functoriality-function-types.lagda.md @@ -10,12 +10,12 @@ module foundation-core.functoriality-function-types where open import foundation.action-on-identifications-functions 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.equivalences +open import foundation-core.function-types open import foundation-core.homotopies open import foundation-core.identity-types ``` diff --git a/src/foundation-core/homotopies.lagda.md b/src/foundation-core/homotopies.lagda.md index 863277ad80..d94bb57a82 100644 --- a/src/foundation-core/homotopies.lagda.md +++ b/src/foundation-core/homotopies.lagda.md @@ -10,10 +10,10 @@ module foundation-core.homotopies where open import foundation.action-on-identifications-dependent-functions open import foundation.action-on-identifications-functions open import foundation.commuting-squares-of-identifications -open import foundation.functions open import foundation.universe-levels -open import foundation-core.dependent-paths +open import foundation-core.dependent-identifications +open import foundation-core.function-types open import foundation-core.identity-types open import foundation-core.transport ``` @@ -39,11 +39,11 @@ module _ {-# INLINE eq-value #-} - map-compute-path-over-eq-value : + map-compute-dependent-identification-eq-value : {x y : X} (p : x ＝ y) (q : eq-value x) (r : eq-value y) → coherence-square-identifications (apd f p) r (ap (tr P p) q) (apd g p) → - path-over eq-value p q r - map-compute-path-over-eq-value refl q r = + dependent-identification eq-value p q r + map-compute-dependent-identification-eq-value refl q r = inv ∘ (concat' r (right-unit ∙ ap-id q)) ``` @@ -59,16 +59,19 @@ module _ {-# INLINE eq-value-function #-} - map-compute-path-over-eq-value-function : + map-compute-dependent-identification-eq-value-function : {x y : X} (p : x ＝ y) (q : eq-value f g x) (r : eq-value f g y) → coherence-square-identifications (ap f p) r q (ap g p) → - path-over eq-value-function p q r - map-compute-path-over-eq-value-function refl q r = inv ∘ concat' r right-unit + dependent-identification eq-value-function p q r + map-compute-dependent-identification-eq-value-function refl q r = + inv ∘ concat' r right-unit -map-compute-path-over-eq-value-id-id : +map-compute-dependent-identification-eq-value-id-id : {l1 : Level} {A : UU l1} {a b : A} (p : a ＝ b) (q : a ＝ a) (r : b ＝ b) → - coherence-square-identifications p r q p → path-over (eq-value id id) p q r -map-compute-path-over-eq-value-id-id refl q r s = inv (s ∙ right-unit) + coherence-square-identifications p r q p → + dependent-identification (eq-value id id) p q r +map-compute-dependent-identification-eq-value-id-id refl q r s = + inv (s ∙ right-unit) ``` ### Homotopies diff --git a/src/foundation-core/identity-types.lagda.md b/src/foundation-core/identity-types.lagda.md index ae02dda23c..7b1665dd02 100644 --- a/src/foundation-core/identity-types.lagda.md +++ b/src/foundation-core/identity-types.lagda.md @@ -53,14 +53,14 @@ identifications in arbitrary types. | ------------------------------------------------ | ------------------------------------------------------------------------------------------------------------------------- | | Action on identifications of binary functions | [`foundation.action-on-identifications-binary-functions`](foundation.action-on-identifications-binary-functions.md) | | Action on identifications of dependent functions | [`foundation.action-on-identifications-dependent-functions`](foundation.action-on-identifications-dependent-functions.md) | -| Action on identifications of functions | [`foundation.action-on-identifications-functions](foundation.action-on-identifications-functions.md) | +| Action on identifications of functions | [`foundation.action-on-identifications-functions`](foundation.action-on-identifications-functions.md) | | Binary transport | [`foundation.binary-transport`](foundation.binary-transport.md) | -| Commuting squares of identifications | [`foundation.commuting-squares-identifications`](foundation.commuting-squares-identifications.md) | -| Dependent paths (foundation) | [`foundation.dependent-paths`](foundation.dependent-paths.md) | -| Dependent paths (foundation-core) | [`foundation-core.dependent-paths`](foundation-core.dependent-paths.md) | +| Commuting squares of identifications | [`foundation.commuting-squares-of-identifications`](foundation.commuting-squares-of-identifications.md) | +| Dependent identifications (foundation) | [`foundation.dependent-identifications`](foundation.dependent-identifications.md) | +| Dependent identifications (foundation-core) | [`foundation-core.dependent-identifications`](foundation-core.dependent-identifications.md) | | The fundamental theorem of identity types | [`foundation.fundamental-theorem-of-identity-types`](foundation.fundamental-theorem-of-identity-types.md) | | Hexagons of identifications | [`foundation.hexagons-of-identifications`](foundation.hexagons-of-identifications.md) | -| Identity systems | [`foundation.identity-sytsemts`](foundation.identity-systems.md) | +| Identity systems | [`foundation.identity-systems`](foundation.identity-systems.md) | | The identity type (foundation) | [`foundation.identity-types`](foundation.identity-types.md) | | The identity type (foundation-core) | [`foundation-core.identity-types`](foundation-core.identity-types.md) | | Large identity types | [`foundation.large-identity-types`](foundation.large-identity-types.md) | diff --git a/src/foundation-core/injective-maps.lagda.md b/src/foundation-core/injective-maps.lagda.md index 0052e87fcd..245b6488e2 100644 --- a/src/foundation-core/injective-maps.lagda.md +++ b/src/foundation-core/injective-maps.lagda.md @@ -9,12 +9,12 @@ module foundation-core.injective-maps where ```agda open import foundation.action-on-identifications-functions 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.embeddings open import foundation-core.equivalences +open import foundation-core.function-types open import foundation-core.homotopies open import foundation-core.identity-types open import foundation-core.propositional-maps diff --git a/src/foundation-core/involutions.lagda.md b/src/foundation-core/involutions.lagda.md index 7b78a45da1..aa4a9deaa5 100644 --- a/src/foundation-core/involutions.lagda.md +++ b/src/foundation-core/involutions.lagda.md @@ -9,7 +9,7 @@ module foundation-core.involutions where ```agda open import foundation.automorphisms open import foundation.dependent-pair-types -open import foundation.functions +open import foundation.function-types open import foundation.universe-levels open import foundation-core.equivalences diff --git a/src/foundation-core/logical-equivalences.lagda.md b/src/foundation-core/logical-equivalences.lagda.md index 2cbd73d4b4..92ed74ffee 100644 --- a/src/foundation-core/logical-equivalences.lagda.md +++ b/src/foundation-core/logical-equivalences.lagda.md @@ -8,11 +8,11 @@ module foundation-core.logical-equivalences where ```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.function-types open import foundation-core.propositions ``` diff --git a/src/foundation-core/path-split-maps.lagda.md b/src/foundation-core/path-split-maps.lagda.md index 55c2207ad0..925e85f1f9 100644 --- a/src/foundation-core/path-split-maps.lagda.md +++ b/src/foundation-core/path-split-maps.lagda.md @@ -9,12 +9,12 @@ module foundation-core.path-split-maps where ```agda open import foundation.action-on-identifications-functions 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.equivalences +open import foundation-core.function-types open import foundation-core.identity-types open import foundation-core.sections ``` diff --git a/src/foundation-core/propositions.lagda.md b/src/foundation-core/propositions.lagda.md index 4b78c27e13..45a5b25af2 100644 --- a/src/foundation-core/propositions.lagda.md +++ b/src/foundation-core/propositions.lagda.md @@ -8,7 +8,6 @@ module foundation-core.propositions where ```agda open import foundation.dependent-pair-types -open import foundation.functions open import foundation.universe-levels open import foundation-core.cartesian-product-types @@ -16,6 +15,7 @@ open import foundation-core.contractible-types open import foundation-core.equality-dependent-pair-types open import foundation-core.equivalences open import foundation-core.function-extensionality +open import foundation-core.function-types open import foundation-core.homotopies open import foundation-core.identity-types open import foundation-core.transport diff --git a/src/foundation-core/pullbacks.lagda.md b/src/foundation-core/pullbacks.lagda.md index 06bb161be0..c1dcf3a086 100644 --- a/src/foundation-core/pullbacks.lagda.md +++ b/src/foundation-core/pullbacks.lagda.md @@ -11,7 +11,6 @@ open import foundation.action-on-identifications-functions 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 @@ -26,6 +25,7 @@ open import foundation-core.diagonal-maps-of-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.function-types open import foundation-core.functoriality-dependent-pair-types open import foundation-core.homotopies open import foundation-core.universal-property-pullbacks diff --git a/src/foundation-core/retractions.lagda.md b/src/foundation-core/retractions.lagda.md index 7f5f621e01..d3527397bf 100644 --- a/src/foundation-core/retractions.lagda.md +++ b/src/foundation-core/retractions.lagda.md @@ -8,10 +8,10 @@ module foundation-core.retractions where ```agda open import foundation.action-on-identifications-functions -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.function-types open import foundation-core.homotopies open import foundation-core.identity-types ``` diff --git a/src/foundation-core/sections.lagda.md b/src/foundation-core/sections.lagda.md index 2173c32fda..7e189e7ae6 100644 --- a/src/foundation-core/sections.lagda.md +++ b/src/foundation-core/sections.lagda.md @@ -7,10 +7,10 @@ module foundation-core.sections where
Imports ```agda -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.function-types open import foundation-core.homotopies ``` diff --git a/src/foundation-core/singleton-induction.lagda.md b/src/foundation-core/singleton-induction.lagda.md index 7fa47ffc31..d698f31ad5 100644 --- a/src/foundation-core/singleton-induction.lagda.md +++ b/src/foundation-core/singleton-induction.lagda.md @@ -9,10 +9,10 @@ module foundation-core.singleton-induction where ```agda open import foundation.action-on-identifications-functions 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.function-types open import foundation-core.homotopies open import foundation-core.identity-types open import foundation-core.sections diff --git a/src/foundation-core/small-types.lagda.md b/src/foundation-core/small-types.lagda.md index e1935fd2aa..2def16cce9 100644 --- a/src/foundation-core/small-types.lagda.md +++ b/src/foundation-core/small-types.lagda.md @@ -15,6 +15,7 @@ open import foundation.identity-types open import foundation.mere-equivalences open import foundation.propositional-truncations open import foundation.raising-universe-levels +open import foundation.transport open import foundation.type-arithmetic-dependent-pair-types open import foundation.unit-type open import foundation.univalence diff --git a/src/foundation-core/subtypes.lagda.md b/src/foundation-core/subtypes.lagda.md index 4887b0238c..0284fb9448 100644 --- a/src/foundation-core/subtypes.lagda.md +++ b/src/foundation-core/subtypes.lagda.md @@ -9,13 +9,13 @@ module foundation-core.subtypes where ```agda open import foundation.action-on-identifications-functions 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.function-types open import foundation-core.functoriality-dependent-pair-types open import foundation-core.identity-types open import foundation-core.logical-equivalences diff --git a/src/foundation-core/transport.lagda.md b/src/foundation-core/transport.lagda.md index a2f0eae2ac..a39e2f1e21 100644 --- a/src/foundation-core/transport.lagda.md +++ b/src/foundation-core/transport.lagda.md @@ -7,7 +7,6 @@ module foundation-core.transport where
Imports ```agda -open import foundation.action-on-identifications-functions open import foundation.universe-levels open import foundation-core.identity-types @@ -35,7 +34,9 @@ tr : tr B refl b = b ``` -### Laws for transport +## Properties + +### Transport preserves concatenation of identifications ```agda module _ @@ -46,36 +47,33 @@ module _ {x y z : A} (p : x ＝ y) (q : y ＝ z) (b : B x) → tr B (p ∙ q) b ＝ tr B q (tr B p b) tr-concat refl q b = refl +``` + +### Transposing transport along the inverse of an identification + +```agda +module _ + {l1 l2 : Level} {A : UU l1} {B : A → UU l2} + where eq-transpose-tr : {x y : A} (p : x ＝ y) {u : B x} {v : B y} → - v ＝ (tr B p u) → (tr B (inv p) v) ＝ u + v ＝ tr B p u → tr B (inv p) v ＝ u eq-transpose-tr refl q = q eq-transpose-tr' : {x y : A} (p : x ＝ y) {u : B x} {v : B y} → - (tr B p u) ＝ v → u ＝ (tr B (inv p) v) + tr B p u ＝ v → u ＝ tr B (inv p) v eq-transpose-tr' refl q = q +``` -tr-ap : - {l1 l2 l3 l4 : Level} {A : UU l1} {B : A → UU l2} {C : UU l3} {D : C → UU l4} - (f : A → C) (g : (x : A) → B x → D (f x)) {x y : A} (p : x ＝ y) (z : B x) → - (tr D (ap f p) (g x z)) ＝ (g y (tr B p z)) -tr-ap f g refl z = refl +### Every family of maps preserves transport +```agda preserves-tr : {l1 l2 l3 : Level} {I : UU l1} {A : I → UU l2} {B : I → UU l3} - (f : (i : I) → A i → B i) {i j : I} (p : i ＝ j) (x : A i) → + (f : (i : I) → A i → B i) + {i j : I} (p : i ＝ j) (x : A i) → f j (tr A p x) ＝ tr B p (f i x) preserves-tr f refl x = refl - -tr-Id-left : - {l : Level} {A : UU l} {a b c : A} (q : b ＝ c) (p : b ＝ a) → - (tr (_＝ a) q p) ＝ ((inv q) ∙ p) -tr-Id-left refl p = refl - -tr-Id-right : - {l : Level} {A : UU l} {a b c : A} (q : b ＝ c) (p : a ＝ b) → - tr (a ＝_) q p ＝ (p ∙ q) -tr-Id-right refl refl = refl ``` diff --git a/src/foundation-core/truncated-maps.lagda.md b/src/foundation-core/truncated-maps.lagda.md index ab737e39d6..8a956c8ad8 100644 --- a/src/foundation-core/truncated-maps.lagda.md +++ b/src/foundation-core/truncated-maps.lagda.md @@ -10,13 +10,13 @@ module foundation-core.truncated-maps where open import foundation.action-on-identifications-functions 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.equivalences open import foundation-core.fibers-of-maps +open import foundation-core.function-types open import foundation-core.functoriality-dependent-pair-types open import foundation-core.homotopies open import foundation-core.identity-types diff --git a/src/foundation-core/universal-property-pullbacks.lagda.md b/src/foundation-core/universal-property-pullbacks.lagda.md index 7c16b8da53..d92c1a4a75 100644 --- a/src/foundation-core/universal-property-pullbacks.lagda.md +++ b/src/foundation-core/universal-property-pullbacks.lagda.md @@ -10,12 +10,12 @@ module foundation-core.universal-property-pullbacks where open import foundation.action-on-identifications-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.contractible-maps open import foundation-core.contractible-types open import foundation-core.equivalences +open import foundation-core.function-types open import foundation-core.functoriality-dependent-pair-types open import foundation-core.functoriality-function-types open import foundation-core.homotopies diff --git a/src/foundation-core/universal-property-truncation.lagda.md b/src/foundation-core/universal-property-truncation.lagda.md index 8acb122ef9..3686de675c 100644 --- a/src/foundation-core/universal-property-truncation.lagda.md +++ b/src/foundation-core/universal-property-truncation.lagda.md @@ -10,12 +10,12 @@ module foundation-core.universal-property-truncation where 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.function-types open import foundation-core.functoriality-dependent-pair-types open import foundation-core.homotopies open import foundation-core.identity-types diff --git a/src/foundation.lagda.md b/src/foundation.lagda.md index 52dae9aa63..a25f0712cb 100644 --- a/src/foundation.lagda.md +++ b/src/foundation.lagda.md @@ -72,8 +72,8 @@ open import foundation.decidable-relations public open import foundation.decidable-subtypes public open import foundation.decidable-types public open import foundation.dependent-binomial-theorem public +open import foundation.dependent-identifications public open import foundation.dependent-pair-types public -open import foundation.dependent-paths public open import foundation.descent-coproduct-types public open import foundation.descent-dependent-pair-types public open import foundation.descent-empty-types public @@ -118,8 +118,8 @@ open import foundation.fibered-maps public open import foundation.fibers-of-maps public open import foundation.full-subtypes public open import foundation.function-extensionality public +open import foundation.function-types public open import foundation.functional-correspondences public -open import foundation.functions public open import foundation.functoriality-cartesian-product-types public open import foundation.functoriality-coproduct-types public open import foundation.functoriality-dependent-function-types public diff --git a/src/foundation/0-connected-types.lagda.md b/src/foundation/0-connected-types.lagda.md index b4f502e21e..95d7acb74c 100644 --- a/src/foundation/0-connected-types.lagda.md +++ b/src/foundation/0-connected-types.lagda.md @@ -11,7 +11,6 @@ open import foundation.action-on-identifications-functions 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 @@ -26,6 +25,7 @@ open import foundation.universe-levels open import foundation-core.cartesian-product-types open import foundation-core.equivalences open import foundation-core.fibers-of-maps +open import foundation-core.function-types open import foundation-core.identity-types open import foundation-core.propositions open import foundation-core.truncated-maps diff --git a/src/foundation/0-maps.lagda.md b/src/foundation/0-maps.lagda.md index a49905c877..2a6319549b 100644 --- a/src/foundation/0-maps.lagda.md +++ b/src/foundation/0-maps.lagda.md @@ -8,10 +8,10 @@ module foundation.0-maps where ```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.function-types open import foundation-core.functoriality-dependent-pair-types open import foundation-core.homotopies open import foundation-core.sets diff --git a/src/foundation/action-on-identifications-functions.lagda.md b/src/foundation/action-on-identifications-functions.lagda.md index 7930cef231..a270aa3f96 100644 --- a/src/foundation/action-on-identifications-functions.lagda.md +++ b/src/foundation/action-on-identifications-functions.lagda.md @@ -7,11 +7,12 @@ module foundation.action-on-identifications-functions where
Imports ```agda -open import foundation.functions open import foundation.universe-levels open import foundation-core.constant-maps +open import foundation-core.function-types open import foundation-core.identity-types +open import foundation-core.transport ```
@@ -96,6 +97,17 @@ ap-const : ap-const b refl = refl ``` +### Transporting along the action on identifications of a function + +```agda +tr-ap : + {l1 l2 l3 l4 : Level} {A : UU l1} {B : A → UU l2} {C : UU l3} {D : C → UU l4} + (f : A → C) (g : (x : A) → B x → D (f x)) + {x y : A} (p : x ＝ y) (z : B x) → + tr D (ap f p) (g x z) ＝ g y (tr B p z) +tr-ap f g refl z = refl +``` + ### The action on identifications of concatenating by `refl` on the right Note that `_∙ refl` is only homotopic to the identity function. Therefore we 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 c2c6ff676a..8e76432c07 100644 --- a/src/foundation/arithmetic-law-coproduct-and-sigma-decompositions.lagda.md +++ b/src/foundation/arithmetic-law-coproduct-and-sigma-decompositions.lagda.md @@ -10,7 +10,6 @@ module foundation.arithmetic-law-coproduct-and-sigma-decompositions where 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 @@ -22,6 +21,7 @@ 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.function-types open import foundation-core.functoriality-dependent-pair-types open import foundation-core.homotopies open import foundation-core.identity-types 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 79918383cb..5a34d34c97 100644 --- a/src/foundation/arithmetic-law-product-and-pi-decompositions.lagda.md +++ b/src/foundation/arithmetic-law-product-and-pi-decompositions.lagda.md @@ -10,7 +10,6 @@ module foundation.arithmetic-law-product-and-pi-decompositions where 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.pi-decompositions open import foundation.product-decompositions @@ -22,6 +21,7 @@ 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.function-types open import foundation-core.functoriality-dependent-pair-types open import foundation-core.homotopies open import foundation-core.identity-types diff --git a/src/foundation/axiom-l.lagda.md b/src/foundation/axiom-l.lagda.md index 3430271a26..f7c903b1bd 100644 --- a/src/foundation/axiom-l.lagda.md +++ b/src/foundation/axiom-l.lagda.md @@ -13,7 +13,6 @@ 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 @@ -23,6 +22,7 @@ open import foundation.universe-levels open import foundation-core.contractible-types open import foundation-core.fibers-of-maps +open import foundation-core.function-types open import foundation-core.functoriality-dependent-pair-types open import foundation-core.identity-types open import foundation-core.injective-maps diff --git a/src/foundation/axiom-of-choice.lagda.md b/src/foundation/axiom-of-choice.lagda.md index ea089242b8..4f8cebe20e 100644 --- a/src/foundation/axiom-of-choice.lagda.md +++ b/src/foundation/axiom-of-choice.lagda.md @@ -9,7 +9,6 @@ module foundation.axiom-of-choice where ```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 @@ -20,6 +19,7 @@ open import foundation.universe-levels open import foundation-core.equivalences open import foundation-core.fibers-of-maps +open import foundation-core.function-types open import foundation-core.functoriality-dependent-pair-types open import foundation-core.homotopies open import foundation-core.identity-types 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 5262453435..8f52b2aee3 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.function-types ```
diff --git a/src/foundation/binary-functoriality-set-quotients.lagda.md b/src/foundation/binary-functoriality-set-quotients.lagda.md index f554097a6c..320fc1cd58 100644 --- a/src/foundation/binary-functoriality-set-quotients.lagda.md +++ b/src/foundation/binary-functoriality-set-quotients.lagda.md @@ -13,7 +13,6 @@ 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 @@ -29,6 +28,7 @@ open import foundation.universe-levels open import foundation-core.contractible-types open import foundation-core.equivalence-relations open import foundation-core.equivalences +open import foundation-core.function-types open import foundation-core.functoriality-dependent-function-types open import foundation-core.functoriality-dependent-pair-types open import foundation-core.propositions diff --git a/src/foundation/binary-transport.lagda.md b/src/foundation/binary-transport.lagda.md index d6cb611fbd..5de9ca3d4f 100644 --- a/src/foundation/binary-transport.lagda.md +++ b/src/foundation/binary-transport.lagda.md @@ -9,10 +9,10 @@ module foundation.binary-transport where ```agda open import foundation.action-on-identifications-functions open import foundation.dependent-pair-types -open import foundation.functions open import foundation.universe-levels open import foundation-core.equivalences +open import foundation-core.function-types open import foundation-core.identity-types open import foundation-core.transport ``` diff --git a/src/foundation/booleans.lagda.md b/src/foundation/booleans.lagda.md index 2c3354a503..cc917fc2f0 100644 --- a/src/foundation/booleans.lagda.md +++ b/src/foundation/booleans.lagda.md @@ -8,7 +8,6 @@ module foundation.booleans where ```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 @@ -17,6 +16,7 @@ open import foundation-core.constant-maps open import foundation-core.coproduct-types open import foundation-core.empty-types open import foundation-core.equivalences +open import foundation-core.function-types open import foundation-core.homotopies open import foundation-core.identity-types open import foundation-core.injective-maps diff --git a/src/foundation/cartesian-product-types.lagda.md b/src/foundation/cartesian-product-types.lagda.md index ee4be138ee..eefc3e3266 100644 --- a/src/foundation/cartesian-product-types.lagda.md +++ b/src/foundation/cartesian-product-types.lagda.md @@ -9,13 +9,27 @@ open import foundation-core.cartesian-product-types public
Imports ```agda +open import foundation.dependent-pair-types open import foundation.subuniverses open import foundation.universe-levels + +open import foundation-core.identity-types +open import foundation-core.transport ```
-## Definitions +## Properties + +### Transport in a family of cartesian products + +```agda +tr-prod : + {l1 l2 : Level} {A : UU l1} {a0 a1 : A} + (B C : A → UU l2) (p : a0 ＝ a1) (u : B a0 × C a0) → + (tr (λ a → B a × C a) p u) ＝ (pair (tr B p (pr1 u)) (tr C p (pr2 u))) +tr-prod B C refl u = refl +``` ### Subuniverses closed under cartesian product types diff --git a/src/foundation/cartesian-products-set-quotients.lagda.md b/src/foundation/cartesian-products-set-quotients.lagda.md index 8de5342e3a..2d985134ba 100644 --- a/src/foundation/cartesian-products-set-quotients.lagda.md +++ b/src/foundation/cartesian-products-set-quotients.lagda.md @@ -11,7 +11,6 @@ open import foundation.action-on-identifications-functions 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 @@ -24,6 +23,7 @@ open import foundation-core.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.function-types open import foundation-core.homotopies open import foundation-core.identity-types open import foundation-core.propositions diff --git a/src/foundation/category-of-sets.lagda.md b/src/foundation/category-of-sets.lagda.md index 553cd91dde..50104496b5 100644 --- a/src/foundation/category-of-sets.lagda.md +++ b/src/foundation/category-of-sets.lagda.md @@ -9,10 +9,10 @@ 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.function-types open import foundation-core.identity-types ``` diff --git a/src/foundation/coherently-invertible-maps.lagda.md b/src/foundation/coherently-invertible-maps.lagda.md index 63a65c1172..41132b88bf 100644 --- a/src/foundation/coherently-invertible-maps.lagda.md +++ b/src/foundation/coherently-invertible-maps.lagda.md @@ -12,7 +12,6 @@ open import foundation-core.coherently-invertible-maps public open import foundation.action-on-identifications-functions 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 @@ -21,6 +20,7 @@ open import foundation.universe-levels open import foundation-core.contractible-maps open import foundation-core.contractible-types open import foundation-core.fibers-of-maps +open import foundation-core.function-types open import foundation-core.functoriality-dependent-pair-types open import foundation-core.homotopies open import foundation-core.propositions diff --git a/src/foundation/commuting-cubes-of-maps.lagda.md b/src/foundation/commuting-cubes-of-maps.lagda.md index 7389275335..6bb15e4748 100644 --- a/src/foundation/commuting-cubes-of-maps.lagda.md +++ b/src/foundation/commuting-cubes-of-maps.lagda.md @@ -11,9 +11,9 @@ open import foundation-core.commuting-cubes-of-maps public ```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.function-types open import foundation-core.homotopies ``` diff --git a/src/foundation/commuting-squares-of-identifications.lagda.md b/src/foundation/commuting-squares-of-identifications.lagda.md index 02a98c3b0f..471bbd126f 100644 --- a/src/foundation/commuting-squares-of-identifications.lagda.md +++ b/src/foundation/commuting-squares-of-identifications.lagda.md @@ -8,9 +8,9 @@ module foundation.commuting-squares-of-identifications where ```agda open import foundation.action-on-identifications-binary-functions -open import foundation.functions open import foundation.universe-levels +open import foundation-core.function-types open import foundation-core.identity-types ``` diff --git a/src/foundation/commuting-squares-of-maps.lagda.md b/src/foundation/commuting-squares-of-maps.lagda.md index 4714295556..894d0be1ba 100644 --- a/src/foundation/commuting-squares-of-maps.lagda.md +++ b/src/foundation/commuting-squares-of-maps.lagda.md @@ -11,10 +11,10 @@ open import foundation-core.commuting-squares-of-maps public ```agda open import foundation.action-on-identifications-functions open import foundation.equivalences -open import foundation.functions open import foundation.functoriality-function-types open import foundation.universe-levels +open import foundation-core.function-types open import foundation-core.identity-types ``` diff --git a/src/foundation/commuting-triangles-of-homotopies.lagda.md b/src/foundation/commuting-triangles-of-homotopies.lagda.md index c4b96cca09..8447a71056 100644 --- a/src/foundation/commuting-triangles-of-homotopies.lagda.md +++ b/src/foundation/commuting-triangles-of-homotopies.lagda.md @@ -8,9 +8,9 @@ module foundation.commuting-triangles-of-homotopies where ```agda open import foundation.action-on-identifications-functions -open import foundation.functions open import foundation.universe-levels +open import foundation-core.function-types open import foundation-core.homotopies open import foundation-core.identity-types ``` diff --git a/src/foundation/complements-subtypes.lagda.md b/src/foundation/complements-subtypes.lagda.md index 4cfd41a52c..441baca4ff 100644 --- a/src/foundation/complements-subtypes.lagda.md +++ b/src/foundation/complements-subtypes.lagda.md @@ -10,12 +10,12 @@ 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.function-types open import foundation-core.subtypes ``` diff --git a/src/foundation/complements.lagda.md b/src/foundation/complements.lagda.md index 0f29fe5564..6a86725f38 100644 --- a/src/foundation/complements.lagda.md +++ b/src/foundation/complements.lagda.md @@ -8,10 +8,10 @@ module foundation.complements where ```agda 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.function-types ```
diff --git a/src/foundation/cones-over-cospans.lagda.md b/src/foundation/cones-over-cospans.lagda.md index a7752ee5d9..40e0417706 100644 --- a/src/foundation/cones-over-cospans.lagda.md +++ b/src/foundation/cones-over-cospans.lagda.md @@ -8,7 +8,6 @@ module foundation.cones-over-cospans where ```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 @@ -17,6 +16,7 @@ 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.function-types open import foundation-core.identity-types open import foundation-core.transport ``` diff --git a/src/foundation/connected-maps.lagda.md b/src/foundation/connected-maps.lagda.md index a2fd7c26ff..9100f3c990 100644 --- a/src/foundation/connected-maps.lagda.md +++ b/src/foundation/connected-maps.lagda.md @@ -9,7 +9,6 @@ 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 @@ -25,6 +24,7 @@ open import foundation-core.contractible-types open import foundation-core.embeddings open import foundation-core.equivalences open import foundation-core.fibers-of-maps +open import foundation-core.function-types open import foundation-core.identity-types open import foundation-core.propositions open import foundation-core.subtypes diff --git a/src/foundation/connected-types.lagda.md b/src/foundation/connected-types.lagda.md index 5c725a0a9b..9a8b21bb10 100644 --- a/src/foundation/connected-types.lagda.md +++ b/src/foundation/connected-types.lagda.md @@ -8,12 +8,12 @@ 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.function-types open import foundation-core.propositions open import foundation-core.truncated-types open import foundation-core.truncation-levels diff --git a/src/foundation/constant-maps.lagda.md b/src/foundation/constant-maps.lagda.md index 0b45fca91f..01b58e58e5 100644 --- a/src/foundation/constant-maps.lagda.md +++ b/src/foundation/constant-maps.lagda.md @@ -12,7 +12,6 @@ open import foundation-core.constant-maps public 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 @@ -23,6 +22,7 @@ open import foundation-core.embeddings open import foundation-core.equivalences open import foundation-core.fibers-of-maps open import foundation-core.function-extensionality +open import foundation-core.function-types open import foundation-core.identity-types open import foundation-core.injective-maps open import foundation-core.propositional-maps diff --git a/src/foundation/constant-type-families.lagda.md b/src/foundation/constant-type-families.lagda.md index 2ab6de981e..77e829f97d 100644 --- a/src/foundation/constant-type-families.lagda.md +++ b/src/foundation/constant-type-families.lagda.md @@ -12,7 +12,7 @@ open import foundation.action-on-identifications-functions open import foundation.dependent-pair-types open import foundation.universe-levels -open import foundation-core.dependent-paths +open import foundation-core.dependent-identifications open import foundation-core.equivalences open import foundation-core.identity-types ``` @@ -70,7 +70,7 @@ pr2 (is-constant-type-family-constant-type-family A B) a = id-equiv ```agda tr-constant-type-family : {l1 l2 : Level} {A : UU l1} {B : UU l2} {x y : A} (p : x ＝ y) (b : B) → - path-over (constant-type-family A B) p b b + dependent-identification (constant-type-family A B) p b b tr-constant-type-family refl b = refl ``` diff --git a/src/foundation/contractible-types.lagda.md b/src/foundation/contractible-types.lagda.md index 020596438f..4d58a6f100 100644 --- a/src/foundation/contractible-types.lagda.md +++ b/src/foundation/contractible-types.lagda.md @@ -12,7 +12,6 @@ open import foundation-core.contractible-types public open import foundation.action-on-identifications-functions 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 @@ -20,6 +19,7 @@ open import foundation.universe-levels open import foundation-core.constant-maps open import foundation-core.contractible-maps open import foundation-core.equivalences +open import foundation-core.function-types open import foundation-core.functoriality-dependent-pair-types open import foundation-core.identity-types open import foundation-core.propositions diff --git a/src/foundation/coproduct-decompositions-subuniverse.lagda.md b/src/foundation/coproduct-decompositions-subuniverse.lagda.md index ecb6f55b78..02f9433697 100644 --- a/src/foundation/coproduct-decompositions-subuniverse.lagda.md +++ b/src/foundation/coproduct-decompositions-subuniverse.lagda.md @@ -12,7 +12,6 @@ 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 @@ -27,6 +26,7 @@ open import foundation.universe-levels open import foundation-core.cartesian-product-types open import foundation-core.contractible-types open import foundation-core.equality-dependent-pair-types +open import foundation-core.function-types open import foundation-core.functoriality-dependent-pair-types open import foundation-core.homotopies open import foundation-core.identity-types diff --git a/src/foundation/coproduct-decompositions.lagda.md b/src/foundation/coproduct-decompositions.lagda.md index 7b30f2872d..1c66e2bc98 100644 --- a/src/foundation/coproduct-decompositions.lagda.md +++ b/src/foundation/coproduct-decompositions.lagda.md @@ -13,9 +13,9 @@ 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.identity-types open import foundation.structure-identity-principle open import foundation.transport open import foundation.type-arithmetic-coproduct-types @@ -29,9 +29,9 @@ open import foundation-core.contractible-types open import foundation-core.coproduct-types open import foundation-core.equality-dependent-pair-types open import foundation-core.fibers-of-maps +open import foundation-core.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 univalent-combinatorics.equality-standard-finite-types @@ -278,7 +278,7 @@ module _ ( λ x → fib f x) ( pr1 a , x , pr2 a)))) ( λ z → pr1 (pr1 z) ＝ x)) - ( eq-pair-Σ p ( tr-Id p (inv p) ∙ left-inv p)) + ( eq-pair-Σ p ( tr-Id-right p (inv p) ∙ left-inv p)) ( ( ( x , (inv p)) , ( ap ( inl) @@ -341,7 +341,7 @@ module _ ( λ x → fib f x) ( pr1 a , x , pr2 a)))) ( λ z → pr1 (pr1 z) ＝ x)) - ( eq-pair-Σ p ( tr-Id p (inv p) ∙ left-inv p)) + ( eq-pair-Σ p ( tr-Id-right p (inv p) ∙ left-inv p)) ( ( ( x , (inv p)) , ( ap ( inr) diff --git a/src/foundation/coproduct-types.lagda.md b/src/foundation/coproduct-types.lagda.md index a8e251fc1f..110dbaf3ab 100644 --- a/src/foundation/coproduct-types.lagda.md +++ b/src/foundation/coproduct-types.lagda.md @@ -11,7 +11,6 @@ open import foundation-core.coproduct-types public ```agda open import foundation.action-on-identifications-functions open import foundation.dependent-pair-types -open import foundation.functions open import foundation.noncontractible-types open import foundation.subuniverses open import foundation.unit-type @@ -20,6 +19,7 @@ open import foundation.universe-levels open import foundation-core.contractible-types open import foundation-core.empty-types open import foundation-core.equivalences +open import foundation-core.function-types open import foundation-core.homotopies open import foundation-core.identity-types open import foundation-core.injective-maps diff --git a/src/foundation/coslice.lagda.md b/src/foundation/coslice.lagda.md index e9038837b9..bc9452e0e2 100644 --- a/src/foundation/coslice.lagda.md +++ b/src/foundation/coslice.lagda.md @@ -9,11 +9,11 @@ module foundation.coslice where ```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.equivalences +open import foundation-core.function-types open import foundation-core.homotopies open import foundation-core.identity-types ``` diff --git a/src/foundation/decidable-dependent-pair-types.lagda.md b/src/foundation/decidable-dependent-pair-types.lagda.md index 6aebc9f8d2..e94c0abda0 100644 --- a/src/foundation/decidable-dependent-pair-types.lagda.md +++ b/src/foundation/decidable-dependent-pair-types.lagda.md @@ -9,7 +9,6 @@ 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 @@ -17,6 +16,7 @@ open import foundation.universe-levels open import foundation-core.coproduct-types open import foundation-core.equivalences +open import foundation-core.function-types open import foundation-core.functoriality-dependent-pair-types ``` diff --git a/src/foundation/decidable-embeddings.lagda.md b/src/foundation/decidable-embeddings.lagda.md index 20e073d0d8..3ec5a9bb95 100644 --- a/src/foundation/decidable-embeddings.lagda.md +++ b/src/foundation/decidable-embeddings.lagda.md @@ -14,7 +14,6 @@ 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 @@ -33,6 +32,7 @@ open import foundation-core.coproduct-types open import foundation-core.decidable-propositions open import foundation-core.empty-types open import foundation-core.fibers-of-maps +open import foundation-core.function-types open import foundation-core.functoriality-dependent-pair-types open import foundation-core.propositions ``` diff --git a/src/foundation/decidable-equivalence-relations.lagda.md b/src/foundation/decidable-equivalence-relations.lagda.md index 6940d8d50f..c60af50e45 100644 --- a/src/foundation/decidable-equivalence-relations.lagda.md +++ b/src/foundation/decidable-equivalence-relations.lagda.md @@ -18,7 +18,6 @@ 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 @@ -38,6 +37,7 @@ 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.function-types open import foundation-core.functoriality-dependent-pair-types open import foundation-core.homotopies open import foundation-core.identity-types diff --git a/src/foundation/decidable-maps.lagda.md b/src/foundation/decidable-maps.lagda.md index e2ca1d0d0a..7645c79310 100644 --- a/src/foundation/decidable-maps.lagda.md +++ b/src/foundation/decidable-maps.lagda.md @@ -11,11 +11,11 @@ open import foundation.action-on-identifications-functions 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.equivalences open import foundation-core.fibers-of-maps +open import foundation-core.function-types open import foundation-core.functoriality-dependent-pair-types open import foundation-core.identity-types open import foundation-core.retractions diff --git a/src/foundation/decidable-propositions.lagda.md b/src/foundation/decidable-propositions.lagda.md index b8a4625ca3..2d492c53c0 100644 --- a/src/foundation/decidable-propositions.lagda.md +++ b/src/foundation/decidable-propositions.lagda.md @@ -16,7 +16,6 @@ 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 @@ -27,6 +26,7 @@ open import foundation.universe-levels open import foundation-core.contractible-types open import foundation-core.coproduct-types +open import foundation-core.function-types open import foundation-core.homotopies open import foundation-core.identity-types open import foundation-core.logical-equivalences diff --git a/src/foundation/decidable-subtypes.lagda.md b/src/foundation/decidable-subtypes.lagda.md index e562439d7e..611ae47d43 100644 --- a/src/foundation/decidable-subtypes.lagda.md +++ b/src/foundation/decidable-subtypes.lagda.md @@ -13,7 +13,6 @@ 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 @@ -21,6 +20,7 @@ open import foundation.universe-levels open import foundation-core.embeddings open import foundation-core.equivalences +open import foundation-core.function-types open import foundation-core.identity-types open import foundation-core.injective-maps open import foundation-core.logical-equivalences diff --git a/src/foundation/decidable-types.lagda.md b/src/foundation/decidable-types.lagda.md index 42fda7ccc7..4028caf9cd 100644 --- a/src/foundation/decidable-types.lagda.md +++ b/src/foundation/decidable-types.lagda.md @@ -11,7 +11,6 @@ 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 @@ -22,6 +21,7 @@ open import foundation.universe-levels open import foundation-core.cartesian-product-types open import foundation-core.equivalences +open import foundation-core.function-types open import foundation-core.propositions open import foundation-core.retractions ``` diff --git a/src/foundation/dependent-binomial-theorem.lagda.md b/src/foundation/dependent-binomial-theorem.lagda.md index 1d88c0ab74..0953e463be 100644 --- a/src/foundation/dependent-binomial-theorem.lagda.md +++ b/src/foundation/dependent-binomial-theorem.lagda.md @@ -11,11 +11,11 @@ open import foundation.action-on-identifications-functions 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 open import foundation.raising-universe-levels +open import foundation.transport open import foundation.type-theoretic-principle-of-choice open import foundation.universal-property-coproduct-types open import foundation.universal-property-dependent-pair-types @@ -24,6 +24,7 @@ open import foundation.universe-levels open import foundation-core.cartesian-product-types open import foundation-core.coproduct-types open import foundation-core.equivalences +open import foundation-core.function-types open import foundation-core.functoriality-dependent-pair-types open import foundation-core.homotopies open import foundation-core.univalence diff --git a/src/foundation/dependent-identifications.lagda.md b/src/foundation/dependent-identifications.lagda.md new file mode 100644 index 0000000000..3defc03867 --- /dev/null +++ b/src/foundation/dependent-identifications.lagda.md @@ -0,0 +1,466 @@ +# Dependent identifications + +```agda +module foundation.dependent-identifications where + +open import foundation-core.dependent-identifications public +``` + +
Imports + +```agda +open import foundation.action-on-identifications-functions +open import foundation.dependent-pair-types +open import foundation.function-extensionality +open import foundation.identity-types +open import foundation.transport +open import foundation.universe-levels + +open import foundation-core.equivalences +open import foundation-core.function-types +open import foundation-core.homotopies +``` + +
+ +## Idea + +We characterize dependent paths in the family of depedent paths; define the +groupoidal operators on dependent paths; define the cohrences paths: prove the +operators are equivalences. + +## Properites + +### Transport in the type family of dependent identifications + +```agda +module _ + {l1 l2 : Level} {A : UU l1} {a0 a1 : A} {p0 p1 : a0 ＝ a1} + (B : A → UU l2) {b0 : B a0} {b1 : B a1} (α : p0 ＝ p1) + where + + tr-dependent-identification : + (q01 : dependent-identification B p0 b0 b1) → + tr (λ t → dependent-identification B t b0 b1) α q01 ＝ + (inv (tr² B α b0) ∙ q01) + tr-dependent-identification q01 = inv (tr-ap {D = (λ x → x ＝ b1)} + (λ t → tr B t b0) (λ x → id) α q01) ∙ tr-Id-left (tr² B α b0) q01 + + tr-inv-dependent-identification : + (q01 : dependent-identification B p1 b0 b1) → + tr (λ t → dependent-identification B t b0 b1) (inv α) q01 ＝ + ((tr² B α b0) ∙ q01) + tr-inv-dependent-identification q01 = + ( inv + ( tr-ap + { D = λ x → x ＝ b1} + ( λ t → tr B t b0) + ( λ x → id) + ( inv α) + ( q01))) ∙ + ( ( tr-Id-left (ap (λ t → tr B t b0) (inv α)) q01) ∙ + ( ( ap (λ t → t ∙ q01) (inv (ap-inv (λ t → tr B t b0) (inv α)))) ∙ + ( ap (λ x → ap (λ t → tr B t b0) x ∙ q01) (inv-inv α)))) + + tr-dependent-identification-eq-inv-tr²-concat : + tr (λ t → dependent-identification B t b0 b1) α ＝ (inv (tr² B α b0) ∙_) + tr-dependent-identification-eq-inv-tr²-concat = + map-inv-equiv + ( htpy-eq , + funext + ( tr + ( λ t → dependent-identification B t b0 b1) α) + ( inv (tr² B α b0) ∙_)) + ( tr-dependent-identification) + + tr-inv-dependent-identification-eq-tr²-concat : + (tr (λ t → dependent-identification B t b0 b1) (inv α)) ＝ ((tr² B α b0) ∙_) + tr-inv-dependent-identification-eq-tr²-concat = + map-inv-equiv + ( htpy-eq , + funext + ( tr + ( λ t → dependent-identification B t b0 b1) + ( inv α)) + ( tr² B α b0 ∙_)) + ( tr-inv-dependent-identification) +``` + +### Characterizing iterated types of dependent identifications + +```agda +module _ + {l1 l2 : Level} {A : UU l1} (B : A → UU l2) + where + + dependent-identification² : + {a0 a1 : A} {p0 p1 : a0 ＝ a1} (α : p0 ＝ p1) + {b0 : B a0} {b1 : B a1} + (q0 : dependent-identification B p0 b0 b1) + (q1 : dependent-identification B p1 b0 b1) → + UU l2 + dependent-identification² α q0 q1 = + q0 ＝ ((tr² B α _) ∙ q1) + + tr-dependent-identification-dependent-identification² : + {a0 a1 : A} {p0 p1 : a0 ＝ a1} + (α : p0 ＝ p1) {b0 : B a0} {b1 : B a1} + (q0 : dependent-identification B p0 b0 b1) + (q1 : dependent-identification B p1 b0 b1) → + dependent-identification² α q0 q1 → + dependent-identification (λ t → dependent-identification B t b0 b1) α q0 q1 + tr-dependent-identification-dependent-identification² + {p0 = refl} refl ._ refl refl = + refl + + dependent-identification²-tr-dependent-identification : + {a0 a1 : A} {p0 p1 : a0 ＝ a1} + (α : p0 ＝ p1) {b0 : B a0} {b1 : B a1} + (q0 : dependent-identification B p0 b0 b1) + (q1 : dependent-identification B p1 b0 b1) → + tr (λ t → dependent-identification B t b0 b1) α q0 ＝ q1 → + dependent-identification² α q0 q1 + dependent-identification²-tr-dependent-identification + {p0 = refl} refl refl ._ refl = + refl + + issec-dependent-identification²-tr-dependent-identification : + {a0 a1 : A} {p0 p1 : a0 ＝ a1} + (α : p0 ＝ p1) {b0 : B a0} {b1 : B a1} + (q0 : dependent-identification B p0 b0 b1) + (q1 : dependent-identification B p1 b0 b1) → + ( tr-dependent-identification-dependent-identification² α q0 q1 ∘ + dependent-identification²-tr-dependent-identification α q0 q1) ~ id + issec-dependent-identification²-tr-dependent-identification + {p0 = refl} refl refl ._ refl = + refl + + isretr-dependent-identification²-tr-dependent-identification : + {a0 a1 : A} {p0 p1 : a0 ＝ a1} + (α : p0 ＝ p1) {b0 : B a0} {b1 : B a1} + (q0 : dependent-identification B p0 b0 b1) + (q1 : dependent-identification B p1 b0 b1) → + ( dependent-identification²-tr-dependent-identification α q0 q1 ∘ + tr-dependent-identification-dependent-identification² α q0 q1) ~ id + isretr-dependent-identification²-tr-dependent-identification + {p0 = refl} refl ._ refl refl = + refl + + is-equiv-tr-dependent-identification-dependent-identification² : + {a0 a1 : A} {p0 p1 : a0 ＝ a1} + (α : p0 ＝ p1) {b0 : B a0} {b1 : B a1} + (q0 : dependent-identification B p0 b0 b1) + (q1 : dependent-identification B p1 b0 b1) → + is-equiv (tr-dependent-identification-dependent-identification² α q0 q1) + is-equiv-tr-dependent-identification-dependent-identification² α q0 q1 = + is-equiv-has-inverse + ( dependent-identification²-tr-dependent-identification α q0 q1) + ( issec-dependent-identification²-tr-dependent-identification α q0 q1) + ( isretr-dependent-identification²-tr-dependent-identification α q0 q1) +``` + +### The groupoidal structure of dependent identifications + +We show that there is groupoidal structure on the dependent identifications. The +statement of the groupoid laws use dependent identifications, due to the +dependent nature of dependent identifications. + +#### Concatenation of dependent identifications + +```agda +module _ + {l1 l2 : Level} {A : UU l1} (B : A → UU l2) + {a0 a1 a2 : A} (p01 : a0 ＝ a1) (p12 : a1 ＝ a2) + {b0 : B a0} {b1 : B a1} {b2 : B a2} + (q01 : dependent-identification B p01 b0 b1) + (q12 : dependent-identification B p12 b1 b2) + where + + concat-dependent-identification : + dependent-identification B (p01 ∙ p12) b0 b2 + concat-dependent-identification = + ( tr-concat {B = B} p01 p12 b0) ∙ + ( ( ap (tr B p12) q01) ∙ + ( q12)) +``` + +#### Inverses of dependent identifications + +```agda +module _ + {l1 l2 : Level} + {A : UU l1} {a0 a1 : A} + (B : A → UU l2) (p01 : a0 ＝ a1) {b0 : B a0} {b1 : B a1} + (q01 : dependent-identification B p01 b0 b1) + where + + inv-dependent-identification : dependent-identification B (inv p01) b1 b0 + inv-dependent-identification = + ( inv (ap (tr B (inv p01)) q01)) ∙ + ( ( inv (tr-concat {B = B} (p01) (inv p01) b0)) ∙ + ( ap (λ t → tr B t b0) (right-inv p01))) +``` + +#### Associativity of concatenation of dependent identifications + +```agda +module _ + {l1 l2 : Level} + {A : UU l1} {a0 a1 : A} (B : A → UU l2) {b0 : B a0} {b1 : B a1} + where + + d-assoc : + {a2 a3 : A} {b2 : B a2} {b3 : B a3} + (p01 : a0 ＝ a1) (q01 : dependent-identification B p01 b0 b1) + (p12 : a1 ＝ a2) (q12 : dependent-identification B p12 b1 b2) + (p23 : a2 ＝ a3) (q23 : dependent-identification B p23 b2 b3) → + dependent-identification² B + ( assoc p01 p12 p23) + ( concat-dependent-identification B + ( p01 ∙ p12) + ( p23) + ( concat-dependent-identification B p01 p12 q01 q12) + ( q23)) + ( concat-dependent-identification B + ( p01) + ( p12 ∙ p23) + ( q01) + ( concat-dependent-identification B p12 p23 q12 q23)) + d-assoc refl refl p12 q12 p23 q23 = refl + + d-assoc' : + {a2 a3 : A} {b2 : B a2} {b3 : B a3} + (p01 : a0 ＝ a1) + (q01 : dependent-identification B p01 b0 b1) (p12 : a1 ＝ a2) + (q12 : dependent-identification B p12 b1 b2) (p23 : a2 ＝ a3) + (q23 : dependent-identification B p23 b2 b3) → + dependent-identification + ( λ t → dependent-identification B t b0 b3) + ( assoc p01 p12 p23) + ( concat-dependent-identification B + ( p01 ∙ p12) + ( p23) + ( concat-dependent-identification B p01 p12 q01 q12) + ( q23)) + ( concat-dependent-identification B + ( p01) + ( p12 ∙ p23) + ( q01) + ( concat-dependent-identification B p12 p23 q12 q23)) + d-assoc' p01 q01 p12 q12 p23 q23 = + tr-dependent-identification-dependent-identification² B + ( assoc p01 p12 p23) + ( concat-dependent-identification B + ( p01 ∙ p12) + ( p23) + ( concat-dependent-identification B p01 p12 q01 q12) + ( q23)) + ( concat-dependent-identification B + ( p01) + ( p12 ∙ p23) + ( q01) + ( concat-dependent-identification B p12 p23 q12 q23)) + ( d-assoc p01 q01 p12 q12 p23 q23) + + d-right-unit : + (p : a0 ＝ a1) + (q : dependent-identification B p b0 b1) → + dependent-identification² + ( B) + ( right-unit {p = p}) + ( concat-dependent-identification B p refl q + ( refl-dependent-identification B)) + ( q) + d-right-unit refl refl = refl + + d-right-unit' : + (p : a0 ＝ a1) (q : dependent-identification B p b0 b1) → + dependent-identification + ( λ t → dependent-identification B t b0 b1) + ( right-unit) + ( concat-dependent-identification B p refl q + ( refl-dependent-identification B)) + ( q) + d-right-unit' p q = + tr-dependent-identification-dependent-identification² B + ( right-unit {p = p}) + ( concat-dependent-identification B p refl q + ( refl-dependent-identification B)) + ( q) + ( d-right-unit p q) + + d-left-unit : + (p : a0 ＝ a1) + (q : dependent-identification B p b0 b1) → + dependent-identification² + ( B) + ( left-unit {p = p}) + ( concat-dependent-identification B refl p + ( refl-dependent-identification B) + ( q)) + ( q) + d-left-unit p q = refl + + d-left-unit' : + (p : a0 ＝ a1) + (q : dependent-identification B p b0 b1) → + dependent-identification + ( λ t → dependent-identification B t b0 b1) + ( left-unit) + ( concat-dependent-identification B refl p + ( refl-dependent-identification B) + ( q)) + ( q) + d-left-unit' p q = + tr-dependent-identification-dependent-identification² B + ( left-unit {p = p}) + ( concat-dependent-identification B refl p + ( refl-dependent-identification B) + ( q)) + ( q) + ( d-left-unit p q) + + d-right-inv : + (p : a0 ＝ a1) (q : dependent-identification B p b0 b1) → + dependent-identification² B + ( right-inv p) + ( concat-dependent-identification B + ( p) + ( inv p) + ( q) + ( inv-dependent-identification B p q)) + ( refl-dependent-identification B) + d-right-inv refl refl = refl + + d-right-inv' : + (p : a0 ＝ a1) (q : dependent-identification B p b0 b1) → + dependent-identification + ( λ t → dependent-identification B t b0 b0) + ( right-inv p) + ( concat-dependent-identification B + ( p) + ( inv p) + ( q) + ( inv-dependent-identification B p q)) + ( refl-dependent-identification B) + d-right-inv' p q = + tr-dependent-identification-dependent-identification² + ( B) + ( right-inv p) + ( concat-dependent-identification B + ( p) + ( inv p) + ( q) + ( inv-dependent-identification B p q)) + ( refl-dependent-identification B) + ( d-right-inv p q) + + d-left-inv : + (p : a0 ＝ a1) (q : dependent-identification B p b0 b1) → + dependent-identification² + ( B) + ( left-inv p) + ( concat-dependent-identification B + ( inv p) + ( p) + ( inv-dependent-identification B p q) + ( q)) + ( refl-dependent-identification B) + d-left-inv refl refl = refl + + d-left-inv' : + (p : a0 ＝ a1) (q : dependent-identification B p b0 b1) → + dependent-identification + ( λ t → dependent-identification B t b1 b1) + ( left-inv p) + ( concat-dependent-identification B + ( inv p) + ( p) + ( inv-dependent-identification B p q) + ( q)) + ( refl-dependent-identification B) + d-left-inv' p q = + tr-dependent-identification-dependent-identification² + ( B) + ( left-inv p) + ( concat-dependent-identification B + ( inv p) + ( p) + ( inv-dependent-identification B p q) + ( q)) + ( refl-dependent-identification B) + ( d-left-inv p q) + + inv-inv-dependent-identification : + (p : a0 ＝ a1) (q : dependent-identification B p b0 b1) → + dependent-identification² B + ( inv-inv p) + ( inv-dependent-identification B + ( inv p) + ( inv-dependent-identification B p q)) + ( q) + inv-inv-dependent-identification refl refl = refl + + inv-inv-dependent-identification' : + (p : a0 ＝ a1) (q : dependent-identification B p b0 b1) → + dependent-identification + ( λ t → dependent-identification B t b0 b1) + ( inv-inv p) + ( inv-dependent-identification B + ( inv p) + ( inv-dependent-identification B p q)) + ( q) + inv-inv-dependent-identification' p q = + tr-dependent-identification-dependent-identification² B + ( inv-inv p) + ( inv-dependent-identification B + ( inv p) + ( inv-dependent-identification B p q)) + ( q) + ( inv-inv-dependent-identification p q) + + distributive-inv-concat-dependent-identification : + {a2 : A} {b2 : B a2} (p01 : a0 ＝ a1) + (q01 : dependent-identification B p01 b0 b1) + (p12 : a1 ＝ a2) (q12 : dependent-identification B p12 b1 b2) → + dependent-identification² B + ( distributive-inv-concat p01 p12) + ( inv-dependent-identification B + ( p01 ∙ p12) + ( concat-dependent-identification B p01 p12 q01 q12)) + ( concat-dependent-identification B + ( inv p12) + ( inv p01) + ( inv-dependent-identification B p12 q12) + ( inv-dependent-identification B p01 q01)) + distributive-inv-concat-dependent-identification refl refl refl refl = refl + + distributive-inv-concat-dependent-identification' : + {a2 : A} {b2 : B a2} (p01 : a0 ＝ a1) + (q01 : dependent-identification B p01 b0 b1) + (p12 : a1 ＝ a2) (q12 : dependent-identification B p12 b1 b2) → + dependent-identification + ( λ t → dependent-identification B t b2 b0) + ( distributive-inv-concat p01 p12) + ( inv-dependent-identification B + ( p01 ∙ p12) + ( concat-dependent-identification B p01 p12 q01 q12)) + ( concat-dependent-identification B + ( inv p12) + ( inv p01) + ( inv-dependent-identification B p12 q12) + ( inv-dependent-identification B p01 q01)) + distributive-inv-concat-dependent-identification' p01 q01 p12 q12 = + tr-dependent-identification-dependent-identification² + ( B) + ( distributive-inv-concat p01 p12) + ( inv-dependent-identification B + ( p01 ∙ p12) + ( concat-dependent-identification B p01 p12 q01 q12)) + ( concat-dependent-identification B + ( inv p12) + ( inv p01) + ( inv-dependent-identification B p12 q12) + ( inv-dependent-identification B p01 q01)) + ( distributive-inv-concat-dependent-identification p01 q01 p12 q12) +``` diff --git a/src/foundation/dependent-pair-types.lagda.md b/src/foundation/dependent-pair-types.lagda.md index b1ce6a4c87..2a0340a775 100644 --- a/src/foundation/dependent-pair-types.lagda.md +++ b/src/foundation/dependent-pair-types.lagda.md @@ -2,12 +2,19 @@ ```agda module foundation.dependent-pair-types where + +open import foundation-core.dependent-pair-types public ```
Imports ```agda open import foundation.universe-levels + +open import foundation-core.dependent-identifications +open import foundation-core.equality-dependent-pair-types +open import foundation-core.identity-types +open import foundation-core.transport ```
@@ -19,58 +26,27 @@ When `B` is a family of types over `A`, then we can form the type of 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 #-} +## Properties -infixr 10 _,_ -pattern _,_ a b = pair a b -``` - -## Constructions +### Transport in a family of dependent pair types ```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 +tr-Σ : + {l1 l2 l3 : Level} {A : UU l1} {a0 a1 : A} {B : A → UU l2} + (C : (x : A) → B x → UU l3) (p : a0 ＝ a1) (z : Σ (B a0) (λ x → C a0 x)) → + tr (λ a → (Σ (B a) (C a))) p z ＝ + pair (tr B p (pr1 z)) (tr (ind-Σ C) (eq-pair-Σ p refl) (pr2 z)) +tr-Σ C refl z = refl ``` -### Families on dependent pair types +### Transport in a family over a dependent pair type ```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 +tr-eq-pair-Σ : + {l1 l2 l3 : Level} {A : UU l1} {a0 a1 : A} + {B : A → UU l2} {b0 : B a0} {b1 : B a1} (C : (Σ A B) → UU l3) + (p : a0 ＝ a1) (q : dependent-identification B p b0 b1) (u : C (a0 , b0)) → + tr C (eq-pair-Σ p q) u ＝ + tr (λ x → C (a1 , x)) q (tr C (eq-pair-Σ p refl) u) +tr-eq-pair-Σ C refl refl u = refl ``` diff --git a/src/foundation/dependent-paths.lagda.md b/src/foundation/dependent-paths.lagda.md deleted file mode 100644 index 85c5f85972..0000000000 --- a/src/foundation/dependent-paths.lagda.md +++ /dev/null @@ -1,380 +0,0 @@ -# Dependent paths - -```agda -module foundation.dependent-paths where - -open import foundation-core.dependent-paths public -``` - -
Imports - -```agda -open import foundation.action-on-identifications-functions -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.equivalences -open import foundation-core.homotopies -open import foundation-core.transport -``` - -
- -## Idea - -We characterize dependent paths in the family of depedent paths; define the -groupoidal operators on dependent paths; define the cohrences paths: prove the -operators are equivalences. - -## Properites - -We characterize dependent paths in the family `λ t → path-over B t b0 b1`. - -```agda -module _ - {l1 l2 : Level} {A : UU l1} {a0 a1 : A} {p0 p1 : a0 ＝ a1} - (B : A → UU l2) - where - - tr² : (α : p0 ＝ p1) (b0 : B a0) → (tr B p0 b0) ＝ (tr B p1 b0) - tr² α b0 = ap (λ t → tr B t b0) α - -module _ - {l1 l2 : Level} {A : UU l1} {a0 a1 : A} {p0 p1 : a0 ＝ a1} - (B : A → UU l2) {b0 : B a0} {b1 : B a1} (α : p0 ＝ p1) - where - - tr-path-over : - (q01 : path-over B p0 b0 b1) → - (tr (λ t → path-over B t b0 b1) α q01) ＝ (inv (tr² B α b0) ∙ q01) - tr-path-over q01 = inv (tr-ap {D = (λ x → x ＝ b1)} - (λ t → tr B t b0) (λ x → id) α q01) ∙ tr-Id-left (tr² B α b0) q01 - - tr-inv-path-over : - (q01 : path-over B p1 b0 b1) → - (tr (λ t → path-over B t b0 b1) (inv α) q01) ＝ ((tr² B α b0) ∙ q01) - tr-inv-path-over q01 = inv (tr-ap {D = λ x → x ＝ b1} - (λ t → tr B t b0) (λ x → id) (inv α) q01) ∙ - (tr-Id-left (ap (λ t → tr B t b0) (inv α)) q01 ∙ - (ap (λ t → t ∙ q01) (inv (ap-inv (λ t → tr B t b0) (inv α))) ∙ - ap (λ x → ap (λ t → tr B t b0) x ∙ q01) (inv-inv α))) - - tr-path-over-eq-inv-tr²-concat : - (tr (λ t → path-over B t b0 b1) α) ＝ (inv (tr² B α b0) ∙_) - tr-path-over-eq-inv-tr²-concat = - map-inv-equiv - ( htpy-eq , - funext (tr (λ t → path-over B t b0 b1) α) (inv (tr² B α b0) ∙_)) - ( tr-path-over) - - tr-inv-path-over-eq-tr²-concat : - (tr (λ t → path-over B t b0 b1) (inv α)) ＝ ((tr² B α b0) ∙_) - tr-inv-path-over-eq-tr²-concat = - map-inv-equiv - ( htpy-eq , - funext (tr (λ t → path-over B t b0 b1) (inv α)) (tr² B α b0 ∙_)) - ( tr-inv-path-over) - -module _ - {l1 l2 : Level} {A : UU l1} {a0 a1 : A} {p0 p1 : a0 ＝ a1} - (B : A → UU l2) (α : p0 ＝ p1) {b0 : B a0} {b1 : B a1} - (q0 : path-over B p0 b0 b1) (q1 : path-over B p1 b0 b1) - where - - path-over² : UU l2 - path-over² = q0 ＝ ((tr² B α b0) ∙ q1) - - tr-path-over-path-over² : - (path-over²) → ((tr (λ t → path-over B t b0 b1) α q0) ＝ q1) - tr-path-over-path-over² z = tr-path-over B α q0 ∙ ( - (map-inv-equiv (equiv-inv-con (inv (tr² B α b0)) q0 q1) - (z ∙ inv (ap (λ t → t ∙ q1) (inv-inv (tr² B α b0)))))) - - path-over²-tr-path-over : - ((tr (λ t → path-over B t b0 b1) α q0) ＝ q1) → (path-over²) - path-over²-tr-path-over z = - ( map-equiv - ( equiv-inv-con (inv (tr² B α b0)) q0 q1) - ( (inv (tr-path-over B α q0)) ∙ z)) ∙ - ( ap (_∙ q1) (inv-inv (tr² B α b0))) - - issec-path-over²-tr-path-over : - ( tr-path-over-path-over² ∘ path-over²-tr-path-over) ~ id - issec-path-over²-tr-path-over z = - ( ap - ( λ x → - tr-path-over B α q0 ∙ - pr1 (pr1 (is-equiv-inv-con (inv (ap (λ t → tr B t b0) α)) q0 q1)) x) - ( assoc - ( inv-con - ( inv (ap (λ t → tr B t b0) α)) - ( q0) - ( q1) - ( inv (tr-path-over B α q0) ∙ z)) - ( ap (_∙ q1) (inv-inv (ap (λ t → tr B t b0) α))) - ( inv (ap (_∙ q1) (inv-inv (ap (λ t → tr B t b0) α)))))) ∙ - ( ( ap - ( λ x → - tr-path-over B α q0 ∙ - pr1 - ( pr1 (is-equiv-inv-con (inv (ap (λ t → tr B t b0) α)) q0 q1)) - ( ( inv-con - ( inv (ap (λ t → tr B t b0) α)) - ( q0) - ( q1) - ( inv (tr-path-over B α q0) ∙ z)) ∙ - ( x))) - ( right-inv (ap (_∙ q1) (inv-inv (ap (λ t → tr B t b0) α))))) ∙ - ( ( ap - ( λ x → - tr-path-over B α q0 ∙ - pr1 - ( pr1 (is-equiv-inv-con (inv (ap (λ t → tr B t b0) α)) q0 q1)) - ( x)) - ( right-unit)) ∙ - ( ( ap - ( λ x → tr-path-over B α q0 ∙ x) - ( isretr-map-inv-equiv - ( equiv-inv-con (inv (ap (λ t → tr B t b0) α)) q0 q1) - ( inv (tr-path-over B α q0) ∙ z))) ∙ - ( ( inv - ( assoc (tr-path-over B α q0) (inv (tr-path-over B α q0)) z)) ∙ - ( ap (_∙ z) (right-inv (tr-path-over B α q0))))))) - - isretr-path-over²-tr-path-over : - ( path-over²-tr-path-over ∘ tr-path-over-path-over²) ~ id - isretr-path-over²-tr-path-over z = - ( ap - ( λ x → - ( inv-con (inv (ap (λ t → tr B t b0) α)) q0 q1 x) ∙ - (ap (_∙ q1) (inv-inv (ap (λ t → tr B t b0) α)))) - ( inv - ( assoc - ( inv (tr-path-over B α q0)) - ( tr-path-over B α q0) - ( pr1 - ( pr1 (is-equiv-inv-con (inv (ap (λ t → tr B t b0) α)) q0 q1)) - ( z ∙ inv (ap (_∙ q1) (inv-inv (ap (λ t → tr B t b0) α)))))))) ∙ - ( ap - ( λ x → - inv-con - ( inv (ap (λ t → tr B t b0) α)) - ( q0) - ( q1) - ( x ∙ - pr1 - ( pr1 (is-equiv-inv-con (inv (ap (λ t → tr B t b0) α)) q0 q1)) - ( z ∙ inv (ap (_∙ q1) (inv-inv (ap (λ t → tr B t b0) α))))) ∙ - ( ap (_∙ q1) (inv-inv (ap (λ t → tr B t b0) α)))) - ( left-inv (tr-path-over B α q0)) ∙ - ( ap - ( _∙ ap (_∙ q1) (inv-inv (ap (λ t → tr B t b0) α))) - ( issec-map-inv-equiv - ( equiv-inv-con (inv (ap (λ t → tr B t b0) α)) q0 q1) - ( z ∙ inv (ap (_∙ q1) (inv-inv (ap (λ t → tr B t b0) α))))) ∙ - ( assoc - ( z) - ( inv (ap (_∙ q1) (inv-inv (ap (λ t → tr B t b0) α)))) - ( ap (_∙ q1) (inv-inv (ap (λ t → tr B t b0) α))) ∙ - ( ap - ( z ∙_) - ( left-inv (ap (_∙ q1) (inv-inv (ap (λ t → tr B t b0) α)))) ∙ - ( right-unit))))) - - is-equiv-tr-path-over-path-over² : - is-equiv tr-path-over-path-over² - is-equiv-tr-path-over-path-over² = - is-equiv-has-inverse path-over²-tr-path-over - issec-path-over²-tr-path-over isretr-path-over²-tr-path-over -``` - -Definition: Groupoidal operators on dependent paths. - -```agda -module _ - {l1 l2 : Level} - {A : UU l1} {a0 a1 a2 : A} - (B : A → UU l2) {b0 : B a0} {b1 : B a1} {b2 : B a2} - (p01 : a0 ＝ a1) (q01 : path-over B p01 b0 b1) - (p12 : a1 ＝ a2) (q12 : path-over B p12 b1 b2) - where - - d-concat : path-over B (p01 ∙ p12) b0 b2 - d-concat = (tr-concat {B = B} p01 p12 b0) ∙ ((ap (tr B p12) q01) ∙ (q12)) - -module _ - {l1 l2 : Level} - {A : UU l1} {a0 a1 : A} - (B : A → UU l2) (p01 : a0 ＝ a1) {b0 : B a0} {b1 : B a1} - (q01 : path-over B p01 b0 b1) - where - - d-inv : path-over B (inv p01) b1 b0 - d-inv = - ( inv (ap (tr B (inv p01)) q01)) ∙ - ( ( inv (tr-concat {B = B} (p01) (inv p01) b0)) ∙ - ( ap (λ t → tr B t b0) (right-inv p01))) -``` - -Now we prove these paths satisfy identities analgous to the usual unit, inverse, -and associativity laws. Though, due to the dependent nature, the naive -identities are not well typed. So these identities involve transporting. - -```agda -module _ - {l1 l2 : Level} - {A : UU l1} {a0 a1 : A} (B : A → UU l2) {b0 : B a0} {b1 : B a1} - where - - d-assoc : - {a2 a3 : A} {b2 : B a2} {b3 : B a3} - (p01 : a0 ＝ a1) (q01 : path-over B p01 b0 b1) - (p12 : a1 ＝ a2) (q12 : path-over B p12 b1 b2) - (p23 : a2 ＝ a3) (q23 : path-over B p23 b2 b3) → - path-over² B (assoc p01 p12 p23) - (d-concat B (p01 ∙ p12) (d-concat B p01 q01 p12 q12) p23 q23) - (d-concat B p01 q01 (p12 ∙ p23) (d-concat B p12 q12 p23 q23)) - d-assoc refl refl p12 q12 p23 q23 = refl - - d-assoc' : - {a2 a3 : A} {b2 : B a2} {b3 : B a3} - (p01 : a0 ＝ a1) - (q01 : path-over B p01 b0 b1) (p12 : a1 ＝ a2) - (q12 : path-over B p12 b1 b2) (p23 : a2 ＝ a3) - (q23 : path-over B p23 b2 b3) → - ( tr - ( λ t → path-over B t b0 b3) - ( assoc p01 p12 p23) - ( d-concat B (p01 ∙ p12) (d-concat B p01 q01 p12 q12) p23 q23)) ＝ - ( d-concat B p01 q01 (p12 ∙ p23) (d-concat B p12 q12 p23 q23)) - d-assoc' p01 q01 p12 q12 p23 q23 = - tr-path-over-path-over² B (assoc p01 p12 p23) - (d-concat B (p01 ∙ p12) (d-concat B p01 q01 p12 q12) p23 q23) - (d-concat B p01 q01 (p12 ∙ p23) (d-concat B p12 q12 p23 q23)) - (d-assoc p01 q01 p12 q12 p23 q23) - - d-right-unit : - (p : a0 ＝ a1) - (q : path-over B p b0 b1) → - path-over² - ( B) - ( right-unit {p = p}) - ( d-concat B p q refl (refl-path-over B)) - ( q) - d-right-unit refl refl = refl - - d-right-unit' : - (p : a0 ＝ a1) (q : path-over B p b0 b1) → - ( tr - ( λ t → path-over B t b0 b1) - ( right-unit) - (d-concat B p q refl (refl-path-over B))) ＝ q - d-right-unit' p q = tr-path-over-path-over² B (right-unit {p = p}) - (d-concat B p q refl (refl-path-over B)) q (d-right-unit p q) - - d-left-unit : - (p : a0 ＝ a1) - (q : path-over B p b0 b1) → - path-over² - ( B) - ( left-unit {p = p}) - ( d-concat B refl (refl-path-over B) p q) - ( q) - d-left-unit p q = refl - - d-left-unit' : - (p : a0 ＝ a1) - (q : path-over B p b0 b1) → - ( tr - ( λ t → path-over B t b0 b1) - ( left-unit) - ( d-concat B refl (refl-path-over B) p q)) ＝ - ( q) - d-left-unit' p q = tr-path-over-path-over² B (left-unit {p = p}) - (d-concat B refl (refl-path-over B) p q) q (d-left-unit p q) - - d-right-inv : - (p : a0 ＝ a1) (q : path-over B p b0 b1) → - path-over² B (right-inv p) (d-concat B p q (inv p) (d-inv B p q)) - (refl-path-over B) - d-right-inv refl refl = refl - - d-right-inv' : - (p : a0 ＝ a1) (q : path-over B p b0 b1) → - ( tr - ( λ t → path-over B t b0 b0) - ( right-inv p) - ( d-concat B p q (inv p) (d-inv B p q))) ＝ - ( refl-path-over B) - d-right-inv' p q = - tr-path-over-path-over² - ( B) - ( right-inv p) - ( d-concat B p q (inv p) (d-inv B p q)) - ( refl-path-over B) - ( d-right-inv p q) - - d-left-inv : - (p : a0 ＝ a1) (q : path-over B p b0 b1) → - path-over² - ( B) - ( left-inv p) - ( d-concat B (inv p) (d-inv B p q) p q) - ( refl-path-over B) - d-left-inv refl refl = refl - - d-left-inv' : - (p : a0 ＝ a1) (q : path-over B p b0 b1) → - ( tr - ( λ t → path-over B t b1 b1) - ( left-inv p) - ( d-concat B (inv p) (d-inv B p q) p q)) ＝ - ( refl-path-over B) - d-left-inv' p q = - tr-path-over-path-over² - ( B) - ( left-inv p) - ( d-concat B (inv p) (d-inv B p q) p q) - ( refl-path-over B) - ( d-left-inv p q) - - d-inv-d-inv : - (p : a0 ＝ a1) (q : path-over B p b0 b1) → - path-over² B (inv-inv p) (d-inv B (inv p) (d-inv B p q)) q - d-inv-d-inv refl refl = refl - - d-inv-d-inv' : - (p : a0 ＝ a1) (q : path-over B p b0 b1) → - ( tr - ( λ t → path-over B t b0 b1) - ( inv-inv p) - ( d-inv B (inv p) (d-inv B p q))) ＝ - ( q) - d-inv-d-inv' p q = tr-path-over-path-over² B (inv-inv p) - (d-inv B (inv p) (d-inv B p q)) q (d-inv-d-inv p q) - - distributive-d-inv-d-concat : - {a2 : A} {b2 : B a2} (p01 : a0 ＝ a1) (q01 : path-over B p01 b0 b1) - (p12 : a1 ＝ a2) (q12 : path-over B p12 b1 b2) → - path-over² B (distributive-inv-concat p01 p12) - (d-inv B (p01 ∙ p12) (d-concat B p01 q01 p12 q12)) - (d-concat B (inv p12) (d-inv B p12 q12) (inv p01) (d-inv B p01 q01)) - distributive-d-inv-d-concat refl refl refl refl = refl - - distributive-d-inv-d-concat' : - {a2 : A} {b2 : B a2} (p01 : a0 ＝ a1) (q01 : path-over B p01 b0 b1) - (p12 : a1 ＝ a2) (q12 : path-over B p12 b1 b2) → - (tr (λ t → path-over B t b2 b0) (distributive-inv-concat p01 p12) ( - (d-inv B (p01 ∙ p12) (d-concat B p01 q01 p12 q12)))) ＝ ( - d-concat B (inv p12) (d-inv B p12 q12) (inv p01) (d-inv B p01 q01)) - distributive-d-inv-d-concat' p01 q01 p12 q12 = - tr-path-over-path-over² - ( B) - ( distributive-inv-concat p01 p12) - ( d-inv B (p01 ∙ p12) (d-concat B p01 q01 p12 q12)) - ( d-concat B (inv p12) (d-inv B p12 q12) (inv p01) (d-inv B p01 q01)) - ( distributive-d-inv-d-concat p01 q01 p12 q12) -``` diff --git a/src/foundation/descent-coproduct-types.lagda.md b/src/foundation/descent-coproduct-types.lagda.md index 5d368ac1f7..706475ea8e 100644 --- a/src/foundation/descent-coproduct-types.lagda.md +++ b/src/foundation/descent-coproduct-types.lagda.md @@ -10,7 +10,6 @@ module foundation.descent-coproduct-types where open import foundation.action-on-identifications-functions 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 @@ -18,6 +17,7 @@ open import foundation.universe-levels open import foundation-core.coproduct-types open import foundation-core.equality-dependent-pair-types open import foundation-core.equivalences +open import foundation-core.function-types open import foundation-core.homotopies open import foundation-core.identity-types open import foundation-core.pullbacks diff --git a/src/foundation/descent-dependent-pair-types.lagda.md b/src/foundation/descent-dependent-pair-types.lagda.md index ef4c7e05f8..c357b1ee13 100644 --- a/src/foundation/descent-dependent-pair-types.lagda.md +++ b/src/foundation/descent-dependent-pair-types.lagda.md @@ -9,11 +9,11 @@ module foundation.descent-dependent-pair-types where ```agda 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.function-types open import foundation-core.functoriality-dependent-pair-types open import foundation-core.homotopies open import foundation-core.identity-types diff --git a/src/foundation/descent-equivalences.lagda.md b/src/foundation/descent-equivalences.lagda.md index 60c172dc23..7edda6820d 100644 --- a/src/foundation/descent-equivalences.lagda.md +++ b/src/foundation/descent-equivalences.lagda.md @@ -9,10 +9,10 @@ module foundation.descent-equivalences where ```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.function-types open import foundation-core.functoriality-dependent-function-types open import foundation-core.pullbacks ``` diff --git a/src/foundation/diagonals-of-maps.lagda.md b/src/foundation/diagonals-of-maps.lagda.md index 030bfb51ee..058f23ae62 100644 --- a/src/foundation/diagonals-of-maps.lagda.md +++ b/src/foundation/diagonals-of-maps.lagda.md @@ -10,13 +10,13 @@ module foundation.diagonals-of-maps where open import foundation.action-on-identifications-functions 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.embeddings open import foundation-core.equivalences open import foundation-core.fibers-of-maps +open import foundation-core.function-types open import foundation-core.homotopies open import foundation-core.identity-types open import foundation-core.propositional-maps diff --git a/src/foundation/discrete-relaxed-sigma-decompositions.lagda.md b/src/foundation/discrete-relaxed-sigma-decompositions.lagda.md index 277c464fe5..5c031f2b25 100644 --- a/src/foundation/discrete-relaxed-sigma-decompositions.lagda.md +++ b/src/foundation/discrete-relaxed-sigma-decompositions.lagda.md @@ -11,13 +11,13 @@ open import foundation.action-on-identifications-functions 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.equality-dependent-pair-types +open import foundation-core.function-types open import foundation-core.identity-types open import foundation-core.propositions open import foundation-core.subtypes diff --git a/src/foundation/discrete-sigma-decompositions.lagda.md b/src/foundation/discrete-sigma-decompositions.lagda.md index f60c27e33f..49230b1666 100644 --- a/src/foundation/discrete-sigma-decompositions.lagda.md +++ b/src/foundation/discrete-sigma-decompositions.lagda.md @@ -11,7 +11,6 @@ open import foundation.action-on-identifications-functions 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 @@ -19,6 +18,7 @@ open import foundation.unit-type open import foundation.universe-levels open import foundation-core.equality-dependent-pair-types +open import foundation-core.function-types open import foundation-core.identity-types open import foundation-core.propositions open import foundation-core.subtypes diff --git a/src/foundation/disjunction.lagda.md b/src/foundation/disjunction.lagda.md index 4b176414df..04ef16ab69 100644 --- a/src/foundation/disjunction.lagda.md +++ b/src/foundation/disjunction.lagda.md @@ -10,13 +10,13 @@ module foundation.disjunction where 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.equivalences +open import foundation-core.function-types open import foundation-core.propositions ``` diff --git a/src/foundation/double-negation-modality.lagda.md b/src/foundation/double-negation-modality.lagda.md index 8387a86427..8420266ab1 100644 --- a/src/foundation/double-negation-modality.lagda.md +++ b/src/foundation/double-negation-modality.lagda.md @@ -8,9 +8,9 @@ module foundation.double-negation-modality where ```agda open import foundation.double-negation -open import foundation.functions open import foundation.universe-levels +open import foundation-core.function-types open import foundation-core.propositions open import foundation-core.transport diff --git a/src/foundation/double-negation.lagda.md b/src/foundation/double-negation.lagda.md index 7fea292347..cb75aac9d0 100644 --- a/src/foundation/double-negation.lagda.md +++ b/src/foundation/double-negation.lagda.md @@ -8,7 +8,6 @@ module foundation.double-negation where ```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 @@ -16,6 +15,7 @@ open import foundation.universe-levels open import foundation-core.cartesian-product-types open import foundation-core.coproduct-types open import foundation-core.empty-types +open import foundation-core.function-types open import foundation-core.propositions ``` diff --git a/src/foundation/embeddings.lagda.md b/src/foundation/embeddings.lagda.md index 5d984b66aa..decd29bc76 100644 --- a/src/foundation/embeddings.lagda.md +++ b/src/foundation/embeddings.lagda.md @@ -14,7 +14,6 @@ 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 @@ -22,6 +21,7 @@ open import foundation.truncated-maps open import foundation.universe-levels open import foundation-core.cartesian-product-types +open import foundation-core.function-types open import foundation-core.functoriality-dependent-pair-types open import foundation-core.homotopies open import foundation-core.propositional-maps diff --git a/src/foundation/empty-types.lagda.md b/src/foundation/empty-types.lagda.md index 7f2ba7f8cd..fdd6b8876b 100644 --- a/src/foundation/empty-types.lagda.md +++ b/src/foundation/empty-types.lagda.md @@ -12,7 +12,6 @@ open import foundation-core.empty-types public 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 @@ -21,6 +20,7 @@ open import foundation.universe-levels open import foundation-core.contractible-types open import foundation-core.equality-dependent-pair-types +open import foundation-core.function-types open import foundation-core.propositions ``` diff --git a/src/foundation/endomorphisms.lagda.md b/src/foundation/endomorphisms.lagda.md index a455eff47a..4e2dae582a 100644 --- a/src/foundation/endomorphisms.lagda.md +++ b/src/foundation/endomorphisms.lagda.md @@ -10,10 +10,10 @@ open import foundation-core.endomorphisms public ```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.function-types open import foundation-core.identity-types open import foundation-core.sets 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 d4960f19a1..95ad386355 100644 --- a/src/foundation/epimorphisms-with-respect-to-truncated-types.lagda.md +++ b/src/foundation/epimorphisms-with-respect-to-truncated-types.lagda.md @@ -9,13 +9,13 @@ 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.function-types open import foundation-core.homotopies open import foundation-core.propositions open import foundation-core.truncated-types diff --git a/src/foundation/epimorphisms.lagda.md b/src/foundation/epimorphisms.lagda.md index 4c841a5fc7..6aad49ca89 100644 --- a/src/foundation/epimorphisms.lagda.md +++ b/src/foundation/epimorphisms.lagda.md @@ -7,10 +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.function-types ```
diff --git a/src/foundation/equality-cartesian-product-types.lagda.md b/src/foundation/equality-cartesian-product-types.lagda.md index 4b38ba9d31..9276ddaaf5 100644 --- a/src/foundation/equality-cartesian-product-types.lagda.md +++ b/src/foundation/equality-cartesian-product-types.lagda.md @@ -9,13 +9,14 @@ module foundation.equality-cartesian-product-types where ```agda open import foundation.action-on-identifications-functions 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.function-types open import foundation-core.homotopies open import foundation-core.identity-types +open import foundation-core.transport ```
@@ -89,8 +90,6 @@ module _ pr2 (equiv-pair-eq s t) = is-equiv-pair-eq s t ``` -## Properties - ### Commuting triangles for `eq-pair` ```agda @@ -137,6 +136,44 @@ ap-pr2-eq-pair : ap-pr2-eq-pair refl refl = refl ``` +#### Computing transport along a path of the form `eq-pair` + +```agda +module _ + {l1 l2 l3 : Level} {A : UU l1} {B : UU l2} {a0 a1 : A} {b0 b1 : B} + where + + tr-eq-pair : + (C : A × B → UU l3) (p : a0 ＝ a1) (q : b0 ＝ b1) (u : C (a0 , b0)) → + tr C (eq-pair p q) u ＝ + tr (λ x → C (a1 , x)) q (tr (λ x → C (x , b0)) p u) + tr-eq-pair C refl refl u = refl +``` + +#### Computing transport along a path of the form `eq-pair` When one of the paths is `refl` + +```agda + left-unit-law-tr-eq-pair : + (C : A × B → UU l3) (q : b0 ＝ b1) (u : C (a0 , b0)) → + (tr C (eq-pair refl q) u) ＝ tr (λ x → C (a0 , x)) q u + left-unit-law-tr-eq-pair C refl u = refl + + right-unit-law-tr-eq-pair : + (C : A × B → UU l3) (p : a0 ＝ a1) (u : C (a0 , b0)) → + (tr C (eq-pair p refl) u) ＝ tr (λ x → C (x , b0)) p u + right-unit-law-tr-eq-pair C refl u = refl +``` + +#### Computing transport along a path of the form `eq-pair` when both paths are identical + +```agda +tr-eq-pair-diagonal : + {l1 l2 : Level} {A : UU l1} {a0 a1 : A} (C : A × A → UU l2) + (p : a0 ＝ a1) (u : C (a0 , a0)) → + tr C (eq-pair p p) u ＝ tr (λ a → C (a , a)) p u +tr-eq-pair-diagonal C refl u = refl +``` + ## See also - Equality proofs in dependent pair types are characterized in diff --git a/src/foundation/equality-coproduct-types.lagda.md b/src/foundation/equality-coproduct-types.lagda.md index 1ed7873279..946e8790a9 100644 --- a/src/foundation/equality-coproduct-types.lagda.md +++ b/src/foundation/equality-coproduct-types.lagda.md @@ -9,7 +9,6 @@ module foundation.equality-coproduct-types where ```agda open import foundation.action-on-identifications-functions open import foundation.dependent-pair-types -open import foundation.functions open import foundation.fundamental-theorem-of-identity-types open import foundation.universe-levels @@ -18,6 +17,7 @@ open import foundation-core.coproduct-types open import foundation-core.embeddings open import foundation-core.empty-types open import foundation-core.equivalences +open import foundation-core.function-types open import foundation-core.functoriality-dependent-pair-types open import foundation-core.homotopies open import foundation-core.identity-types diff --git a/src/foundation/equality-dependent-pair-types.lagda.md b/src/foundation/equality-dependent-pair-types.lagda.md index 9680fb805a..86918c69f1 100644 --- a/src/foundation/equality-dependent-pair-types.lagda.md +++ b/src/foundation/equality-dependent-pair-types.lagda.md @@ -10,11 +10,11 @@ open import foundation-core.equality-dependent-pair-types public ```agda open import foundation.action-on-identifications-functions -open import foundation.dependent-pair-types -open import foundation.identity-types +open import foundation.transport open import foundation.universe-levels -open import foundation-core.transport +open import foundation-core.dependent-pair-types +open import foundation-core.identity-types ```
diff --git a/src/foundation/equality-fibers-of-maps.lagda.md b/src/foundation/equality-fibers-of-maps.lagda.md index db879e2fcc..1e45c6ebde 100644 --- a/src/foundation/equality-fibers-of-maps.lagda.md +++ b/src/foundation/equality-fibers-of-maps.lagda.md @@ -9,16 +9,16 @@ module foundation.equality-fibers-of-maps where ```agda open import foundation.action-on-identifications-functions open import foundation.dependent-pair-types -open import foundation.functions open import foundation.identity-types +open import foundation.transport 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.function-types open import foundation-core.functoriality-dependent-pair-types open import foundation-core.homotopies -open import foundation-core.transport ```
diff --git a/src/foundation/equivalence-extensionality.lagda.md b/src/foundation/equivalence-extensionality.lagda.md index 9be466d333..b3c231ac85 100644 --- a/src/foundation/equivalence-extensionality.lagda.md +++ b/src/foundation/equivalence-extensionality.lagda.md @@ -9,7 +9,6 @@ module foundation.equivalence-extensionality where ```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 @@ -19,6 +18,7 @@ open import foundation-core.contractible-maps open import foundation-core.contractible-types open import foundation-core.equivalences open import foundation-core.fibers-of-maps +open import foundation-core.function-types open import foundation-core.functoriality-dependent-function-types open import foundation-core.functoriality-dependent-pair-types open import foundation-core.homotopies diff --git a/src/foundation/equivalence-induction.lagda.md b/src/foundation/equivalence-induction.lagda.md index a0932212d4..fc3708e59c 100644 --- a/src/foundation/equivalence-induction.lagda.md +++ b/src/foundation/equivalence-induction.lagda.md @@ -10,11 +10,11 @@ open import foundation-core.equivalence-induction public ```agda open import foundation.dependent-pair-types -open import foundation.functions open import foundation.univalence open import foundation.universe-levels open import foundation-core.equivalences +open import foundation-core.function-types open import foundation-core.sections ``` diff --git a/src/foundation/equivalence-relations.lagda.md b/src/foundation/equivalence-relations.lagda.md index 5059bed565..737b5d650c 100644 --- a/src/foundation/equivalence-relations.lagda.md +++ b/src/foundation/equivalence-relations.lagda.md @@ -15,7 +15,6 @@ 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 @@ -34,6 +33,7 @@ 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.function-types open import foundation-core.functoriality-dependent-pair-types open import foundation-core.identity-types open import foundation-core.logical-equivalences diff --git a/src/foundation/equivalences-maybe.lagda.md b/src/foundation/equivalences-maybe.lagda.md index a59f91bb1c..cd80b115f2 100644 --- a/src/foundation/equivalences-maybe.lagda.md +++ b/src/foundation/equivalences-maybe.lagda.md @@ -12,7 +12,6 @@ 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 @@ -23,6 +22,7 @@ open import foundation-core.coproduct-types open import foundation-core.embeddings open import foundation-core.empty-types open import foundation-core.equality-dependent-pair-types +open import foundation-core.function-types open import foundation-core.homotopies open import foundation-core.identity-types open import foundation-core.injective-maps diff --git a/src/foundation/equivalences.lagda.md b/src/foundation/equivalences.lagda.md index 9218a84d1b..8e5d3a26f1 100644 --- a/src/foundation/equivalences.lagda.md +++ b/src/foundation/equivalences.lagda.md @@ -14,7 +14,6 @@ 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 @@ -26,6 +25,7 @@ open import foundation-core.contractible-maps open import foundation-core.contractible-types open import foundation-core.embeddings open import foundation-core.fibers-of-maps +open import foundation-core.function-types open import foundation-core.functoriality-dependent-function-types open import foundation-core.functoriality-dependent-pair-types open import foundation-core.propositions diff --git a/src/foundation/exponents-set-quotients.lagda.md b/src/foundation/exponents-set-quotients.lagda.md index f6cfa3cbf5..e35ee51f05 100644 --- a/src/foundation/exponents-set-quotients.lagda.md +++ b/src/foundation/exponents-set-quotients.lagda.md @@ -12,7 +12,6 @@ module foundation.exponents-set-quotients where 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 @@ -23,6 +22,7 @@ open import foundation.universe-levels open import foundation-core.contractible-types open import foundation-core.embeddings open import foundation-core.equivalence-relations +open import foundation-core.function-types open import foundation-core.homotopies open import foundation-core.identity-types open import foundation-core.propositions diff --git a/src/foundation/faithful-maps.lagda.md b/src/foundation/faithful-maps.lagda.md index 086f2d89fa..e3c6615418 100644 --- a/src/foundation/faithful-maps.lagda.md +++ b/src/foundation/faithful-maps.lagda.md @@ -10,11 +10,11 @@ module foundation.faithful-maps where open import foundation.0-maps open import foundation.action-on-identifications-functions 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.function-types open import foundation-core.functoriality-dependent-pair-types open import foundation-core.homotopies open import foundation-core.identity-types diff --git a/src/foundation/fiber-inclusions.lagda.md b/src/foundation/fiber-inclusions.lagda.md index 5e6781bb93..074540711a 100644 --- a/src/foundation/fiber-inclusions.lagda.md +++ b/src/foundation/fiber-inclusions.lagda.md @@ -13,6 +13,7 @@ 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.transport open import foundation.type-arithmetic-dependent-pair-types open import foundation.unit-type open import foundation.universe-levels diff --git a/src/foundation/fibered-equivalences.lagda.md b/src/foundation/fibered-equivalences.lagda.md index 693664444c..3c6cf37845 100644 --- a/src/foundation/fibered-equivalences.lagda.md +++ b/src/foundation/fibered-equivalences.lagda.md @@ -11,13 +11,13 @@ 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.fibers-of-maps +open import foundation-core.function-types open import foundation-core.homotopies open import foundation-core.identity-types open import foundation-core.propositions diff --git a/src/foundation/fibered-involutions.lagda.md b/src/foundation/fibered-involutions.lagda.md index fc0e8297b7..d6f0e5c1b4 100644 --- a/src/foundation/fibered-involutions.lagda.md +++ b/src/foundation/fibered-involutions.lagda.md @@ -9,11 +9,11 @@ module foundation.fibered-involutions where ```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.equivalences +open import foundation-core.function-types open import foundation-core.homotopies open import foundation-core.involutions ``` diff --git a/src/foundation/fibered-maps.lagda.md b/src/foundation/fibered-maps.lagda.md index e565dc079d..86d0782353 100644 --- a/src/foundation/fibered-maps.lagda.md +++ b/src/foundation/fibered-maps.lagda.md @@ -10,7 +10,6 @@ module foundation.fibered-maps where 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 @@ -22,6 +21,7 @@ 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.function-types open import foundation-core.identity-types open import foundation-core.small-types open import foundation-core.truncated-types diff --git a/src/foundation/fibers-of-maps.lagda.md b/src/foundation/fibers-of-maps.lagda.md index 78d282ebb1..74d9573d4c 100644 --- a/src/foundation/fibers-of-maps.lagda.md +++ b/src/foundation/fibers-of-maps.lagda.md @@ -11,12 +11,12 @@ open import foundation-core.fibers-of-maps public ```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.constant-maps +open import foundation-core.function-types open import foundation-core.functoriality-dependent-pair-types open import foundation-core.homotopies open import foundation-core.identity-types diff --git a/src/foundation/function-extensionality.lagda.md b/src/foundation/function-extensionality.lagda.md index f245ccf139..53383aac4c 100644 --- a/src/foundation/function-extensionality.lagda.md +++ b/src/foundation/function-extensionality.lagda.md @@ -12,10 +12,10 @@ open import foundation-core.function-extensionality public open import foundation.action-on-identifications-binary-functions open import foundation.action-on-identifications-functions open import foundation.dependent-pair-types -open import foundation.functions open import foundation.universe-levels open import foundation-core.equivalences +open import foundation-core.function-types open import foundation-core.homotopies open import foundation-core.identity-types ``` diff --git a/src/foundation/function-types.lagda.md b/src/foundation/function-types.lagda.md new file mode 100644 index 0000000000..a14e912eb0 --- /dev/null +++ b/src/foundation/function-types.lagda.md @@ -0,0 +1,69 @@ +# Function types + +```agda +module foundation.function-types where + +open import foundation-core.function-types public +``` + +
Imports + +```agda +open import foundation.action-on-identifications-dependent-functions +open import foundation.function-extensionality +open import foundation.universe-levels + +open import foundation-core.equivalences +open import foundation-core.homotopies +open import foundation-core.identity-types +open import foundation-core.transport +``` + +
+ +## Properties + +### Transport in a family of function types + +```agda +module _ + {l1 l2 l3 : Level} {A : UU l1} {x y : A} (B : A → UU l2) (C : A → UU l3) + where + + tr-function-type : + (p : x ＝ y) (f : B x → C x) → + tr (λ a → B a → C a) p f ＝ (tr C p ∘ (f ∘ tr B (inv p))) + tr-function-type refl f = refl + + compute-dependent-identification-function-type : + (p : x ＝ y) (f : B x → C x) (g : B y → C y) → + ((b : B x) → tr C p (f b) ＝ g (tr B p b)) ≃ + (tr (λ a → B a → C a) p f ＝ g) + compute-dependent-identification-function-type refl f g = + inv-equiv equiv-funext + + map-compute-dependent-identification-function-type : + (p : x ＝ y) (f : B x → C x) (g : B y → C y) → + ((b : B x) → tr C p (f b) ＝ g (tr B p b)) → + (tr (λ a → B a → C a) p f ＝ g) + map-compute-dependent-identification-function-type p f g = + map-equiv (compute-dependent-identification-function-type p f g) +``` + +Relation between`compute-dependent-identification-function-type` and +`preserves-tr` + +```agda +module _ + {l1 l2 l3 : Level} {I : UU l1} {i0 i1 : I} {A : I → UU l2} {B : I → UU l3} + (f : (i : I) → A i → B i) + where + + preserves-tr-apd-function : + (p : i0 ＝ i1) (a : A i0) → + map-inv-equiv + ( compute-dependent-identification-function-type A B p (f i0) (f i1)) + ( apd f p) a ＝ + inv-htpy (preserves-tr f p) a + preserves-tr-apd-function refl = refl-htpy +``` diff --git a/src/foundation/functoriality-cartesian-product-types.lagda.md b/src/foundation/functoriality-cartesian-product-types.lagda.md index ad4e66e6ff..b9cf84522f 100644 --- a/src/foundation/functoriality-cartesian-product-types.lagda.md +++ b/src/foundation/functoriality-cartesian-product-types.lagda.md @@ -9,7 +9,6 @@ module foundation.functoriality-cartesian-product-types where ```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 @@ -17,6 +16,7 @@ open import foundation-core.contractible-maps open import foundation-core.contractible-types open import foundation-core.equivalences open import foundation-core.fibers-of-maps +open import foundation-core.function-types open import foundation-core.homotopies open import foundation-core.identity-types ``` diff --git a/src/foundation/functoriality-coproduct-types.lagda.md b/src/foundation/functoriality-coproduct-types.lagda.md index 4bfd896cae..257fc0ac34 100644 --- a/src/foundation/functoriality-coproduct-types.lagda.md +++ b/src/foundation/functoriality-coproduct-types.lagda.md @@ -15,7 +15,6 @@ 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 @@ -29,6 +28,7 @@ open import foundation-core.cartesian-product-types open import foundation-core.contractible-types open import foundation-core.empty-types open import foundation-core.fibers-of-maps +open import foundation-core.function-types open import foundation-core.functoriality-dependent-function-types open import foundation-core.functoriality-dependent-pair-types open import foundation-core.identity-types diff --git a/src/foundation/functoriality-dependent-function-types.lagda.md b/src/foundation/functoriality-dependent-function-types.lagda.md index ac233d09ce..fb61f2910e 100644 --- a/src/foundation/functoriality-dependent-function-types.lagda.md +++ b/src/foundation/functoriality-dependent-function-types.lagda.md @@ -14,8 +14,8 @@ 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.transport open import foundation.type-theoretic-principle-of-choice open import foundation.unit-type open import foundation.universal-property-unit-type @@ -25,10 +25,10 @@ open import foundation-core.commuting-squares-of-maps open import foundation-core.constant-maps open import foundation-core.embeddings open import foundation-core.fibers-of-maps +open import foundation-core.function-types open import foundation-core.functoriality-dependent-pair-types open import foundation-core.homotopies open import foundation-core.propositional-maps -open import foundation-core.transport open import foundation-core.truncated-maps open import foundation-core.truncated-types open import foundation-core.truncation-levels diff --git a/src/foundation/functoriality-dependent-pair-types.lagda.md b/src/foundation/functoriality-dependent-pair-types.lagda.md index 0a40ffce4a..595ee6e06d 100644 --- a/src/foundation/functoriality-dependent-pair-types.lagda.md +++ b/src/foundation/functoriality-dependent-pair-types.lagda.md @@ -11,12 +11,12 @@ open import foundation-core.functoriality-dependent-pair-types public ```agda 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.function-types open import foundation-core.homotopies open import foundation-core.identity-types open import foundation-core.pullbacks diff --git a/src/foundation/functoriality-fibers-of-maps.lagda.md b/src/foundation/functoriality-fibers-of-maps.lagda.md index c69f4cfd22..2bd2eb7825 100644 --- a/src/foundation/functoriality-fibers-of-maps.lagda.md +++ b/src/foundation/functoriality-fibers-of-maps.lagda.md @@ -10,11 +10,11 @@ module foundation.functoriality-fibers-of-maps where open import foundation.action-on-identifications-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.equality-dependent-pair-types open import foundation-core.fibers-of-maps +open import foundation-core.function-types open import foundation-core.functoriality-dependent-pair-types open import foundation-core.homotopies open import foundation-core.identity-types diff --git a/src/foundation/functoriality-function-types.lagda.md b/src/foundation/functoriality-function-types.lagda.md index 1013de3cf3..d55a9df047 100644 --- a/src/foundation/functoriality-function-types.lagda.md +++ b/src/foundation/functoriality-function-types.lagda.md @@ -10,12 +10,12 @@ 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.function-types open import foundation-core.homotopies open import foundation-core.truncated-maps open import foundation-core.truncation-levels diff --git a/src/foundation/functoriality-propositional-truncation.lagda.md b/src/foundation/functoriality-propositional-truncation.lagda.md index cb18a2174b..a311111dfe 100644 --- a/src/foundation/functoriality-propositional-truncation.lagda.md +++ b/src/foundation/functoriality-propositional-truncation.lagda.md @@ -9,13 +9,13 @@ module foundation.functoriality-propositional-truncation where ```agda open import foundation.action-on-identifications-functions 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.equivalences open import foundation-core.function-extensionality +open import foundation-core.function-types open import foundation-core.homotopies open import foundation-core.propositions ``` diff --git a/src/foundation/functoriality-set-quotients.lagda.md b/src/foundation/functoriality-set-quotients.lagda.md index 1195dd2744..9b120e3cfe 100644 --- a/src/foundation/functoriality-set-quotients.lagda.md +++ b/src/foundation/functoriality-set-quotients.lagda.md @@ -12,7 +12,6 @@ module foundation.functoriality-set-quotients where open import foundation.action-on-identifications-functions 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 @@ -27,6 +26,7 @@ open import foundation-core.commuting-squares-of-maps open import foundation-core.contractible-types open import foundation-core.equivalence-relations open import foundation-core.equivalences +open import foundation-core.function-types open import foundation-core.identity-types open import foundation-core.logical-equivalences open import foundation-core.propositions diff --git a/src/foundation/functoriality-set-truncation.lagda.md b/src/foundation/functoriality-set-truncation.lagda.md index bf80ab43aa..43b6cecb62 100644 --- a/src/foundation/functoriality-set-truncation.lagda.md +++ b/src/foundation/functoriality-set-truncation.lagda.md @@ -9,7 +9,6 @@ module foundation.functoriality-set-truncation where ```agda open import foundation.action-on-identifications-functions open import foundation.dependent-pair-types -open import foundation.functions open import foundation.functoriality-truncation open import foundation.images open import foundation.propositional-truncations @@ -26,6 +25,7 @@ open import foundation-core.contractible-types open import foundation-core.embeddings open import foundation-core.equivalences open import foundation-core.fibers-of-maps +open import foundation-core.function-types open import foundation-core.functoriality-dependent-pair-types open import foundation-core.homotopies open import foundation-core.identity-types diff --git a/src/foundation/functoriality-truncation.lagda.md b/src/foundation/functoriality-truncation.lagda.md index 13d69626a2..7919065e69 100644 --- a/src/foundation/functoriality-truncation.lagda.md +++ b/src/foundation/functoriality-truncation.lagda.md @@ -9,7 +9,6 @@ module foundation.functoriality-truncation where ```agda open import foundation.action-on-identifications-functions open import foundation.dependent-pair-types -open import foundation.functions open import foundation.truncations open import foundation.universe-levels @@ -17,6 +16,7 @@ open import foundation-core.commuting-squares-of-maps open import foundation-core.contractible-types open import foundation-core.equivalences open import foundation-core.function-extensionality +open import foundation-core.function-types open import foundation-core.homotopies open import foundation-core.truncation-levels ``` diff --git a/src/foundation/hilberts-epsilon-operators.lagda.md b/src/foundation/hilberts-epsilon-operators.lagda.md index da62df48e8..6ad95cd052 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.function-types ```
diff --git a/src/foundation/homotopies.lagda.md b/src/foundation/homotopies.lagda.md index 231e7765c0..da8305cbf5 100644 --- a/src/foundation/homotopies.lagda.md +++ b/src/foundation/homotopies.lagda.md @@ -13,13 +13,13 @@ open import foundation.action-on-identifications-dependent-functions open import foundation.action-on-identifications-functions 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.equivalences +open import foundation-core.function-types open import foundation-core.functoriality-dependent-function-types open import foundation-core.functoriality-dependent-pair-types open import foundation-core.sections @@ -217,60 +217,60 @@ module _ equiv-con-inv-htpy = pair (con-inv-htpy H K L) is-equiv-con-inv-htpy ``` -### Computing path-overs in the type family `eq-value` of dependent functions +### Computing dependent-identifications in the type family `eq-value` of dependent functions ```agda module _ {l1 l2 : Level} {A : UU l1} {B : A → UU l2} (f g : (x : A) → B x) where - is-equiv-map-compute-path-over-eq-value : + is-equiv-map-compute-dependent-identification-eq-value : {x y : A} (p : x ＝ y) (q : eq-value f g x) (r : eq-value f g y) → - is-equiv (map-compute-path-over-eq-value f g p q r) - is-equiv-map-compute-path-over-eq-value refl q r = + is-equiv (map-compute-dependent-identification-eq-value f g p q r) + is-equiv-map-compute-dependent-identification-eq-value refl q r = is-equiv-comp ( inv) ( concat' r (right-unit ∙ ap-id q)) ( is-equiv-concat' r (right-unit ∙ ap-id q)) ( is-equiv-inv r q) - compute-path-over-eq-value : + compute-dependent-identification-eq-value : {x y : A} (p : x ＝ y) (q : eq-value f g x) (r : eq-value f g y) → (((apd f p) ∙ r) ＝ ((ap (tr B p) q) ∙ (apd g p))) ≃ (tr (eq-value f g) p q ＝ r) - pr1 (compute-path-over-eq-value p q r) = - map-compute-path-over-eq-value f g p q r - pr2 (compute-path-over-eq-value p q r) = - is-equiv-map-compute-path-over-eq-value p q r + pr1 (compute-dependent-identification-eq-value p q r) = + map-compute-dependent-identification-eq-value f g p q r + pr2 (compute-dependent-identification-eq-value p q r) = + is-equiv-map-compute-dependent-identification-eq-value p q r ``` -### Computing path-overs in the type family `eq-value` of ordinary functions +### Computing dependent-identifications in the type family `eq-value` of ordinary functions ```agda module _ {l1 l2 : Level} {A : UU l1} {B : UU l2} (f g : A → B) where - is-equiv-map-compute-path-over-eq-value-function : + is-equiv-map-compute-dependent-identification-eq-value-function : {x y : A} (p : x ＝ y) (q : eq-value f g x) (q' : eq-value f g y) → - is-equiv (map-compute-path-over-eq-value-function f g p q q') - is-equiv-map-compute-path-over-eq-value-function refl q q' = + is-equiv (map-compute-dependent-identification-eq-value-function f g p q q') + is-equiv-map-compute-dependent-identification-eq-value-function refl q q' = is-equiv-comp ( inv) ( concat' q' right-unit) ( is-equiv-concat' q' right-unit) ( is-equiv-inv q' q) - compute-path-over-eq-value-function : + compute-dependent-identification-eq-value-function : {a0 a1 : A} (p : a0 ＝ a1) (q : f a0 ＝ g a0) (q' : f a1 ＝ g a1) → (((ap f p) ∙ q') ＝ (q ∙ (ap g p))) ≃ ((tr (eq-value f g) p q) ＝ q') - pr1 (compute-path-over-eq-value-function p q q') = - map-compute-path-over-eq-value-function f g p q q' - pr2 (compute-path-over-eq-value-function p q q') = - is-equiv-map-compute-path-over-eq-value-function p q q' + pr1 (compute-dependent-identification-eq-value-function p q q') = + map-compute-dependent-identification-eq-value-function f g p q q' + pr2 (compute-dependent-identification-eq-value-function p q q') = + is-equiv-map-compute-dependent-identification-eq-value-function p q q' ``` -### Relation between between `compute-path-over-eq-value-function` and `nat-htpy` +### Relation between between `compute-dependent-identification-eq-value-function` and `nat-htpy` ```agda module _ @@ -280,16 +280,20 @@ module _ nat-htpy-apd-htpy : (p : a0 ＝ a1) → - (map-inv-equiv (compute-path-over-eq-value-function + (map-inv-equiv (compute-dependent-identification-eq-value-function f g p (H a0) (H a1))) (apd H p) ＝ inv (nat-htpy H p) nat-htpy-apd-htpy refl = inv ( ap ( map-inv-equiv - ( compute-path-over-eq-value-function f g refl (H a0) (H a0))) + ( compute-dependent-identification-eq-value-function f g refl + ( H a0) + ( H a0))) ( ap inv (left-inv right-unit))) ∙ ( isretr-map-inv-equiv - ( compute-path-over-eq-value-function f g refl (H a0) (H a1)) + ( compute-dependent-identification-eq-value-function f g refl + ( H a0) + ( H a1)) ( inv right-unit)) ``` diff --git a/src/foundation/identity-types.lagda.md b/src/foundation/identity-types.lagda.md index 4f86af8718..c4f5025b6b 100644 --- a/src/foundation/identity-types.lagda.md +++ b/src/foundation/identity-types.lagda.md @@ -12,12 +12,13 @@ open import foundation-core.identity-types public open import foundation.action-on-identifications-functions open import foundation.binary-equivalences open import foundation.dependent-pair-types +open import foundation.equality-cartesian-product-types open import foundation.equivalence-extensionality open import foundation.function-extensionality -open import foundation.functions open import foundation.universe-levels open import foundation-core.equivalences +open import foundation-core.function-types open import foundation-core.homotopies open import foundation-core.transport ``` @@ -41,14 +42,14 @@ identifications in arbitrary types. | ------------------------------------------------ | ------------------------------------------------------------------------------------------------------------------------- | | Action on identifications of binary functions | [`foundation.action-on-identifications-binary-functions`](foundation.action-on-identifications-binary-functions.md) | | Action on identifications of dependent functions | [`foundation.action-on-identifications-dependent-functions`](foundation.action-on-identifications-dependent-functions.md) | -| Action on identifications of functions | [`foundation.action-on-identifications-functions](foundation.action-on-identifications-functions.md) | +| Action on identifications of functions | [`foundation.action-on-identifications-functions`](foundation.action-on-identifications-functions.md) | | Binary transport | [`foundation.binary-transport`](foundation.binary-transport.md) | -| Commuting squares of identifications | [`foundation.commuting-squares-identifications`](foundation.commuting-squares-identifications.md) | -| Dependent paths (foundation) | [`foundation.dependent-paths`](foundation.dependent-paths.md) | -| Dependent paths (foundation-core) | [`foundation-core.dependent-paths`](foundation-core.dependent-paths.md) | +| Commuting squares of identifications | [`foundation.commuting-squares-of-identifications`](foundation.commuting-squares-of-identifications.md) | +| Dependent identifications (foundation) | [`foundation.dependent-identifications`](foundation.dependent-identifications.md) | +| Dependent identifications (foundation-core) | [`foundation-core.dependent-identifications`](foundation-core.dependent-identifications.md) | | The fundamental theorem of identity types | [`foundation.fundamental-theorem-of-identity-types`](foundation.fundamental-theorem-of-identity-types.md) | | Hexagons of identifications | [`foundation.hexagons-of-identifications`](foundation.hexagons-of-identifications.md) | -| Identity systems | [`foundation.identity-sytsemts`](foundation.identity-systems.md) | +| Identity systems | [`foundation.identity-systems`](foundation.identity-systems.md) | | The identity type (foundation) | [`foundation.identity-types`](foundation.identity-types.md) | | The identity type (foundation-core) | [`foundation-core.identity-types`](foundation-core.identity-types.md) | | Large identity types | [`foundation.large-identity-types`](foundation.large-identity-types.md) | @@ -240,35 +241,29 @@ module _ pr2 (equiv-con-inv p q r) = is-equiv-con-inv p q r ``` -### Transport is an equivalence +### Computing transport in the type family of identifications with a fixed target ```agda -module _ - {l1 l2 : Level} {A : UU l1} (B : A → UU l2) {x y : A} - where +tr-Id-left : + {l : Level} {A : UU l} {a b c : A} (q : b ＝ c) (p : b ＝ a) → + tr (_＝ a) q p ＝ ((inv q) ∙ p) +tr-Id-left refl p = refl +``` - inv-tr : x ＝ y → B y → B x - inv-tr p = tr B (inv p) +### Computing transport in the type family of identifications with a fixed source - isretr-inv-tr : (p : x ＝ y) → ((inv-tr p) ∘ (tr B p)) ~ id - isretr-inv-tr refl b = refl +```agda +tr-Id-right : + {l : Level} {A : UU l} {a b c : A} (q : b ＝ c) (p : a ＝ b) → + tr (a ＝_) q p ＝ (p ∙ q) +tr-Id-right refl refl = refl +``` - issec-inv-tr : (p : x ＝ y) → ((tr B p) ∘ (inv-tr p)) ~ id - issec-inv-tr refl b = refl +### Computing transport of loops - is-equiv-tr : (p : x ＝ y) → is-equiv (tr B p) - is-equiv-tr p = - is-equiv-has-inverse - ( inv-tr p) - ( issec-inv-tr p) - ( isretr-inv-tr p) - - equiv-tr : x ＝ y → (B x) ≃ (B y) - pr1 (equiv-tr p) = tr B p - pr2 (equiv-tr p) = is-equiv-tr p - -equiv-tr-refl : - {l1 l2 : Level} {A : UU l1} (B : A → UU l2) {x : A} → - equiv-tr B refl ＝ id-equiv {A = B x} -equiv-tr-refl B = eq-htpy-equiv (λ _ → refl) +```agda +tr-loop : + {l1 : Level} {A : UU l1} {a0 a1 : A} (p : a0 ＝ a1) (l : a0 ＝ a0) → + (tr (λ y → y ＝ y) p l) ＝ ((inv p ∙ l) ∙ p) +tr-loop refl l = inv right-unit ``` diff --git a/src/foundation/images-subtypes.lagda.md b/src/foundation/images-subtypes.lagda.md index 3e247a71db..620c7d57ae 100644 --- a/src/foundation/images-subtypes.lagda.md +++ b/src/foundation/images-subtypes.lagda.md @@ -7,10 +7,10 @@ module foundation.images-subtypes where
Imports ```agda -open import foundation.functions open import foundation.images open import foundation.universe-levels +open import foundation-core.function-types open import foundation-core.subtypes ``` diff --git a/src/foundation/images.lagda.md b/src/foundation/images.lagda.md index 93346c8b46..bfb1b168a7 100644 --- a/src/foundation/images.lagda.md +++ b/src/foundation/images.lagda.md @@ -8,7 +8,6 @@ module foundation.images where ```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 @@ -21,6 +20,7 @@ open import foundation-core.contractible-types open import foundation-core.embeddings open import foundation-core.equivalences open import foundation-core.fibers-of-maps +open import foundation-core.function-types open import foundation-core.homotopies open import foundation-core.identity-types open import foundation-core.injective-maps diff --git a/src/foundation/involutions.lagda.md b/src/foundation/involutions.lagda.md index 6985c8820a..32cf391b53 100644 --- a/src/foundation/involutions.lagda.md +++ b/src/foundation/involutions.lagda.md @@ -12,9 +12,9 @@ open import foundation-core.involutions public 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.function-types open import foundation-core.homotopies open import foundation-core.identity-types open import foundation-core.injective-maps diff --git a/src/foundation/isolated-points.lagda.md b/src/foundation/isolated-points.lagda.md index 249de1822a..f469609c92 100644 --- a/src/foundation/isolated-points.lagda.md +++ b/src/foundation/isolated-points.lagda.md @@ -15,7 +15,6 @@ 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 @@ -29,6 +28,7 @@ open import foundation-core.coproduct-types open import foundation-core.decidable-propositions open import foundation-core.empty-types open import foundation-core.equivalences +open import foundation-core.function-types open import foundation-core.functoriality-dependent-pair-types open import foundation-core.homotopies open import foundation-core.injective-maps diff --git a/src/foundation/isomorphisms-of-sets.lagda.md b/src/foundation/isomorphisms-of-sets.lagda.md index 3f98b19122..8d36d44308 100644 --- a/src/foundation/isomorphisms-of-sets.lagda.md +++ b/src/foundation/isomorphisms-of-sets.lagda.md @@ -11,11 +11,11 @@ open import foundation.action-on-identifications-functions 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.function-types open import foundation-core.identity-types open import foundation-core.propositions open import foundation-core.subtypes diff --git a/src/foundation/iterated-cartesian-product-types.lagda.md b/src/foundation/iterated-cartesian-product-types.lagda.md index b6a6b222f4..9264006bc9 100644 --- a/src/foundation/iterated-cartesian-product-types.lagda.md +++ b/src/foundation/iterated-cartesian-product-types.lagda.md @@ -13,7 +13,6 @@ open import finite-group-theory.permutations-standard-finite-types open import foundation.action-on-identifications-functions 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 @@ -28,6 +27,7 @@ open import foundation-core.cartesian-product-types open import foundation-core.contractible-types open import foundation-core.coproduct-types open import foundation-core.equivalences +open import foundation-core.function-types open import foundation-core.identity-types open import lists.arrays diff --git a/src/foundation/iterating-automorphisms.lagda.md b/src/foundation/iterating-automorphisms.lagda.md index ce820fefd3..067029fb17 100644 --- a/src/foundation/iterating-automorphisms.lagda.md +++ b/src/foundation/iterating-automorphisms.lagda.md @@ -13,12 +13,12 @@ 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.coproduct-types open import foundation-core.equivalences +open import foundation-core.function-types open import foundation-core.homotopies ``` diff --git a/src/foundation/large-locale-of-propositions.lagda.md b/src/foundation/large-locale-of-propositions.lagda.md index 58fdcc595a..1f9a6d2b41 100644 --- a/src/foundation/large-locale-of-propositions.lagda.md +++ b/src/foundation/large-locale-of-propositions.lagda.md @@ -9,11 +9,11 @@ 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.function-types open import foundation-core.propositions open import order-theory.large-frames diff --git a/src/foundation/locally-small-types.lagda.md b/src/foundation/locally-small-types.lagda.md index 8184c4ebba..dfcfb2b8d9 100644 --- a/src/foundation/locally-small-types.lagda.md +++ b/src/foundation/locally-small-types.lagda.md @@ -9,7 +9,6 @@ module foundation.locally-small-types where ```agda open import foundation.dependent-pair-types open import foundation.function-extensionality -open import foundation.functions open import foundation.inhabited-subtypes open import foundation.subuniverses open import foundation.univalence @@ -17,6 +16,7 @@ open import foundation.universe-levels open import foundation-core.equality-dependent-pair-types open import foundation-core.equivalences +open import foundation-core.function-types open import foundation-core.homotopies open import foundation-core.identity-types open import foundation-core.propositions diff --git a/src/foundation/logical-equivalences.lagda.md b/src/foundation/logical-equivalences.lagda.md index 45351d2ef7..96c146e8a7 100644 --- a/src/foundation/logical-equivalences.lagda.md +++ b/src/foundation/logical-equivalences.lagda.md @@ -10,10 +10,10 @@ open import foundation-core.logical-equivalences public ```agda open import foundation.dependent-pair-types -open import foundation.functions open import foundation.universe-levels open import foundation-core.equivalences +open import foundation-core.function-types open import foundation-core.identity-types open import foundation-core.propositions ``` diff --git a/src/foundation/monomorphisms.lagda.md b/src/foundation/monomorphisms.lagda.md index ab12304219..922ee2bed2 100644 --- a/src/foundation/monomorphisms.lagda.md +++ b/src/foundation/monomorphisms.lagda.md @@ -10,11 +10,11 @@ module foundation.monomorphisms where open import foundation.action-on-identifications-functions 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.equivalences +open import foundation-core.function-types open import foundation-core.identity-types open import foundation-core.propositional-maps open import foundation-core.propositions diff --git a/src/foundation/morphisms-cospans.lagda.md b/src/foundation/morphisms-cospans.lagda.md index 916e09488d..c63c822306 100644 --- a/src/foundation/morphisms-cospans.lagda.md +++ b/src/foundation/morphisms-cospans.lagda.md @@ -8,10 +8,10 @@ module foundation.morphisms-cospans where ```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.function-types open import foundation-core.homotopies ``` diff --git a/src/foundation/multivariable-functoriality-set-quotients.lagda.md b/src/foundation/multivariable-functoriality-set-quotients.lagda.md index 940d7a71de..ce554985de 100644 --- a/src/foundation/multivariable-functoriality-set-quotients.lagda.md +++ b/src/foundation/multivariable-functoriality-set-quotients.lagda.md @@ -9,13 +9,13 @@ 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.function-types open import foundation-core.homotopies open import linear-algebra.vectors diff --git a/src/foundation/multivariable-operations.lagda.md b/src/foundation/multivariable-operations.lagda.md index 09f934bf30..0d1defbf4f 100644 --- a/src/foundation/multivariable-operations.lagda.md +++ b/src/foundation/multivariable-operations.lagda.md @@ -12,7 +12,6 @@ open import elementary-number-theory.natural-numbers open import foundation.action-on-identifications-functions 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 @@ -20,6 +19,7 @@ open import foundation.universe-levels open import foundation-core.cartesian-product-types open import foundation-core.coproduct-types open import foundation-core.equivalences +open import foundation-core.function-types open import foundation-core.homotopies open import foundation-core.identity-types diff --git a/src/foundation/noncontractible-types.lagda.md b/src/foundation/noncontractible-types.lagda.md index 77c76afa04..081cf913a9 100644 --- a/src/foundation/noncontractible-types.lagda.md +++ b/src/foundation/noncontractible-types.lagda.md @@ -10,11 +10,11 @@ module foundation.noncontractible-types where 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.empty-types +open import foundation-core.function-types open import foundation-core.identity-types open import foundation-core.negation ``` diff --git a/src/foundation/partitions.lagda.md b/src/foundation/partitions.lagda.md index 12824ab3bc..23a67f49e5 100644 --- a/src/foundation/partitions.lagda.md +++ b/src/foundation/partitions.lagda.md @@ -14,7 +14,6 @@ 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 @@ -27,11 +26,13 @@ open import foundation.small-types open import foundation.subtype-identity-principle open import foundation.subtypes open import foundation.surjective-maps +open import foundation.transport open import foundation.type-arithmetic-dependent-pair-types open import foundation.universe-levels open import foundation-core.cartesian-product-types open import foundation-core.fibers-of-maps +open import foundation-core.function-types open import foundation-core.functoriality-dependent-function-types open import foundation-core.functoriality-dependent-pair-types open import foundation-core.homotopies diff --git a/src/foundation/path-algebra.lagda.md b/src/foundation/path-algebra.lagda.md index eb5753097e..33276428f6 100644 --- a/src/foundation/path-algebra.lagda.md +++ b/src/foundation/path-algebra.lagda.md @@ -12,11 +12,11 @@ open import foundation.action-on-identifications-functions 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.function-types open import foundation-core.homotopies ``` diff --git a/src/foundation/perfect-images.lagda.md b/src/foundation/perfect-images.lagda.md index 8bf2a1509c..ada8d389fe 100644 --- a/src/foundation/perfect-images.lagda.md +++ b/src/foundation/perfect-images.lagda.md @@ -13,7 +13,6 @@ open import foundation.action-on-identifications-functions 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 @@ -24,6 +23,7 @@ open import foundation-core.coproduct-types open import foundation-core.embeddings open import foundation-core.empty-types open import foundation-core.fibers-of-maps +open import foundation-core.function-types open import foundation-core.identity-types open import foundation-core.injective-maps open import foundation-core.propositional-maps diff --git a/src/foundation/pi-decompositions.lagda.md b/src/foundation/pi-decompositions.lagda.md index 1e476b7df8..b1cc04c869 100644 --- a/src/foundation/pi-decompositions.lagda.md +++ b/src/foundation/pi-decompositions.lagda.md @@ -12,7 +12,6 @@ module foundation.pi-decompositions where 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.fundamental-theorem-of-identity-types open import foundation.structure-identity-principle @@ -21,6 +20,7 @@ open import foundation.univalence open import foundation.universe-levels open import foundation-core.contractible-types +open import foundation-core.function-types open import foundation-core.functoriality-dependent-pair-types open import foundation-core.homotopies open import foundation-core.identity-types diff --git a/src/foundation/preidempotent-maps.lagda.md b/src/foundation/preidempotent-maps.lagda.md index 343ccb584e..921ad4b0e8 100644 --- a/src/foundation/preidempotent-maps.lagda.md +++ b/src/foundation/preidempotent-maps.lagda.md @@ -8,9 +8,9 @@ module foundation.preidempotent-maps where ```agda open import foundation.dependent-pair-types -open import foundation.functions open import foundation.universe-levels +open import foundation-core.function-types open import foundation-core.homotopies open import foundation-core.propositions open import foundation-core.sets diff --git a/src/foundation/products-unordered-pairs-of-types.lagda.md b/src/foundation/products-unordered-pairs-of-types.lagda.md index c9013920c4..3728979e02 100644 --- a/src/foundation/products-unordered-pairs-of-types.lagda.md +++ b/src/foundation/products-unordered-pairs-of-types.lagda.md @@ -8,17 +8,18 @@ module foundation.products-unordered-pairs-of-types where ```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.transport 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.equivalences +open import foundation-core.function-types open import univalent-combinatorics.2-element-types open import univalent-combinatorics.universal-property-standard-finite-types diff --git a/src/foundation/projective-types.lagda.md b/src/foundation/projective-types.lagda.md index a158fc310c..55148c4ed8 100644 --- a/src/foundation/projective-types.lagda.md +++ b/src/foundation/projective-types.lagda.md @@ -10,11 +10,11 @@ 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.function-types open import foundation-core.sets open import foundation-core.truncated-types ``` diff --git a/src/foundation/propositional-extensionality.lagda.md b/src/foundation/propositional-extensionality.lagda.md index 624a72a527..bae2cfb441 100644 --- a/src/foundation/propositional-extensionality.lagda.md +++ b/src/foundation/propositional-extensionality.lagda.md @@ -16,6 +16,7 @@ open import foundation.logical-equivalences open import foundation.negation open import foundation.raising-universe-levels open import foundation.subtype-identity-principle +open import foundation.transport open import foundation.type-arithmetic-cartesian-product-types open import foundation.unit-type open import foundation.univalence diff --git a/src/foundation/propositional-truncations.lagda.md b/src/foundation/propositional-truncations.lagda.md index 2a9f2457d9..8342afce15 100644 --- a/src/foundation/propositional-truncations.lagda.md +++ b/src/foundation/propositional-truncations.lagda.md @@ -9,7 +9,6 @@ module foundation.propositional-truncations where ```agda open import foundation.action-on-identifications-functions 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 @@ -20,6 +19,7 @@ open import foundation-core.cartesian-product-types open import foundation-core.contractible-types open import foundation-core.coproduct-types open import foundation-core.equivalences +open import foundation-core.function-types open import foundation-core.homotopies open import foundation-core.identity-types open import foundation-core.sets diff --git a/src/foundation/pullbacks.lagda.md b/src/foundation/pullbacks.lagda.md index 4d53f63daf..989722d99d 100644 --- a/src/foundation/pullbacks.lagda.md +++ b/src/foundation/pullbacks.lagda.md @@ -16,7 +16,6 @@ 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 @@ -29,6 +28,7 @@ open import foundation-core.constant-maps open import foundation-core.contractible-types open import foundation-core.diagonal-maps-of-types open import foundation-core.equality-dependent-pair-types +open import foundation-core.function-types open import foundation-core.functoriality-dependent-function-types open import foundation-core.propositions open import foundation-core.transport diff --git a/src/foundation/raising-universe-levels.lagda.md b/src/foundation/raising-universe-levels.lagda.md index 05d35ee4cc..3b9a34fef4 100644 --- a/src/foundation/raising-universe-levels.lagda.md +++ b/src/foundation/raising-universe-levels.lagda.md @@ -9,10 +9,10 @@ module foundation.raising-universe-levels where ```agda open import foundation.action-on-identifications-functions open import foundation.dependent-pair-types -open import foundation.functions open import foundation.universe-levels open import foundation-core.equivalences +open import foundation-core.function-types open import foundation-core.homotopies open import foundation-core.identity-types open import foundation-core.propositions diff --git a/src/foundation/relaxed-sigma-decompositions.lagda.md b/src/foundation/relaxed-sigma-decompositions.lagda.md index 2a52cb45bc..f2f11dcaaf 100644 --- a/src/foundation/relaxed-sigma-decompositions.lagda.md +++ b/src/foundation/relaxed-sigma-decompositions.lagda.md @@ -13,7 +13,6 @@ open import foundation.action-on-identifications-functions 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 @@ -22,6 +21,7 @@ open import foundation.univalence open import foundation.universe-levels open import foundation-core.contractible-types +open import foundation-core.function-types open import foundation-core.functoriality-dependent-pair-types open import foundation-core.homotopies open import foundation-core.identity-types diff --git a/src/foundation/repetitions-of-values.lagda.md b/src/foundation/repetitions-of-values.lagda.md index b202b36ccd..f7c3e99b13 100644 --- a/src/foundation/repetitions-of-values.lagda.md +++ b/src/foundation/repetitions-of-values.lagda.md @@ -11,12 +11,12 @@ open import foundation.action-on-identifications-functions 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.function-types open import foundation-core.functoriality-dependent-pair-types open import foundation-core.homotopies open import foundation-core.injective-maps diff --git a/src/foundation/repetitions-sequences.lagda.md b/src/foundation/repetitions-sequences.lagda.md index 9c38ae242c..5c6019bfe9 100644 --- a/src/foundation/repetitions-sequences.lagda.md +++ b/src/foundation/repetitions-sequences.lagda.md @@ -12,7 +12,6 @@ open import elementary-number-theory.strictly-ordered-pairs-of-natural-numbers open import foundation.action-on-identifications-functions 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 @@ -21,6 +20,7 @@ open import foundation.universe-levels open import foundation-core.embeddings open import foundation-core.empty-types +open import foundation-core.function-types open import foundation-core.functoriality-dependent-pair-types open import foundation-core.identity-types open import foundation-core.injective-maps diff --git a/src/foundation/retractions.lagda.md b/src/foundation/retractions.lagda.md index d845eaae7f..9495577172 100644 --- a/src/foundation/retractions.lagda.md +++ b/src/foundation/retractions.lagda.md @@ -12,10 +12,10 @@ open import foundation-core.retractions public open import foundation.action-on-identifications-functions open import foundation.coslice open import foundation.dependent-pair-types -open import foundation.functions open import foundation.universe-levels open import foundation-core.equivalences +open import foundation-core.function-types open import foundation-core.homotopies open import foundation-core.identity-types open import foundation-core.injective-maps diff --git a/src/foundation/sections.lagda.md b/src/foundation/sections.lagda.md index 904053da9a..fdbd9f0136 100644 --- a/src/foundation/sections.lagda.md +++ b/src/foundation/sections.lagda.md @@ -12,7 +12,6 @@ open import foundation-core.sections public open import foundation.action-on-identifications-functions 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 @@ -21,6 +20,7 @@ open import foundation.universe-levels open import foundation-core.contractible-types open import foundation-core.equivalences open import foundation-core.fibers-of-maps +open import foundation-core.function-types open import foundation-core.functoriality-dependent-pair-types open import foundation-core.homotopies open import foundation-core.identity-types diff --git a/src/foundation/sequences.lagda.md b/src/foundation/sequences.lagda.md index 7cf6e83bff..e2fa18053f 100644 --- a/src/foundation/sequences.lagda.md +++ b/src/foundation/sequences.lagda.md @@ -9,8 +9,9 @@ module foundation.sequences where ```agda open import elementary-number-theory.natural-numbers -open import foundation.functions open import foundation.universe-levels + +open import foundation-core.function-types ```
diff --git a/src/foundation/set-presented-types.lagda.md b/src/foundation/set-presented-types.lagda.md index fc03b8d896..ee7d3b9858 100644 --- a/src/foundation/set-presented-types.lagda.md +++ b/src/foundation/set-presented-types.lagda.md @@ -8,11 +8,11 @@ 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.function-types open import foundation-core.propositions open import foundation-core.sets ``` diff --git a/src/foundation/set-quotients.lagda.md b/src/foundation/set-quotients.lagda.md index f2acaafc79..0f1c904aee 100644 --- a/src/foundation/set-quotients.lagda.md +++ b/src/foundation/set-quotients.lagda.md @@ -13,7 +13,6 @@ 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 @@ -27,6 +26,7 @@ open import foundation.universe-levels open import foundation-core.equivalence-relations open import foundation-core.function-extensionality +open import foundation-core.function-types open import foundation-core.functoriality-dependent-function-types open import foundation-core.homotopies open import foundation-core.propositions diff --git a/src/foundation/set-truncations.lagda.md b/src/foundation/set-truncations.lagda.md index f01a9b6bf9..7f7ae14a5e 100644 --- a/src/foundation/set-truncations.lagda.md +++ b/src/foundation/set-truncations.lagda.md @@ -12,7 +12,6 @@ module foundation.set-truncations where 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 @@ -36,6 +35,7 @@ open import foundation-core.coproduct-types open import foundation-core.embeddings open import foundation-core.empty-types open import foundation-core.equivalences +open import foundation-core.function-types open import foundation-core.functoriality-dependent-function-types open import foundation-core.functoriality-dependent-pair-types open import foundation-core.functoriality-function-types diff --git a/src/foundation/sets.lagda.md b/src/foundation/sets.lagda.md index 25a85e39eb..7c9e85027f 100644 --- a/src/foundation/sets.lagda.md +++ b/src/foundation/sets.lagda.md @@ -11,7 +11,6 @@ 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 @@ -20,6 +19,7 @@ open import foundation-core.1-types open import foundation-core.cartesian-product-types open import foundation-core.embeddings open import foundation-core.equivalences +open import foundation-core.function-types open import foundation-core.identity-types open import foundation-core.propositional-maps open import foundation-core.propositions diff --git a/src/foundation/sigma-decompositions.lagda.md b/src/foundation/sigma-decompositions.lagda.md index bbe5c7dae0..ff68eb2de9 100644 --- a/src/foundation/sigma-decompositions.lagda.md +++ b/src/foundation/sigma-decompositions.lagda.md @@ -13,7 +13,6 @@ open import foundation.action-on-identifications-functions 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 @@ -25,6 +24,7 @@ open import foundation.univalence open import foundation.universe-levels open import foundation-core.contractible-types +open import foundation-core.function-types open import foundation-core.functoriality-dependent-pair-types open import foundation-core.homotopies open import foundation-core.identity-types diff --git a/src/foundation/singleton-induction.lagda.md b/src/foundation/singleton-induction.lagda.md index 53ff85821d..aacc8691d9 100644 --- a/src/foundation/singleton-induction.lagda.md +++ b/src/foundation/singleton-induction.lagda.md @@ -9,10 +9,10 @@ module foundation.singleton-induction where ```agda open import foundation.action-on-identifications-functions 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.function-types open import foundation-core.homotopies open import foundation-core.identity-types open import foundation-core.sections diff --git a/src/foundation/slice.lagda.md b/src/foundation/slice.lagda.md index 88f7059f22..96015ef626 100644 --- a/src/foundation/slice.lagda.md +++ b/src/foundation/slice.lagda.md @@ -10,7 +10,6 @@ module foundation.slice where 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 @@ -23,6 +22,7 @@ open import foundation-core.contractible-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.function-types open import foundation-core.functoriality-dependent-pair-types open import foundation-core.identity-types open import foundation-core.injective-maps diff --git a/src/foundation/split-surjective-maps.lagda.md b/src/foundation/split-surjective-maps.lagda.md index abf38b8ce1..11c3227eef 100644 --- a/src/foundation/split-surjective-maps.lagda.md +++ b/src/foundation/split-surjective-maps.lagda.md @@ -8,13 +8,13 @@ module foundation.split-surjective-maps where ```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.equivalences open import foundation-core.fibers-of-maps +open import foundation-core.function-types open import foundation-core.injective-maps open import foundation-core.retractions open import foundation-core.sections diff --git a/src/foundation/structured-type-duality.lagda.md b/src/foundation/structured-type-duality.lagda.md index 895d88c9a9..8e3dd60f53 100644 --- a/src/foundation/structured-type-duality.lagda.md +++ b/src/foundation/structured-type-duality.lagda.md @@ -9,7 +9,6 @@ module foundation.structured-type-duality where ```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 @@ -18,6 +17,7 @@ open import foundation.univalence open import foundation.universe-levels open import foundation-core.fibers-of-maps +open import foundation-core.function-types open import foundation-core.functoriality-dependent-function-types open import foundation-core.functoriality-dependent-pair-types ``` diff --git a/src/foundation/subterminal-types.lagda.md b/src/foundation/subterminal-types.lagda.md index 78e8db23d5..fc23e64bed 100644 --- a/src/foundation/subterminal-types.lagda.md +++ b/src/foundation/subterminal-types.lagda.md @@ -8,13 +8,13 @@ module foundation.subterminal-types where ```agda open import foundation.action-on-identifications-functions -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.function-types open import foundation-core.identity-types open import foundation-core.propositions ``` diff --git a/src/foundation/subtype-identity-principle.lagda.md b/src/foundation/subtype-identity-principle.lagda.md index 0c3f0c7e63..5b7189520e 100644 --- a/src/foundation/subtype-identity-principle.lagda.md +++ b/src/foundation/subtype-identity-principle.lagda.md @@ -8,13 +8,13 @@ module foundation.subtype-identity-principle where ```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.function-types open import foundation-core.identity-types open import foundation-core.propositions ``` diff --git a/src/foundation/subtypes.lagda.md b/src/foundation/subtypes.lagda.md index b9e24b7b97..a958dea8a8 100644 --- a/src/foundation/subtypes.lagda.md +++ b/src/foundation/subtypes.lagda.md @@ -11,7 +11,6 @@ open import foundation-core.subtypes public ```agda open import foundation.dependent-pair-types open import foundation.equality-dependent-function-types -open import foundation.functions open import foundation.fundamental-theorem-of-identity-types open import foundation.propositional-extensionality open import foundation.universe-levels @@ -19,6 +18,7 @@ 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.function-types open import foundation-core.homotopies open import foundation-core.identity-types open import foundation-core.injective-maps diff --git a/src/foundation/surjective-maps.lagda.md b/src/foundation/surjective-maps.lagda.md index 986843b240..89018d69a3 100644 --- a/src/foundation/surjective-maps.lagda.md +++ b/src/foundation/surjective-maps.lagda.md @@ -13,7 +13,6 @@ 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 @@ -30,6 +29,7 @@ open import foundation-core.constant-maps open import foundation-core.contractible-maps open import foundation-core.equivalences open import foundation-core.fibers-of-maps +open import foundation-core.function-types open import foundation-core.functoriality-dependent-function-types open import foundation-core.propositional-maps open import foundation-core.propositions diff --git a/src/foundation/symmetric-difference.lagda.md b/src/foundation/symmetric-difference.lagda.md index 564e20addc..f1fa00fd8a 100644 --- a/src/foundation/symmetric-difference.lagda.md +++ b/src/foundation/symmetric-difference.lagda.md @@ -11,7 +11,6 @@ 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 @@ -19,6 +18,7 @@ open import foundation.universe-levels open import foundation-core.coproduct-types open import foundation-core.decidable-propositions open import foundation-core.equivalences +open import foundation-core.function-types open import foundation-core.propositions open import foundation-core.subtypes open import foundation-core.transport diff --git a/src/foundation/symmetric-identity-types.lagda.md b/src/foundation/symmetric-identity-types.lagda.md index c4d818c448..0ec898ddc3 100644 --- a/src/foundation/symmetric-identity-types.lagda.md +++ b/src/foundation/symmetric-identity-types.lagda.md @@ -10,7 +10,6 @@ module foundation.symmetric-identity-types where open import foundation.action-on-identifications-functions 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 @@ -23,6 +22,7 @@ open import foundation-core.contractible-types open import foundation-core.coproduct-types open import foundation-core.equality-dependent-pair-types open import foundation-core.equivalences +open import foundation-core.function-types open import foundation-core.functoriality-dependent-pair-types open import univalent-combinatorics.standard-finite-types diff --git a/src/foundation/transport.lagda.md b/src/foundation/transport.lagda.md index d3c1725f8c..c4ee6b2b6f 100644 --- a/src/foundation/transport.lagda.md +++ b/src/foundation/transport.lagda.md @@ -9,18 +9,12 @@ open import foundation-core.transport public
Imports ```agda -open import foundation.action-on-identifications-dependent-functions open import foundation.action-on-identifications-functions -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-paths -open import foundation-core.equality-dependent-pair-types +open import foundation-core.dependent-pair-types open import foundation-core.equivalences +open import foundation-core.function-types open import foundation-core.homotopies open import foundation-core.identity-types ``` @@ -29,170 +23,63 @@ open import foundation-core.identity-types ## Idea -This file records interactions between transport (`tr`) and other constructions. +Given a type family `B` over `A`, an +[identification](foundation-core.identity-types.md) `p : x ＝ y` in `A` and an +element `b : B x`, we can [**transport**](foundation-core.transport.md) the +element `b` along the identification `p` to obtain an element `tr B p b : B y`. -## Properties - -### Transport - -```agda -module _ - {l1 l2 l3 : Level} {X : UU l1} {A : X → UU l2} - {x0 x1 : X} (p : x0 ＝ x1) - where - - preserves-tr-id : - (p : x0 ＝ x1) → preserves-tr (λ x → id {A = A x}) p ~ refl-htpy - preserves-tr-id refl = refl-htpy -``` - -### Transport in a family of cartesian products - -```agda -tr-prod : - {l1 l2 : Level} {A : UU l1} {a0 a1 : A} - (B C : A → UU l2) (p : a0 ＝ a1) (u : B a0 × C a0) → - (tr (λ a → B a × C a) p u) ＝ (pair (tr B p (pr1 u)) (tr C p (pr2 u))) -tr-prod B C refl u = refl -``` +The fact that `tr B p` is an [equivalence](foundation-core.equivalences.md) is +recorded in this file. -### Transport in a family over a cartesian product +## Definitions -#### Computing transport along a path of the form `eq-pair` +### The action on identifications of transport ```agda module _ - {l1 l2 l3 : Level} {A : UU l1} {B : UU l2} {a0 a1 : A} {b0 b1 : B} + {l1 l2 : Level} {A : UU l1} {a0 a1 : A} {p0 p1 : a0 ＝ a1} + (B : A → UU l2) where - tr-eq-pair : - (C : A × B → UU l3) (p : a0 ＝ a1) (q : b0 ＝ b1) (u : C (a0 , b0)) → - tr C (eq-pair p q) u ＝ - tr (λ x → C (a1 , x)) q (tr (λ x → C (x , b0)) p u) - tr-eq-pair C refl refl u = refl -``` - -#### Computing transport along a path of the form `eq-pair` When one of the paths is `refl` - -```agda - left-unit-law-tr-eq-pair : - (C : A × B → UU l3) (q : b0 ＝ b1) (u : C (a0 , b0)) → - (tr C (eq-pair refl q) u) ＝ tr (λ x → C (a0 , x)) q u - left-unit-law-tr-eq-pair C refl u = refl - - right-unit-law-tr-eq-pair : - (C : A × B → UU l3) (p : a0 ＝ a1) (u : C (a0 , b0)) → - (tr C (eq-pair p refl) u) ＝ tr (λ x → C (x , b0)) p u - right-unit-law-tr-eq-pair C refl u = refl -``` - -#### Computing transport along a path of the form `eq-pair` when both paths are identical - -```agda -tr-eq-pair-diagonal : - {l1 l2 : Level} {A : UU l1} {a0 a1 : A} (C : A × A → UU l2) - (p : a0 ＝ a1) (u : C (a0 , a0)) → - tr C (eq-pair p p) u ＝ tr (λ a → C (a , a)) p u -tr-eq-pair-diagonal C refl u = refl + tr² : (α : p0 ＝ p1) (b0 : B a0) → (tr B p0 b0) ＝ (tr B p1 b0) + tr² α b0 = ap (λ t → tr B t b0) α ``` -### Transport in a family of dependent pair types - -```agda -tr-Σ : - {l1 l2 l3 : Level} {A : UU l1} {a0 a1 : A} {B : A → UU l2} - (C : (x : A) → B x → UU l3) (p : a0 ＝ a1) (z : Σ (B a0) (λ x → C a0 x)) → - tr (λ a → (Σ (B a) (C a))) p z ＝ - pair (tr B p (pr1 z)) (tr (ind-Σ C) (eq-pair-Σ p refl) (pr2 z)) -tr-Σ C refl z = refl -``` - -### Transport in a family over a dependent pair type - -```agda -tr-eq-pair-Σ : - {l1 l2 l3 : Level} {A : UU l1} {a0 a1 : A} - {B : A → UU l2} {b0 : B a0} {b1 : B a1} (C : (Σ A B) → UU l3) - (p : a0 ＝ a1) (q : path-over B p b0 b1) (u : C (a0 , b0)) → - tr C (eq-pair-Σ p q) u ＝ - tr (λ x → C (a1 , x)) q (tr C (eq-pair-Σ p refl) u) -tr-eq-pair-Σ C refl refl u = refl -``` - -### Transport in a family of function types - -```agda -module _ - {l1 l2 l3 : Level} {A : UU l1} {x y : A} (B : A → UU l2) (C : A → UU l3) - where - - tr-function-type : - (p : x ＝ y) (f : B x → C x) → - tr (λ a → B a → C a) p f ＝ (tr C p ∘ (f ∘ tr B (inv p))) - tr-function-type refl f = refl - - compute-path-over-function-type : - (p : x ＝ y) (f : B x → C x) (g : B y → C y) → - ((b : B x) → tr C p (f b) ＝ g (tr B p b)) ≃ - (tr (λ a → B a → C a) p f ＝ g) - compute-path-over-function-type refl f g = inv-equiv equiv-funext - - map-compute-path-over-function-type : - (p : x ＝ y) (f : B x → C x) (g : B y → C y) → - ((b : B x) → tr C p (f b) ＝ g (tr B p b)) → - (tr (λ a → B a → C a) p f ＝ g) - map-compute-path-over-function-type p f g = - map-equiv (compute-path-over-function-type p f g) -``` +## Properties -Relation between`compute-path-over-function-type` and `preserves-tr` +### Transport is an equivalence ```agda module _ - {l1 l2 l3 : Level} {I : UU l1} {i0 i1 : I} {A : I → UU l2} {B : I → UU l3} - (f : (i : I) → A i → B i) + {l1 l2 : Level} {A : UU l1} (B : A → UU l2) {x y : A} where - preserves-tr-apd-function : - (p : i0 ＝ i1) (a : A i0) → - (map-inv-equiv (compute-path-over-function-type - A B p (f i0) (f i1)) (apd f p) a) ＝ - inv-htpy (preserves-tr f p) a - preserves-tr-apd-function refl = refl-htpy -``` + inv-tr : x ＝ y → B y → B x + inv-tr p = tr B (inv p) -### Transport in identity types + isretr-inv-tr : (p : x ＝ y) → ((inv-tr p) ∘ (tr B p)) ~ id + isretr-inv-tr refl b = refl -```agda -tr-fx＝gy : - {l1 l2 l3 : Level} {A : UU l1} {B : UU l2} {C : UU l3} - {a0 a1 : A} {b0 b1 : B} (f : A → C) (g : B → C) - (p : a0 ＝ a1) (q : b0 ＝ b1) (s : f a0 ＝ g b0) → - (tr (λ z → (f (pr1 z)) ＝ (g (pr2 z))) (eq-pair p q) s) ＝ - (((inv (ap f p)) ∙ s) ∙ (ap g q)) -tr-fx＝gy f g refl refl s = inv right-unit - -tr-x＝y : - {l1 : Level} {A : UU l1} {a0 a1 a2 a3 : A} - (p : a0 ＝ a1) (q : a2 ＝ a3) (s : a0 ＝ a2) → - (tr (λ z → (pr1 z) ＝ (pr2 z)) (eq-pair p q) s) ＝ ((inv p) ∙ (s ∙ q)) -tr-x＝y refl refl s = inv right-unit -``` + issec-inv-tr : (p : x ＝ y) → ((tr B p) ∘ (inv-tr p)) ~ id + issec-inv-tr refl b = refl -### Transport in the family of loops + is-equiv-tr : (p : x ＝ y) → is-equiv (tr B p) + is-equiv-tr p = + is-equiv-has-inverse + ( inv-tr p) + ( issec-inv-tr p) + ( isretr-inv-tr p) -```agda -tr-loop : - {l1 : Level} {A : UU l1} {a0 a1 : A} (p : a0 ＝ a1) (l : a0 ＝ a0) → - (tr (λ y → y ＝ y) p l) ＝ (((inv p) ∙ l) ∙ p) -tr-loop refl l = inv right-unit + equiv-tr : x ＝ y → (B x) ≃ (B y) + pr1 (equiv-tr p) = tr B p + pr2 (equiv-tr p) = is-equiv-tr p ``` -### Transport of identifications +### Transporting along `refl` is the identity equivalence ```agda -tr-Id : - {l1 : Level} {A : UU l1} {a0 a1 a2 : A} (p : a1 ＝ a2) (l : a0 ＝ a1) → - (tr (a0 ＝_) p l) ＝ (l ∙ p) -tr-Id refl refl = refl +equiv-tr-refl : + {l1 l2 : Level} {A : UU l1} (B : A → UU l2) {x : A} → + equiv-tr B refl ＝ id-equiv {A = B x} +equiv-tr-refl B = refl ``` diff --git a/src/foundation/trivial-relaxed-sigma-decompositions.lagda.md b/src/foundation/trivial-relaxed-sigma-decompositions.lagda.md index 68e25d86e3..a502cb303d 100644 --- a/src/foundation/trivial-relaxed-sigma-decompositions.lagda.md +++ b/src/foundation/trivial-relaxed-sigma-decompositions.lagda.md @@ -10,13 +10,13 @@ module foundation.trivial-relaxed-sigma-decompositions where 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.equality-dependent-pair-types +open import foundation-core.function-types open import foundation-core.identity-types open import foundation-core.propositions open import foundation-core.subtypes diff --git a/src/foundation/trivial-sigma-decompositions.lagda.md b/src/foundation/trivial-sigma-decompositions.lagda.md index 4fca2baec6..85d29fdebb 100644 --- a/src/foundation/trivial-sigma-decompositions.lagda.md +++ b/src/foundation/trivial-sigma-decompositions.lagda.md @@ -10,7 +10,6 @@ module foundation.trivial-sigma-decompositions where 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 @@ -21,6 +20,7 @@ open import foundation.universe-levels open import foundation-core.empty-types open import foundation-core.equality-dependent-pair-types +open import foundation-core.function-types open import foundation-core.identity-types open import foundation-core.propositions open import foundation-core.subtypes diff --git a/src/foundation/truncation-equivalences.lagda.md b/src/foundation/truncation-equivalences.lagda.md index b994fd2464..572a23aa50 100644 --- a/src/foundation/truncation-equivalences.lagda.md +++ b/src/foundation/truncation-equivalences.lagda.md @@ -10,11 +10,11 @@ module foundation.truncation-equivalences where 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.function-types open import foundation-core.homotopies open import foundation-core.truncated-types open import foundation-core.truncation-levels diff --git a/src/foundation/truncation-levels.lagda.md b/src/foundation/truncation-levels.lagda.md index 807afc90b7..d31c673da4 100644 --- a/src/foundation/truncation-levels.lagda.md +++ b/src/foundation/truncation-levels.lagda.md @@ -12,8 +12,8 @@ open import foundation-core.truncation-levels public open import elementary-number-theory.natural-numbers open import foundation.action-on-identifications-functions -open import foundation.functions +open import foundation-core.function-types open import foundation-core.identity-types ``` diff --git a/src/foundation/truncation-modalities.lagda.md b/src/foundation/truncation-modalities.lagda.md index 8f1ca352bc..a11c518ea6 100644 --- a/src/foundation/truncation-modalities.lagda.md +++ b/src/foundation/truncation-modalities.lagda.md @@ -7,10 +7,10 @@ module foundation.truncation-modalities where
Imports ```agda -open import foundation.functions open import foundation.truncations open import foundation.universe-levels +open import foundation-core.function-types open import foundation-core.truncation-levels open import orthogonal-factorization-systems.modal-operators diff --git a/src/foundation/truncations.lagda.md b/src/foundation/truncations.lagda.md index 2cc3669a9f..fd1ad457d0 100644 --- a/src/foundation/truncations.lagda.md +++ b/src/foundation/truncations.lagda.md @@ -10,7 +10,6 @@ module foundation.truncations where open import foundation.action-on-identifications-functions 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 @@ -23,6 +22,7 @@ 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.function-types open import foundation-core.functoriality-dependent-pair-types open import foundation-core.homotopies open import foundation-core.propositions diff --git a/src/foundation/type-arithmetic-booleans.lagda.md b/src/foundation/type-arithmetic-booleans.lagda.md index 86598e064b..f6a11c83f2 100644 --- a/src/foundation/type-arithmetic-booleans.lagda.md +++ b/src/foundation/type-arithmetic-booleans.lagda.md @@ -9,11 +9,11 @@ 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.equivalences +open import foundation-core.function-types open import foundation-core.homotopies open import foundation-core.identity-types ``` diff --git a/src/foundation/type-arithmetic-cartesian-product-types.lagda.md b/src/foundation/type-arithmetic-cartesian-product-types.lagda.md index 505154ed7e..e6c48948ff 100644 --- a/src/foundation/type-arithmetic-cartesian-product-types.lagda.md +++ b/src/foundation/type-arithmetic-cartesian-product-types.lagda.md @@ -9,13 +9,13 @@ module foundation.type-arithmetic-cartesian-product-types where ```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.function-types open import foundation-core.homotopies open import foundation-core.identity-types open import foundation-core.propositions diff --git a/src/foundation/type-arithmetic-coproduct-types.lagda.md b/src/foundation/type-arithmetic-coproduct-types.lagda.md index 45cbf5b8f1..57e6cbb6c3 100644 --- a/src/foundation/type-arithmetic-coproduct-types.lagda.md +++ b/src/foundation/type-arithmetic-coproduct-types.lagda.md @@ -10,13 +10,13 @@ module foundation.type-arithmetic-coproduct-types where 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.empty-types open import foundation-core.equivalences +open import foundation-core.function-types open import foundation-core.homotopies open import foundation-core.identity-types ``` diff --git a/src/foundation/type-arithmetic-dependent-pair-types.lagda.md b/src/foundation/type-arithmetic-dependent-pair-types.lagda.md index 044fc6f365..0985d9b8d1 100644 --- a/src/foundation/type-arithmetic-dependent-pair-types.lagda.md +++ b/src/foundation/type-arithmetic-dependent-pair-types.lagda.md @@ -9,7 +9,6 @@ module foundation.type-arithmetic-dependent-pair-types where ```agda open import foundation.action-on-identifications-functions open import foundation.dependent-pair-types -open import foundation.functions open import foundation.universe-levels open import foundation-core.cartesian-product-types @@ -18,6 +17,7 @@ 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.function-types open import foundation-core.homotopies open import foundation-core.identity-types open import foundation-core.singleton-induction diff --git a/src/foundation/type-arithmetic-empty-type.lagda.md b/src/foundation/type-arithmetic-empty-type.lagda.md index ef42003019..e527b764d5 100644 --- a/src/foundation/type-arithmetic-empty-type.lagda.md +++ b/src/foundation/type-arithmetic-empty-type.lagda.md @@ -9,7 +9,6 @@ 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 @@ -17,6 +16,7 @@ open import foundation-core.contractible-maps open import foundation-core.contractible-types open import foundation-core.empty-types open import foundation-core.equivalences +open import foundation-core.function-types open import foundation-core.homotopies open import foundation-core.identity-types ``` diff --git a/src/foundation/type-arithmetic-unit-type.lagda.md b/src/foundation/type-arithmetic-unit-type.lagda.md index c46342ab4c..cceebb7cda 100644 --- a/src/foundation/type-arithmetic-unit-type.lagda.md +++ b/src/foundation/type-arithmetic-unit-type.lagda.md @@ -9,12 +9,12 @@ module foundation.type-arithmetic-unit-type where ```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.equivalences +open import foundation-core.function-types open import foundation-core.homotopies open import foundation-core.identity-types ``` diff --git a/src/foundation/type-duality.lagda.md b/src/foundation/type-duality.lagda.md index b3bcc41232..20e66ba9e5 100644 --- a/src/foundation/type-duality.lagda.md +++ b/src/foundation/type-duality.lagda.md @@ -11,7 +11,6 @@ open import foundation.action-on-identifications-functions 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 @@ -24,6 +23,7 @@ open import foundation-core.contractible-maps open import foundation-core.contractible-types open import foundation-core.embeddings open import foundation-core.fibers-of-maps +open import foundation-core.function-types open import foundation-core.functoriality-dependent-function-types open import foundation-core.functoriality-dependent-pair-types open import foundation-core.homotopies diff --git a/src/foundation/type-theoretic-principle-of-choice.lagda.md b/src/foundation/type-theoretic-principle-of-choice.lagda.md index 74d0579edb..a7cf748864 100644 --- a/src/foundation/type-theoretic-principle-of-choice.lagda.md +++ b/src/foundation/type-theoretic-principle-of-choice.lagda.md @@ -9,12 +9,12 @@ module foundation.type-theoretic-principle-of-choice where ```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.equality-dependent-pair-types open import foundation-core.equivalences +open import foundation-core.function-types open import foundation-core.homotopies open import foundation-core.identity-types open import foundation-core.transport diff --git a/src/foundation/uniqueness-image.lagda.md b/src/foundation/uniqueness-image.lagda.md index aac1422bbb..b143ced3e1 100644 --- a/src/foundation/uniqueness-image.lagda.md +++ b/src/foundation/uniqueness-image.lagda.md @@ -9,7 +9,6 @@ module foundation.uniqueness-image where ```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 @@ -18,6 +17,7 @@ open import foundation.universe-levels open import foundation-core.contractible-types open import foundation-core.embeddings +open import foundation-core.function-types open import foundation-core.functoriality-dependent-pair-types open import foundation-core.homotopies open import foundation-core.propositions diff --git a/src/foundation/uniqueness-set-quotients.lagda.md b/src/foundation/uniqueness-set-quotients.lagda.md index ba7b14b5df..93c37976a6 100644 --- a/src/foundation/uniqueness-set-quotients.lagda.md +++ b/src/foundation/uniqueness-set-quotients.lagda.md @@ -10,7 +10,6 @@ module foundation.uniqueness-set-quotients where open import foundation.action-on-identifications-functions 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 @@ -20,6 +19,7 @@ open import foundation.universe-levels open import foundation-core.contractible-types open import foundation-core.equivalence-relations open import foundation-core.function-extensionality +open import foundation-core.function-types open import foundation-core.homotopies open import foundation-core.identity-types open import foundation-core.injective-maps diff --git a/src/foundation/uniqueness-set-truncations.lagda.md b/src/foundation/uniqueness-set-truncations.lagda.md index 78b16d8588..8161102eb5 100644 --- a/src/foundation/uniqueness-set-truncations.lagda.md +++ b/src/foundation/uniqueness-set-truncations.lagda.md @@ -8,7 +8,6 @@ module foundation.uniqueness-set-truncations where ```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 @@ -17,6 +16,7 @@ open import foundation.universe-levels open import foundation-core.contractible-types open import foundation-core.equivalences +open import foundation-core.function-types open import foundation-core.homotopies ``` diff --git a/src/foundation/uniqueness-truncation.lagda.md b/src/foundation/uniqueness-truncation.lagda.md index 78d89d7df5..aa8a4313a2 100644 --- a/src/foundation/uniqueness-truncation.lagda.md +++ b/src/foundation/uniqueness-truncation.lagda.md @@ -7,9 +7,9 @@ module foundation.uniqueness-truncation where
Imports ```agda -open import foundation.functions open import foundation.universe-levels +open import foundation-core.function-types open import foundation-core.homotopies open import foundation-core.truncated-types open import foundation-core.truncation-levels diff --git a/src/foundation/univalence-action-on-equivalences.lagda.md b/src/foundation/univalence-action-on-equivalences.lagda.md index e8ef470bc5..cd6fed4427 100644 --- a/src/foundation/univalence-action-on-equivalences.lagda.md +++ b/src/foundation/univalence-action-on-equivalences.lagda.md @@ -13,6 +13,7 @@ open import foundation.function-extensionality open import foundation.identity-types open import foundation.sets open import foundation.subuniverses +open import foundation.transport open import foundation.univalence open import foundation.universe-levels @@ -22,7 +23,6 @@ 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.transport ```
diff --git a/src/foundation/univalence-implies-function-extensionality.lagda.md b/src/foundation/univalence-implies-function-extensionality.lagda.md index 51aac4d836..c9a0bbb6bd 100644 --- a/src/foundation/univalence-implies-function-extensionality.lagda.md +++ b/src/foundation/univalence-implies-function-extensionality.lagda.md @@ -9,7 +9,6 @@ module foundation.univalence-implies-function-extensionality where ```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 @@ -18,6 +17,7 @@ open import foundation-core.contractible-maps open import foundation-core.contractible-types open import foundation-core.fibers-of-maps open import foundation-core.function-extensionality +open import foundation-core.function-types open import foundation-core.homotopies open import foundation-core.identity-types open import foundation-core.transport diff --git a/src/foundation/univalence.lagda.md b/src/foundation/univalence.lagda.md index 85c79d71ca..1a1741facd 100644 --- a/src/foundation/univalence.lagda.md +++ b/src/foundation/univalence.lagda.md @@ -13,11 +13,11 @@ open import foundation.action-on-identifications-functions 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.function-types open import foundation-core.functoriality-dependent-pair-types open import foundation-core.homotopies open import foundation-core.identity-types diff --git a/src/foundation/univalent-type-families.lagda.md b/src/foundation/univalent-type-families.lagda.md index f2da3797cb..eb65000fa4 100644 --- a/src/foundation/univalent-type-families.lagda.md +++ b/src/foundation/univalent-type-families.lagda.md @@ -8,6 +8,7 @@ module foundation.univalent-type-families where ```agda open import foundation.identity-types +open import foundation.transport open import foundation.universe-levels open import foundation-core.equivalences diff --git a/src/foundation/universal-property-booleans.lagda.md b/src/foundation/universal-property-booleans.lagda.md index 267b3dedce..37c901143d 100644 --- a/src/foundation/universal-property-booleans.lagda.md +++ b/src/foundation/universal-property-booleans.lagda.md @@ -11,11 +11,11 @@ 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.equivalences +open import foundation-core.function-types open import foundation-core.homotopies open import foundation-core.identity-types ``` diff --git a/src/foundation/universal-property-cartesian-product-types.lagda.md b/src/foundation/universal-property-cartesian-product-types.lagda.md index d4ffa19392..57c3af9196 100644 --- a/src/foundation/universal-property-cartesian-product-types.lagda.md +++ b/src/foundation/universal-property-cartesian-product-types.lagda.md @@ -9,7 +9,6 @@ module foundation.universal-property-cartesian-product-types where ```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 @@ -18,6 +17,7 @@ open import foundation-core.constant-maps open import foundation-core.contractible-types open import foundation-core.equality-dependent-pair-types open import foundation-core.equivalences +open import foundation-core.function-types open import foundation-core.homotopies open import foundation-core.identity-types open import foundation-core.pullbacks diff --git a/src/foundation/universal-property-coproduct-types.lagda.md b/src/foundation/universal-property-coproduct-types.lagda.md index bf1d98547b..1d1b757f45 100644 --- a/src/foundation/universal-property-coproduct-types.lagda.md +++ b/src/foundation/universal-property-coproduct-types.lagda.md @@ -11,11 +11,11 @@ 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.function-types open import foundation-core.identity-types ``` diff --git a/src/foundation/universal-property-empty-type.lagda.md b/src/foundation/universal-property-empty-type.lagda.md index d178d004f8..a8520ea604 100644 --- a/src/foundation/universal-property-empty-type.lagda.md +++ b/src/foundation/universal-property-empty-type.lagda.md @@ -10,11 +10,11 @@ module foundation.universal-property-empty-type where 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.empty-types +open import foundation-core.function-types ```
diff --git a/src/foundation/universal-property-fiber-products.lagda.md b/src/foundation/universal-property-fiber-products.lagda.md index fff71c9a53..14c82c1e4c 100644 --- a/src/foundation/universal-property-fiber-products.lagda.md +++ b/src/foundation/universal-property-fiber-products.lagda.md @@ -10,13 +10,13 @@ module foundation.universal-property-fiber-products where 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.equality-dependent-pair-types open import foundation-core.equivalences open import foundation-core.fibers-of-maps +open import foundation-core.function-types open import foundation-core.functoriality-dependent-pair-types open import foundation-core.homotopies open import foundation-core.identity-types diff --git a/src/foundation/universal-property-image.lagda.md b/src/foundation/universal-property-image.lagda.md index 99f2c23a36..a560cb5a38 100644 --- a/src/foundation/universal-property-image.lagda.md +++ b/src/foundation/universal-property-image.lagda.md @@ -9,18 +9,19 @@ module foundation.universal-property-image where ```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.transport open import foundation.universe-levels open import foundation-core.contractible-maps open import foundation-core.contractible-types open import foundation-core.equivalences open import foundation-core.fibers-of-maps +open import foundation-core.function-types open import foundation-core.functoriality-dependent-function-types open import foundation-core.functoriality-dependent-pair-types open import foundation-core.homotopies diff --git a/src/foundation/universal-property-maybe.lagda.md b/src/foundation/universal-property-maybe.lagda.md index 6644e2b569..dc31007dfa 100644 --- a/src/foundation/universal-property-maybe.lagda.md +++ b/src/foundation/universal-property-maybe.lagda.md @@ -9,13 +9,13 @@ module foundation.universal-property-maybe where ```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.equivalences +open import foundation-core.function-types open import foundation-core.homotopies open import foundation-core.identity-types ``` 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 da62b81e96..98cda762c6 100644 --- a/src/foundation/universal-property-propositional-truncation-into-sets.lagda.md +++ b/src/foundation/universal-property-propositional-truncation-into-sets.lagda.md @@ -10,13 +10,13 @@ module foundation.universal-property-propositional-truncation-into-sets where open import foundation.action-on-identifications-functions 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.equivalences open import foundation-core.fibers-of-maps +open import foundation-core.function-types open import foundation-core.homotopies open import foundation-core.identity-types open import foundation-core.propositions diff --git a/src/foundation/universal-property-propositional-truncation.lagda.md b/src/foundation/universal-property-propositional-truncation.lagda.md index 209b1c3cd2..31c5843201 100644 --- a/src/foundation/universal-property-propositional-truncation.lagda.md +++ b/src/foundation/universal-property-propositional-truncation.lagda.md @@ -10,7 +10,6 @@ module foundation.universal-property-propositional-truncation where 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 @@ -20,6 +19,7 @@ open import foundation.universe-levels open import foundation-core.contractible-maps open import foundation-core.contractible-types +open import foundation-core.function-types open import foundation-core.functoriality-dependent-function-types open import foundation-core.functoriality-dependent-pair-types open import foundation-core.homotopies diff --git a/src/foundation/universal-property-set-quotients.lagda.md b/src/foundation/universal-property-set-quotients.lagda.md index d5c19a0e69..dbea2f13c5 100644 --- a/src/foundation/universal-property-set-quotients.lagda.md +++ b/src/foundation/universal-property-set-quotients.lagda.md @@ -16,7 +16,6 @@ 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 @@ -26,6 +25,7 @@ open import foundation.propositional-truncations open import foundation.reflecting-maps-equivalence-relations open import foundation.sets open import foundation.surjective-maps +open import foundation.transport open import foundation.universal-property-image open import foundation.universe-levels @@ -36,6 +36,7 @@ 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.function-types open import foundation-core.functoriality-dependent-pair-types open import foundation-core.homotopies open import foundation-core.injective-maps @@ -43,7 +44,6 @@ open import foundation-core.propositional-maps open import foundation-core.propositions open import foundation-core.small-types open import foundation-core.subtypes -open import foundation-core.transport open import foundation-core.univalence ``` diff --git a/src/foundation/universal-property-set-truncation.lagda.md b/src/foundation/universal-property-set-truncation.lagda.md index d76ce9015f..c79125f305 100644 --- a/src/foundation/universal-property-set-truncation.lagda.md +++ b/src/foundation/universal-property-set-truncation.lagda.md @@ -10,7 +10,6 @@ module foundation.universal-property-set-truncation where 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 @@ -21,6 +20,7 @@ open import foundation.universe-levels open import foundation-core.contractible-maps open import foundation-core.contractible-types +open import foundation-core.function-types open import foundation-core.functoriality-dependent-pair-types open import foundation-core.homotopies open import foundation-core.identity-types diff --git a/src/foundation/universal-property-unit-type.lagda.md b/src/foundation/universal-property-unit-type.lagda.md index f736576930..d351b55dff 100644 --- a/src/foundation/universal-property-unit-type.lagda.md +++ b/src/foundation/universal-property-unit-type.lagda.md @@ -10,11 +10,11 @@ module foundation.universal-property-unit-type where 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.function-types open import foundation-core.homotopies ``` diff --git a/src/foundation/unordered-pairs.lagda.md b/src/foundation/unordered-pairs.lagda.md index 2dc3cf1764..0300d10cab 100644 --- a/src/foundation/unordered-pairs.lagda.md +++ b/src/foundation/unordered-pairs.lagda.md @@ -12,7 +12,6 @@ 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 @@ -24,6 +23,7 @@ open import foundation-core.contractible-types open import foundation-core.coproduct-types open import foundation-core.embeddings open import foundation-core.equivalences +open import foundation-core.function-types open import foundation-core.functoriality-dependent-pair-types open import foundation-core.functoriality-function-types open import foundation-core.identity-types diff --git a/src/foundation/unordered-tuples.lagda.md b/src/foundation/unordered-tuples.lagda.md index e2a25a4d49..d91eb8693e 100644 --- a/src/foundation/unordered-tuples.lagda.md +++ b/src/foundation/unordered-tuples.lagda.md @@ -11,7 +11,6 @@ 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 @@ -19,6 +18,7 @@ open import foundation.universe-levels open import foundation-core.contractible-types open import foundation-core.equivalences +open import foundation-core.function-types open import foundation-core.functoriality-dependent-pair-types open import foundation-core.functoriality-function-types open import foundation-core.identity-types diff --git a/src/foundation/vectors-set-quotients.lagda.md b/src/foundation/vectors-set-quotients.lagda.md index 6150d9cd45..3fca944e95 100644 --- a/src/foundation/vectors-set-quotients.lagda.md +++ b/src/foundation/vectors-set-quotients.lagda.md @@ -16,7 +16,6 @@ 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 @@ -32,6 +31,7 @@ open import foundation-core.coproduct-types open import foundation-core.equality-dependent-pair-types open import foundation-core.equivalence-relations open import foundation-core.equivalences +open import foundation-core.function-types open import foundation-core.homotopies open import foundation-core.identity-types open import foundation-core.propositions diff --git a/src/graph-theory/directed-graphs.lagda.md b/src/graph-theory/directed-graphs.lagda.md index 76bf02f584..e75e6571d2 100644 --- a/src/graph-theory/directed-graphs.lagda.md +++ b/src/graph-theory/directed-graphs.lagda.md @@ -9,7 +9,7 @@ module graph-theory.directed-graphs where ```agda open import foundation.cartesian-product-types open import foundation.dependent-pair-types -open import foundation.functions +open import foundation.function-types open import foundation.identity-types open import foundation.universe-levels ``` diff --git a/src/graph-theory/enriched-undirected-graphs.lagda.md b/src/graph-theory/enriched-undirected-graphs.lagda.md index e3c663db83..85ede25fd6 100644 --- a/src/graph-theory/enriched-undirected-graphs.lagda.md +++ b/src/graph-theory/enriched-undirected-graphs.lagda.md @@ -11,7 +11,7 @@ open import foundation.action-on-identifications-functions open import foundation.connected-components open import foundation.dependent-pair-types open import foundation.equivalences -open import foundation.functions +open import foundation.function-types open import foundation.homotopies open import foundation.identity-types open import foundation.universe-levels diff --git a/src/graph-theory/equivalences-directed-graphs.lagda.md b/src/graph-theory/equivalences-directed-graphs.lagda.md index e7adf75d01..8b1ae9e66f 100644 --- a/src/graph-theory/equivalences-directed-graphs.lagda.md +++ b/src/graph-theory/equivalences-directed-graphs.lagda.md @@ -16,7 +16,7 @@ open import foundation.dependent-pair-types open import foundation.equality-dependent-function-types open import foundation.equivalence-extensionality open import foundation.equivalences -open import foundation.functions +open import foundation.function-types open import foundation.functoriality-dependent-function-types open import foundation.functoriality-dependent-pair-types open import foundation.fundamental-theorem-of-identity-types diff --git a/src/graph-theory/equivalences-enriched-undirected-graphs.lagda.md b/src/graph-theory/equivalences-enriched-undirected-graphs.lagda.md index 09613aea7f..309e845787 100644 --- a/src/graph-theory/equivalences-enriched-undirected-graphs.lagda.md +++ b/src/graph-theory/equivalences-enriched-undirected-graphs.lagda.md @@ -12,11 +12,12 @@ open import foundation.dependent-pair-types open import foundation.equality-dependent-function-types open import foundation.equivalence-extensionality open import foundation.equivalences -open import foundation.functions +open import foundation.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.transport open import foundation.universe-levels open import graph-theory.enriched-undirected-graphs diff --git a/src/graph-theory/equivalences-undirected-graphs.lagda.md b/src/graph-theory/equivalences-undirected-graphs.lagda.md index e037d97d1e..050660708e 100644 --- a/src/graph-theory/equivalences-undirected-graphs.lagda.md +++ b/src/graph-theory/equivalences-undirected-graphs.lagda.md @@ -13,7 +13,7 @@ open import foundation.dependent-pair-types open import foundation.equality-dependent-function-types open import foundation.equivalence-extensionality open import foundation.equivalences -open import foundation.functions +open import foundation.function-types open import foundation.functoriality-dependent-function-types open import foundation.functoriality-dependent-pair-types open import foundation.fundamental-theorem-of-identity-types diff --git a/src/graph-theory/finite-graphs.lagda.md b/src/graph-theory/finite-graphs.lagda.md index dad45d2acb..d870b440f2 100644 --- a/src/graph-theory/finite-graphs.lagda.md +++ b/src/graph-theory/finite-graphs.lagda.md @@ -10,7 +10,7 @@ module graph-theory.finite-graphs where open import foundation.dependent-pair-types open import foundation.equivalences open import foundation.fibers-of-maps -open import foundation.functions +open import foundation.function-types open import foundation.homotopies open import foundation.universe-levels open import foundation.unordered-pairs diff --git a/src/graph-theory/geometric-realizations-undirected-graphs.lagda.md b/src/graph-theory/geometric-realizations-undirected-graphs.lagda.md index 1154143846..220556ead9 100644 --- a/src/graph-theory/geometric-realizations-undirected-graphs.lagda.md +++ b/src/graph-theory/geometric-realizations-undirected-graphs.lagda.md @@ -9,7 +9,7 @@ module graph-theory.geometric-realizations-undirected-graphs where ```agda open import foundation.dependent-pair-types open import foundation.equivalences -open import foundation.functions +open import foundation.function-types open import foundation.symmetric-identity-types open import foundation.universe-levels diff --git a/src/graph-theory/morphisms-directed-graphs.lagda.md b/src/graph-theory/morphisms-directed-graphs.lagda.md index 2671c7e245..83db7e2a3d 100644 --- a/src/graph-theory/morphisms-directed-graphs.lagda.md +++ b/src/graph-theory/morphisms-directed-graphs.lagda.md @@ -12,7 +12,7 @@ open import foundation.contractible-types open import foundation.dependent-pair-types open import foundation.equality-dependent-function-types open import foundation.equivalences -open import foundation.functions +open import foundation.function-types open import foundation.functoriality-dependent-pair-types open import foundation.fundamental-theorem-of-identity-types open import foundation.homotopies diff --git a/src/graph-theory/morphisms-undirected-graphs.lagda.md b/src/graph-theory/morphisms-undirected-graphs.lagda.md index 085e324388..de1a35bfe2 100644 --- a/src/graph-theory/morphisms-undirected-graphs.lagda.md +++ b/src/graph-theory/morphisms-undirected-graphs.lagda.md @@ -12,7 +12,7 @@ open import foundation.contractible-types open import foundation.dependent-pair-types open import foundation.equality-dependent-function-types open import foundation.equivalences -open import foundation.functions +open import foundation.function-types open import foundation.functoriality-dependent-function-types open import foundation.functoriality-dependent-pair-types open import foundation.fundamental-theorem-of-identity-types diff --git a/src/graph-theory/neighbors-undirected-graphs.lagda.md b/src/graph-theory/neighbors-undirected-graphs.lagda.md index ab665b0df2..1131fd1d64 100644 --- a/src/graph-theory/neighbors-undirected-graphs.lagda.md +++ b/src/graph-theory/neighbors-undirected-graphs.lagda.md @@ -10,7 +10,7 @@ module graph-theory.neighbors-undirected-graphs where open import foundation.dependent-pair-types open import foundation.equality-dependent-pair-types open import foundation.equivalences -open import foundation.functions +open import foundation.function-types open import foundation.functoriality-dependent-pair-types open import foundation.homotopies open import foundation.identity-types diff --git a/src/graph-theory/stereoisomerism-enriched-undirected-graphs.lagda.md b/src/graph-theory/stereoisomerism-enriched-undirected-graphs.lagda.md index 08cba53352..629fcaf718 100644 --- a/src/graph-theory/stereoisomerism-enriched-undirected-graphs.lagda.md +++ b/src/graph-theory/stereoisomerism-enriched-undirected-graphs.lagda.md @@ -8,7 +8,7 @@ module graph-theory.stereoisomerism-enriched-undirected-graphs where ```agda open import foundation.dependent-pair-types -open import foundation.functions +open import foundation.function-types open import foundation.homotopies open import foundation.universe-levels diff --git a/src/graph-theory/walks-directed-graphs.lagda.md b/src/graph-theory/walks-directed-graphs.lagda.md index 945ea76c76..8dc556f777 100644 --- a/src/graph-theory/walks-directed-graphs.lagda.md +++ b/src/graph-theory/walks-directed-graphs.lagda.md @@ -16,7 +16,7 @@ open import foundation.contractible-types open import foundation.coproduct-types open import foundation.dependent-pair-types open import foundation.equivalences -open import foundation.functions +open import foundation.function-types open import foundation.functoriality-cartesian-product-types open import foundation.functoriality-dependent-pair-types open import foundation.homotopies diff --git a/src/graph-theory/walks-undirected-graphs.lagda.md b/src/graph-theory/walks-undirected-graphs.lagda.md index 9e852bd693..73fdd1a21c 100644 --- a/src/graph-theory/walks-undirected-graphs.lagda.md +++ b/src/graph-theory/walks-undirected-graphs.lagda.md @@ -17,7 +17,7 @@ open import foundation.decidable-types open import foundation.dependent-pair-types open import foundation.empty-types open import foundation.equivalences -open import foundation.functions +open import foundation.function-types open import foundation.functoriality-coproduct-types open import foundation.fundamental-theorem-of-identity-types open import foundation.identity-types diff --git a/src/group-theory/abelian-groups.lagda.md b/src/group-theory/abelian-groups.lagda.md index 0edc7ff07a..1bfef88b84 100644 --- a/src/group-theory/abelian-groups.lagda.md +++ b/src/group-theory/abelian-groups.lagda.md @@ -14,7 +14,7 @@ open import foundation.binary-equivalences open import foundation.dependent-pair-types open import foundation.embeddings open import foundation.equivalences -open import foundation.functions +open import foundation.function-types open import foundation.homotopies open import foundation.identity-types open import foundation.injective-maps diff --git a/src/group-theory/concrete-group-actions.lagda.md b/src/group-theory/concrete-group-actions.lagda.md index 3bc36916f1..bc5108da26 100644 --- a/src/group-theory/concrete-group-actions.lagda.md +++ b/src/group-theory/concrete-group-actions.lagda.md @@ -7,7 +7,7 @@ module group-theory.concrete-group-actions where
Imports ```agda -open import foundation.functions +open import foundation.function-types open import foundation.sets open import foundation.transport open import foundation.universe-levels diff --git a/src/group-theory/decidable-subgroups.lagda.md b/src/group-theory/decidable-subgroups.lagda.md index 3e2560d939..a3ac6cf692 100644 --- a/src/group-theory/decidable-subgroups.lagda.md +++ b/src/group-theory/decidable-subgroups.lagda.md @@ -13,7 +13,7 @@ open import foundation.dependent-pair-types open import foundation.embeddings open import foundation.equivalence-relations open import foundation.equivalences -open import foundation.functions +open import foundation.function-types open import foundation.identity-types open import foundation.propositions open import foundation.sets diff --git a/src/group-theory/equivalences-semigroups.lagda.md b/src/group-theory/equivalences-semigroups.lagda.md index 1f1063e00f..931c471cbd 100644 --- a/src/group-theory/equivalences-semigroups.lagda.md +++ b/src/group-theory/equivalences-semigroups.lagda.md @@ -11,7 +11,7 @@ open import foundation.contractible-types open import foundation.dependent-pair-types open import foundation.equivalences open import foundation.function-extensionality -open import foundation.functions +open import foundation.function-types open import foundation.identity-types open import foundation.propositions open import foundation.sets diff --git a/src/group-theory/group-actions.lagda.md b/src/group-theory/group-actions.lagda.md index c19ee78481..af5965d7b5 100644 --- a/src/group-theory/group-actions.lagda.md +++ b/src/group-theory/group-actions.lagda.md @@ -11,7 +11,7 @@ open import foundation.action-on-identifications-functions open import foundation.dependent-pair-types open import foundation.equivalences open import foundation.function-extensionality -open import foundation.functions +open import foundation.function-types open import foundation.homotopies open import foundation.identity-types open import foundation.sets diff --git a/src/group-theory/groups.lagda.md b/src/group-theory/groups.lagda.md index fa16f6f4a8..092e6396a1 100644 --- a/src/group-theory/groups.lagda.md +++ b/src/group-theory/groups.lagda.md @@ -16,7 +16,7 @@ open import foundation.dependent-pair-types open import foundation.embeddings open import foundation.equivalences open import foundation.function-extensionality -open import foundation.functions +open import foundation.function-types open import foundation.homotopies open import foundation.identity-types open import foundation.injective-maps diff --git a/src/group-theory/homomorphisms-abelian-groups.lagda.md b/src/group-theory/homomorphisms-abelian-groups.lagda.md index 0f951fa0f2..83d772abab 100644 --- a/src/group-theory/homomorphisms-abelian-groups.lagda.md +++ b/src/group-theory/homomorphisms-abelian-groups.lagda.md @@ -10,7 +10,7 @@ module group-theory.homomorphisms-abelian-groups where open import foundation.contractible-types open import foundation.dependent-pair-types open import foundation.equivalences -open import foundation.functions +open import foundation.function-types open import foundation.identity-types open import foundation.sets open import foundation.universe-levels diff --git a/src/group-theory/homomorphisms-concrete-group-actions.lagda.md b/src/group-theory/homomorphisms-concrete-group-actions.lagda.md index 9ae00a9864..b86eae7b09 100644 --- a/src/group-theory/homomorphisms-concrete-group-actions.lagda.md +++ b/src/group-theory/homomorphisms-concrete-group-actions.lagda.md @@ -11,7 +11,7 @@ open import foundation.0-connected-types open import foundation.equality-dependent-function-types open import foundation.equivalences open import foundation.function-extensionality -open import foundation.functions +open import foundation.function-types open import foundation.homotopies open import foundation.identity-types open import foundation.propositions diff --git a/src/group-theory/homomorphisms-group-actions.lagda.md b/src/group-theory/homomorphisms-group-actions.lagda.md index 4caa7c971c..af9f5ac4dd 100644 --- a/src/group-theory/homomorphisms-group-actions.lagda.md +++ b/src/group-theory/homomorphisms-group-actions.lagda.md @@ -11,7 +11,7 @@ open import foundation.commuting-squares-of-maps open import foundation.contractible-types open import foundation.dependent-pair-types open import foundation.equivalences -open import foundation.functions +open import foundation.function-types open import foundation.fundamental-theorem-of-identity-types open import foundation.homotopies open import foundation.identity-types diff --git a/src/group-theory/homomorphisms-semigroups.lagda.md b/src/group-theory/homomorphisms-semigroups.lagda.md index 8ef57b6ef5..364a109ea6 100644 --- a/src/group-theory/homomorphisms-semigroups.lagda.md +++ b/src/group-theory/homomorphisms-semigroups.lagda.md @@ -11,7 +11,7 @@ open import foundation.action-on-identifications-functions open import foundation.contractible-types open import foundation.dependent-pair-types open import foundation.equivalences -open import foundation.functions +open import foundation.function-types open import foundation.fundamental-theorem-of-identity-types open import foundation.homotopies open import foundation.identity-types diff --git a/src/group-theory/isomorphisms-abelian-groups.lagda.md b/src/group-theory/isomorphisms-abelian-groups.lagda.md index b457bad296..f5704bb05b 100644 --- a/src/group-theory/isomorphisms-abelian-groups.lagda.md +++ b/src/group-theory/isomorphisms-abelian-groups.lagda.md @@ -10,7 +10,7 @@ module group-theory.isomorphisms-abelian-groups where open import foundation.contractible-types open import foundation.dependent-pair-types open import foundation.equivalences -open import foundation.functions +open import foundation.function-types open import foundation.functoriality-dependent-pair-types open import foundation.fundamental-theorem-of-identity-types open import foundation.homotopies diff --git a/src/group-theory/iterated-cartesian-products-concrete-groups.lagda.md b/src/group-theory/iterated-cartesian-products-concrete-groups.lagda.md index a04a94966f..5e88782537 100644 --- a/src/group-theory/iterated-cartesian-products-concrete-groups.lagda.md +++ b/src/group-theory/iterated-cartesian-products-concrete-groups.lagda.md @@ -15,7 +15,7 @@ open import foundation.contractible-types open import foundation.coproduct-types open import foundation.dependent-pair-types open import foundation.equivalences -open import foundation.functions +open import foundation.function-types open import foundation.functoriality-cartesian-product-types open import foundation.identity-types open import foundation.iterated-cartesian-product-types diff --git a/src/group-theory/loop-groups-sets.lagda.md b/src/group-theory/loop-groups-sets.lagda.md index f28c3da3e0..370401f4fd 100644 --- a/src/group-theory/loop-groups-sets.lagda.md +++ b/src/group-theory/loop-groups-sets.lagda.md @@ -12,7 +12,7 @@ open import foundation.dependent-pair-types open import foundation.equality-dependent-pair-types open import foundation.equivalences open import foundation.function-extensionality -open import foundation.functions +open import foundation.function-types open import foundation.identity-truncated-types open import foundation.identity-types open import foundation.propositional-truncations diff --git a/src/group-theory/normal-subgroups.lagda.md b/src/group-theory/normal-subgroups.lagda.md index a5a164e7a0..97f5000ac1 100644 --- a/src/group-theory/normal-subgroups.lagda.md +++ b/src/group-theory/normal-subgroups.lagda.md @@ -14,7 +14,7 @@ open import foundation.dependent-pair-types open import foundation.embeddings open import foundation.equivalence-relations open import foundation.equivalences -open import foundation.functions +open import foundation.function-types open import foundation.identity-types open import foundation.propositions open import foundation.subtype-identity-principle diff --git a/src/group-theory/normal-submonoids-commutative-monoids.lagda.md b/src/group-theory/normal-submonoids-commutative-monoids.lagda.md index 4780000efb..a9049cfff7 100644 --- a/src/group-theory/normal-submonoids-commutative-monoids.lagda.md +++ b/src/group-theory/normal-submonoids-commutative-monoids.lagda.md @@ -11,7 +11,7 @@ open import foundation.binary-relations open import foundation.dependent-pair-types open import foundation.equivalence-relations open import foundation.equivalences -open import foundation.functions +open import foundation.function-types open import foundation.identity-types open import foundation.logical-equivalences open import foundation.propositions diff --git a/src/group-theory/normal-submonoids.lagda.md b/src/group-theory/normal-submonoids.lagda.md index f9c3414203..66a6ede1dc 100644 --- a/src/group-theory/normal-submonoids.lagda.md +++ b/src/group-theory/normal-submonoids.lagda.md @@ -12,7 +12,7 @@ open import foundation.binary-relations open import foundation.dependent-pair-types open import foundation.equivalence-relations open import foundation.equivalences -open import foundation.functions +open import foundation.function-types open import foundation.identity-types open import foundation.logical-equivalences open import foundation.propositions diff --git a/src/group-theory/orbits-concrete-group-actions.lagda.md b/src/group-theory/orbits-concrete-group-actions.lagda.md index c28d37950d..6935dd1b90 100644 --- a/src/group-theory/orbits-concrete-group-actions.lagda.md +++ b/src/group-theory/orbits-concrete-group-actions.lagda.md @@ -8,7 +8,7 @@ module group-theory.orbits-concrete-group-actions where ```agda open import foundation.dependent-pair-types -open import foundation.functions +open import foundation.function-types open import foundation.sets open import foundation.universe-levels diff --git a/src/group-theory/products-of-tuples-of-elements-commutative-monoids.lagda.md b/src/group-theory/products-of-tuples-of-elements-commutative-monoids.lagda.md index 732bb101ef..120ca6b54c 100644 --- a/src/group-theory/products-of-tuples-of-elements-commutative-monoids.lagda.md +++ b/src/group-theory/products-of-tuples-of-elements-commutative-monoids.lagda.md @@ -10,7 +10,7 @@ module group-theory.products-of-tuples-of-elements-commutative-monoids where open import elementary-number-theory.natural-numbers open import foundation.coproduct-types -open import foundation.functions +open import foundation.function-types open import foundation.unit-type open import foundation.universe-levels diff --git a/src/group-theory/stabilizer-groups-concrete-group-actions.lagda.md b/src/group-theory/stabilizer-groups-concrete-group-actions.lagda.md index 3b7600c42d..71ea73ff4f 100644 --- a/src/group-theory/stabilizer-groups-concrete-group-actions.lagda.md +++ b/src/group-theory/stabilizer-groups-concrete-group-actions.lagda.md @@ -9,7 +9,7 @@ module group-theory.stabilizer-groups-concrete-group-actions where ```agda open import foundation.0-connected-types open import foundation.dependent-pair-types -open import foundation.functions +open import foundation.function-types open import foundation.mere-equality open import foundation.propositional-truncations open import foundation.sets diff --git a/src/group-theory/subgroups-generated-by-subsets-groups.lagda.md b/src/group-theory/subgroups-generated-by-subsets-groups.lagda.md index 54a2eacf08..4b1d251b12 100644 --- a/src/group-theory/subgroups-generated-by-subsets-groups.lagda.md +++ b/src/group-theory/subgroups-generated-by-subsets-groups.lagda.md @@ -12,7 +12,7 @@ open import foundation.cartesian-product-types open import foundation.coproduct-types open import foundation.dependent-pair-types open import foundation.fibers-of-maps -open import foundation.functions +open import foundation.function-types open import foundation.identity-types open import foundation.propositional-truncations open import foundation.propositions diff --git a/src/group-theory/subgroups.lagda.md b/src/group-theory/subgroups.lagda.md index 5baa031c40..0d830f7d4f 100644 --- a/src/group-theory/subgroups.lagda.md +++ b/src/group-theory/subgroups.lagda.md @@ -11,7 +11,7 @@ open import foundation.dependent-pair-types open import foundation.embeddings open import foundation.equivalence-relations open import foundation.equivalences -open import foundation.functions +open import foundation.function-types open import foundation.identity-types open import foundation.injective-maps open import foundation.propositions diff --git a/src/group-theory/submonoids.lagda.md b/src/group-theory/submonoids.lagda.md index 35802dbd4a..5fc06a9077 100644 --- a/src/group-theory/submonoids.lagda.md +++ b/src/group-theory/submonoids.lagda.md @@ -9,7 +9,7 @@ module group-theory.submonoids where ```agda open import foundation.dependent-pair-types open import foundation.equivalences -open import foundation.functions +open import foundation.function-types open import foundation.identity-types open import foundation.propositions open import foundation.sets diff --git a/src/group-theory/subsemigroups.lagda.md b/src/group-theory/subsemigroups.lagda.md index 93ecd65a35..0ae47dce4e 100644 --- a/src/group-theory/subsemigroups.lagda.md +++ b/src/group-theory/subsemigroups.lagda.md @@ -9,7 +9,7 @@ module group-theory.subsemigroups where ```agda open import foundation.dependent-pair-types open import foundation.equivalences -open import foundation.functions +open import foundation.function-types open import foundation.identity-types open import foundation.propositions open import foundation.sets diff --git a/src/group-theory/symmetric-groups.lagda.md b/src/group-theory/symmetric-groups.lagda.md index aaba122b1a..b9cf7294ae 100644 --- a/src/group-theory/symmetric-groups.lagda.md +++ b/src/group-theory/symmetric-groups.lagda.md @@ -13,7 +13,7 @@ open import foundation.dependent-pair-types open import foundation.equality-dependent-pair-types open import foundation.equivalences open import foundation.function-extensionality -open import foundation.functions +open import foundation.function-types open import foundation.identity-types open import foundation.propositions open import foundation.sets diff --git a/src/group-theory/torsors.lagda.md b/src/group-theory/torsors.lagda.md index 2157720589..79bdaef78b 100644 --- a/src/group-theory/torsors.lagda.md +++ b/src/group-theory/torsors.lagda.md @@ -12,7 +12,7 @@ open import foundation.action-on-identifications-functions open import foundation.contractible-types open import foundation.dependent-pair-types open import foundation.equivalences -open import foundation.functions +open import foundation.function-types open import foundation.fundamental-theorem-of-identity-types open import foundation.homotopies open import foundation.identity-types diff --git a/src/group-theory/transitive-concrete-group-actions.lagda.md b/src/group-theory/transitive-concrete-group-actions.lagda.md index bd831e4fcc..dd3eac590f 100644 --- a/src/group-theory/transitive-concrete-group-actions.lagda.md +++ b/src/group-theory/transitive-concrete-group-actions.lagda.md @@ -14,7 +14,7 @@ open import foundation.dependent-pair-types open import foundation.equality-dependent-pair-types open import foundation.equivalences open import foundation.existential-quantification -open import foundation.functions +open import foundation.function-types open import foundation.identity-types open import foundation.inhabited-types open import foundation.propositional-truncations diff --git a/src/higher-group-theory/equivalences-higher-groups.lagda.md b/src/higher-group-theory/equivalences-higher-groups.lagda.md index 8783d5d3f3..fd0dd1c34f 100644 --- a/src/higher-group-theory/equivalences-higher-groups.lagda.md +++ b/src/higher-group-theory/equivalences-higher-groups.lagda.md @@ -11,7 +11,7 @@ open import foundation.0-connected-types open import foundation.contractible-types open import foundation.dependent-pair-types open import foundation.equivalences -open import foundation.functions +open import foundation.function-types open import foundation.fundamental-theorem-of-identity-types open import foundation.identity-types open import foundation.subtype-identity-principle diff --git a/src/higher-group-theory/iterated-cartesian-products-higher-groups.lagda.md b/src/higher-group-theory/iterated-cartesian-products-higher-groups.lagda.md index 6bb71dbc7a..3ce51d4723 100644 --- a/src/higher-group-theory/iterated-cartesian-products-higher-groups.lagda.md +++ b/src/higher-group-theory/iterated-cartesian-products-higher-groups.lagda.md @@ -14,7 +14,7 @@ open import foundation.contractible-types open import foundation.coproduct-types open import foundation.dependent-pair-types open import foundation.equivalences -open import foundation.functions +open import foundation.function-types open import foundation.functoriality-cartesian-product-types open import foundation.identity-types open import foundation.iterated-cartesian-product-types diff --git a/src/higher-group-theory/subgroups-higher-groups.lagda.md b/src/higher-group-theory/subgroups-higher-groups.lagda.md index 78fbf494bd..14e6b1ebce 100644 --- a/src/higher-group-theory/subgroups-higher-groups.lagda.md +++ b/src/higher-group-theory/subgroups-higher-groups.lagda.md @@ -10,7 +10,7 @@ module higher-group-theory.subgroups-higher-groups where open import foundation.0-connected-types open import foundation.cartesian-product-types open import foundation.dependent-pair-types -open import foundation.functions +open import foundation.function-types open import foundation.sets open import foundation.universe-levels diff --git a/src/linear-algebra/functoriality-vectors.lagda.md b/src/linear-algebra/functoriality-vectors.lagda.md index 711e1d91a7..0eed673e8c 100644 --- a/src/linear-algebra/functoriality-vectors.lagda.md +++ b/src/linear-algebra/functoriality-vectors.lagda.md @@ -12,7 +12,7 @@ open import elementary-number-theory.natural-numbers open import foundation.action-on-identifications-binary-functions open import foundation.dependent-pair-types open import foundation.function-extensionality -open import foundation.functions +open import foundation.function-types open import foundation.homotopies open import foundation.identity-types open import foundation.universe-levels diff --git a/src/linear-algebra/vectors.lagda.md b/src/linear-algebra/vectors.lagda.md index 0d950c9d1d..24ae6b5f4a 100644 --- a/src/linear-algebra/vectors.lagda.md +++ b/src/linear-algebra/vectors.lagda.md @@ -16,7 +16,7 @@ open import foundation.coproduct-types open import foundation.dependent-pair-types open import foundation.equivalences open import foundation.function-extensionality -open import foundation.functions +open import foundation.function-types open import foundation.homotopies open import foundation.identity-types open import foundation.raising-universe-levels diff --git a/src/lists/arrays.lagda.md b/src/lists/arrays.lagda.md index a641c5de81..cd06173415 100644 --- a/src/lists/arrays.lagda.md +++ b/src/lists/arrays.lagda.md @@ -15,7 +15,7 @@ open import foundation.dependent-pair-types open import foundation.empty-types open import foundation.equality-dependent-pair-types open import foundation.equivalences -open import foundation.functions +open import foundation.function-types open import foundation.homotopies open import foundation.identity-types open import foundation.propositions diff --git a/src/lists/concatenation-lists.lagda.md b/src/lists/concatenation-lists.lagda.md index 85e805275b..3607cc2fa2 100644 --- a/src/lists/concatenation-lists.lagda.md +++ b/src/lists/concatenation-lists.lagda.md @@ -12,7 +12,7 @@ open import elementary-number-theory.natural-numbers open import foundation.action-on-identifications-functions open import foundation.dependent-pair-types -open import foundation.functions +open import foundation.function-types open import foundation.identity-types open import foundation.sets open import foundation.universe-levels diff --git a/src/lists/functoriality-lists.lagda.md b/src/lists/functoriality-lists.lagda.md index 1425317bfb..62b9b8f9c9 100644 --- a/src/lists/functoriality-lists.lagda.md +++ b/src/lists/functoriality-lists.lagda.md @@ -12,7 +12,7 @@ open import elementary-number-theory.natural-numbers open import foundation.action-on-identifications-functions open import foundation.dependent-pair-types open import foundation.equality-dependent-pair-types -open import foundation.functions +open import foundation.function-types open import foundation.homotopies open import foundation.identity-types open import foundation.propositions diff --git a/src/lists/lists-discrete-types.lagda.md b/src/lists/lists-discrete-types.lagda.md index 80b7e408bb..203d586fe6 100644 --- a/src/lists/lists-discrete-types.lagda.md +++ b/src/lists/lists-discrete-types.lagda.md @@ -14,7 +14,7 @@ open import foundation.decidable-equality open import foundation.decidable-types open import foundation.dependent-pair-types open import foundation.empty-types -open import foundation.functions +open import foundation.function-types open import foundation.identity-types open import foundation.raising-universe-levels open import foundation.unit-type diff --git a/src/lists/lists.lagda.md b/src/lists/lists.lagda.md index b085dfedff..39f9678c60 100644 --- a/src/lists/lists.lagda.md +++ b/src/lists/lists.lagda.md @@ -16,7 +16,7 @@ open import foundation.coproduct-types open import foundation.dependent-pair-types open import foundation.empty-types open import foundation.equivalences -open import foundation.functions +open import foundation.function-types open import foundation.functoriality-dependent-pair-types open import foundation.homotopies open import foundation.identity-types diff --git a/src/lists/permutation-vectors.lagda.md b/src/lists/permutation-vectors.lagda.md index 97fd754b92..dbaa198d2c 100644 --- a/src/lists/permutation-vectors.lagda.md +++ b/src/lists/permutation-vectors.lagda.md @@ -18,7 +18,7 @@ open import foundation.coproduct-types open import foundation.dependent-pair-types open import foundation.equivalence-extensionality open import foundation.equivalences -open import foundation.functions +open import foundation.function-types open import foundation.functoriality-coproduct-types open import foundation.identity-types open import foundation.negation diff --git a/src/lists/reversing-lists.lagda.md b/src/lists/reversing-lists.lagda.md index 57a556f11f..551bed29c5 100644 --- a/src/lists/reversing-lists.lagda.md +++ b/src/lists/reversing-lists.lagda.md @@ -10,7 +10,7 @@ module lists.reversing-lists where open import elementary-number-theory.natural-numbers open import foundation.action-on-identifications-functions -open import foundation.functions +open import foundation.function-types open import foundation.identity-types open import foundation.universe-levels diff --git a/src/lists/sorted-vectors.lagda.md b/src/lists/sorted-vectors.lagda.md index 00b8edbe46..027ae8e900 100644 --- a/src/lists/sorted-vectors.lagda.md +++ b/src/lists/sorted-vectors.lagda.md @@ -14,7 +14,7 @@ open import finite-group-theory.permutations-standard-finite-types open import foundation.coproduct-types open import foundation.dependent-pair-types open import foundation.equivalences -open import foundation.functions +open import foundation.function-types open import foundation.propositions open import foundation.unit-type open import foundation.universe-levels diff --git a/src/online-encyclopedia-of-integer-sequences/oeis.lagda.md b/src/online-encyclopedia-of-integer-sequences/oeis.lagda.md index 661ee14d22..abb14f10d9 100644 --- a/src/online-encyclopedia-of-integer-sequences/oeis.lagda.md +++ b/src/online-encyclopedia-of-integer-sequences/oeis.lagda.md @@ -21,7 +21,7 @@ open import elementary-number-theory.pisano-periods open import finite-group-theory.finite-groups -open import foundation.functions +open import foundation.function-types ```
diff --git a/src/order-theory/closure-operators-large-locales.lagda.md b/src/order-theory/closure-operators-large-locales.lagda.md index 69d07b0411..0de3e9948b 100644 --- a/src/order-theory/closure-operators-large-locales.lagda.md +++ b/src/order-theory/closure-operators-large-locales.lagda.md @@ -9,7 +9,7 @@ module order-theory.closure-operators-large-locales where ```agda open import foundation.action-on-identifications-functions open import foundation.dependent-pair-types -open import foundation.functions +open import foundation.function-types open import foundation.identity-types open import foundation.logical-equivalences open import foundation.propositions diff --git a/src/order-theory/closure-operators-large-posets.lagda.md b/src/order-theory/closure-operators-large-posets.lagda.md index 8fe9de3455..fde3a411e8 100644 --- a/src/order-theory/closure-operators-large-posets.lagda.md +++ b/src/order-theory/closure-operators-large-posets.lagda.md @@ -7,7 +7,7 @@ module order-theory.closure-operators-large-posets where
Imports ```agda -open import foundation.functions +open import foundation.function-types open import foundation.identity-types open import foundation.propositions open import foundation.sets diff --git a/src/order-theory/decidable-total-preorders.lagda.md b/src/order-theory/decidable-total-preorders.lagda.md index b49e65f89b..0ef58b49a2 100644 --- a/src/order-theory/decidable-total-preorders.lagda.md +++ b/src/order-theory/decidable-total-preorders.lagda.md @@ -13,7 +13,7 @@ open import foundation.decidable-propositions open import foundation.decidable-types open import foundation.dependent-pair-types open import foundation.empty-types -open import foundation.functions +open import foundation.function-types open import foundation.identity-types open import foundation.propositional-truncations open import foundation.propositions diff --git a/src/order-theory/finitely-graded-posets.lagda.md b/src/order-theory/finitely-graded-posets.lagda.md index a764744b79..167076d2a6 100644 --- a/src/order-theory/finitely-graded-posets.lagda.md +++ b/src/order-theory/finitely-graded-posets.lagda.md @@ -16,7 +16,7 @@ open import foundation.dependent-pair-types open import foundation.embeddings open import foundation.empty-types open import foundation.equality-dependent-pair-types -open import foundation.functions +open import foundation.function-types open import foundation.identity-types open import foundation.injective-maps open import foundation.propositional-truncations diff --git a/src/order-theory/homomorphisms-frames.lagda.md b/src/order-theory/homomorphisms-frames.lagda.md index 5e988b400e..2695c9e520 100644 --- a/src/order-theory/homomorphisms-frames.lagda.md +++ b/src/order-theory/homomorphisms-frames.lagda.md @@ -9,7 +9,7 @@ module order-theory.homomorphisms-frames where ```agda open import foundation.cartesian-product-types open import foundation.dependent-pair-types -open import foundation.functions +open import foundation.function-types open import foundation.universe-levels open import order-theory.frames diff --git a/src/order-theory/homomorphisms-large-meet-semilattices.lagda.md b/src/order-theory/homomorphisms-large-meet-semilattices.lagda.md index 1bb8fd59cf..7a32fe95be 100644 --- a/src/order-theory/homomorphisms-large-meet-semilattices.lagda.md +++ b/src/order-theory/homomorphisms-large-meet-semilattices.lagda.md @@ -7,7 +7,7 @@ module order-theory.homomorphisms-large-meet-semilattices where
Imports ```agda -open import foundation.functions +open import foundation.function-types open import foundation.identity-types open import foundation.universe-levels diff --git a/src/order-theory/homomorphisms-large-suplattices.lagda.md b/src/order-theory/homomorphisms-large-suplattices.lagda.md index 30129f3e0e..e0bdab126f 100644 --- a/src/order-theory/homomorphisms-large-suplattices.lagda.md +++ b/src/order-theory/homomorphisms-large-suplattices.lagda.md @@ -7,7 +7,7 @@ module order-theory.homomorphisms-large-suplattices where
Imports ```agda -open import foundation.functions +open import foundation.function-types open import foundation.identity-types open import foundation.universe-levels diff --git a/src/order-theory/homomorphisms-meet-sup-lattices.lagda.md b/src/order-theory/homomorphisms-meet-sup-lattices.lagda.md index 23ca9f24ea..2414f0191a 100644 --- a/src/order-theory/homomorphisms-meet-sup-lattices.lagda.md +++ b/src/order-theory/homomorphisms-meet-sup-lattices.lagda.md @@ -9,7 +9,7 @@ module order-theory.homomorphisms-meet-sup-lattices where ```agda open import foundation.cartesian-product-types open import foundation.dependent-pair-types -open import foundation.functions +open import foundation.function-types open import foundation.universe-levels open import order-theory.homomorphisms-meet-semilattices diff --git a/src/order-theory/homomorphisms-sup-lattices.lagda.md b/src/order-theory/homomorphisms-sup-lattices.lagda.md index 0517d3273a..dd8a89ca01 100644 --- a/src/order-theory/homomorphisms-sup-lattices.lagda.md +++ b/src/order-theory/homomorphisms-sup-lattices.lagda.md @@ -9,7 +9,7 @@ module order-theory.homomorphisms-sup-lattices where ```agda open import foundation.cartesian-product-types open import foundation.dependent-pair-types -open import foundation.functions +open import foundation.function-types open import foundation.universe-levels open import order-theory.least-upper-bounds-posets diff --git a/src/order-theory/large-subframes.lagda.md b/src/order-theory/large-subframes.lagda.md index 118ec50f90..ca4d10d2c4 100644 --- a/src/order-theory/large-subframes.lagda.md +++ b/src/order-theory/large-subframes.lagda.md @@ -8,7 +8,7 @@ module order-theory.large-subframes where ```agda open import foundation.dependent-pair-types -open import foundation.functions +open import foundation.function-types open import foundation.identity-types open import foundation.propositions open import foundation.subtypes diff --git a/src/order-theory/nuclei-large-locales.lagda.md b/src/order-theory/nuclei-large-locales.lagda.md index 83f6f7e569..db1eeefc21 100644 --- a/src/order-theory/nuclei-large-locales.lagda.md +++ b/src/order-theory/nuclei-large-locales.lagda.md @@ -9,7 +9,7 @@ module order-theory.nuclei-large-locales where ```agda open import foundation.action-on-identifications-functions open import foundation.dependent-pair-types -open import foundation.functions +open import foundation.function-types open import foundation.identity-types open import foundation.logical-equivalences open import foundation.propositions diff --git a/src/order-theory/order-preserving-maps-posets.lagda.md b/src/order-theory/order-preserving-maps-posets.lagda.md index 08f136e3ae..3f6a07dd3b 100644 --- a/src/order-theory/order-preserving-maps-posets.lagda.md +++ b/src/order-theory/order-preserving-maps-posets.lagda.md @@ -10,7 +10,7 @@ module order-theory.order-preserving-maps-posets where open import foundation.contractible-types open import foundation.dependent-pair-types open import foundation.equivalences -open import foundation.functions +open import foundation.function-types open import foundation.identity-types open import foundation.propositions open import foundation.sets diff --git a/src/order-theory/order-preserving-maps-preorders.lagda.md b/src/order-theory/order-preserving-maps-preorders.lagda.md index d5534c7b4b..0b0b94470a 100644 --- a/src/order-theory/order-preserving-maps-preorders.lagda.md +++ b/src/order-theory/order-preserving-maps-preorders.lagda.md @@ -10,7 +10,7 @@ module order-theory.order-preserving-maps-preorders where open import foundation.contractible-types open import foundation.dependent-pair-types open import foundation.equivalences -open import foundation.functions +open import foundation.function-types open import foundation.fundamental-theorem-of-identity-types open import foundation.homotopies open import foundation.identity-types diff --git a/src/order-theory/preorders.lagda.md b/src/order-theory/preorders.lagda.md index fba1611606..87542eb0e7 100644 --- a/src/order-theory/preorders.lagda.md +++ b/src/order-theory/preorders.lagda.md @@ -9,7 +9,7 @@ module order-theory.preorders where ```agda open import foundation.cartesian-product-types open import foundation.dependent-pair-types -open import foundation.functions +open import foundation.function-types open import foundation.identity-types open import foundation.negation open import foundation.propositions diff --git a/src/order-theory/subpreorders.lagda.md b/src/order-theory/subpreorders.lagda.md index 39b88abf50..83d8a2888e 100644 --- a/src/order-theory/subpreorders.lagda.md +++ b/src/order-theory/subpreorders.lagda.md @@ -8,7 +8,7 @@ module order-theory.subpreorders where ```agda open import foundation.dependent-pair-types -open import foundation.functions +open import foundation.function-types open import foundation.identity-types open import foundation.propositions open import foundation.subtypes diff --git a/src/orthogonal-factorization-systems/closed-modalities.lagda.md b/src/orthogonal-factorization-systems/closed-modalities.lagda.md index cdf017adc4..1005074b0f 100644 --- a/src/orthogonal-factorization-systems/closed-modalities.lagda.md +++ b/src/orthogonal-factorization-systems/closed-modalities.lagda.md @@ -11,7 +11,7 @@ open import foundation.contractible-maps open import foundation.contractible-types open import foundation.dependent-pair-types open import foundation.equivalences -open import foundation.functions +open import foundation.function-types open import foundation.functoriality-dependent-pair-types open import foundation.identity-types open import foundation.propositions diff --git a/src/orthogonal-factorization-systems/extensions-of-maps.lagda.md b/src/orthogonal-factorization-systems/extensions-of-maps.lagda.md index 588503fc23..a72943a8c0 100644 --- a/src/orthogonal-factorization-systems/extensions-of-maps.lagda.md +++ b/src/orthogonal-factorization-systems/extensions-of-maps.lagda.md @@ -16,7 +16,7 @@ open import foundation.embeddings open import foundation.equivalences open import foundation.fibers-of-maps open import foundation.function-extensionality -open import foundation.functions +open import foundation.function-types open import foundation.functoriality-dependent-function-types open import foundation.functoriality-dependent-pair-types open import foundation.fundamental-theorem-of-identity-types diff --git a/src/orthogonal-factorization-systems/factorization-operations-function-classes.lagda.md b/src/orthogonal-factorization-systems/factorization-operations-function-classes.lagda.md index 13cbd34753..afa20e426e 100644 --- a/src/orthogonal-factorization-systems/factorization-operations-function-classes.lagda.md +++ b/src/orthogonal-factorization-systems/factorization-operations-function-classes.lagda.md @@ -8,7 +8,7 @@ module orthogonal-factorization-systems.factorization-operations-function-classe ```agda open import foundation.contractible-types -open import foundation.functions +open import foundation.function-types open import foundation.inhabited-types open import foundation.propositions open import foundation.universe-levels diff --git a/src/orthogonal-factorization-systems/factorizations-of-maps.lagda.md b/src/orthogonal-factorization-systems/factorizations-of-maps.lagda.md index 2412d4e5c4..8c554ff154 100644 --- a/src/orthogonal-factorization-systems/factorizations-of-maps.lagda.md +++ b/src/orthogonal-factorization-systems/factorizations-of-maps.lagda.md @@ -11,7 +11,7 @@ open import foundation.conjunction open import foundation.contractible-types open import foundation.dependent-pair-types open import foundation.equivalences -open import foundation.functions +open import foundation.function-types open import foundation.fundamental-theorem-of-identity-types open import foundation.homotopies open import foundation.identity-types diff --git a/src/orthogonal-factorization-systems/function-classes.lagda.md b/src/orthogonal-factorization-systems/function-classes.lagda.md index dad0e382f2..dde480c486 100644 --- a/src/orthogonal-factorization-systems/function-classes.lagda.md +++ b/src/orthogonal-factorization-systems/function-classes.lagda.md @@ -9,7 +9,7 @@ module orthogonal-factorization-systems.function-classes where ```agda open import foundation.dependent-pair-types open import foundation.equivalences -open import foundation.functions +open import foundation.function-types open import foundation.propositions open import foundation.pullback-squares open import foundation.universe-levels diff --git a/src/orthogonal-factorization-systems/higher-modalities.lagda.md b/src/orthogonal-factorization-systems/higher-modalities.lagda.md index ea97d47ba6..f0151edfb0 100644 --- a/src/orthogonal-factorization-systems/higher-modalities.lagda.md +++ b/src/orthogonal-factorization-systems/higher-modalities.lagda.md @@ -12,7 +12,7 @@ open import foundation.cartesian-product-types open import foundation.dependent-pair-types open import foundation.equivalences open import foundation.function-extensionality -open import foundation.functions +open import foundation.function-types open import foundation.homotopies open import foundation.identity-types open import foundation.small-types diff --git a/src/orthogonal-factorization-systems/identity-modality.lagda.md b/src/orthogonal-factorization-systems/identity-modality.lagda.md index bc8747608d..c72bc79321 100644 --- a/src/orthogonal-factorization-systems/identity-modality.lagda.md +++ b/src/orthogonal-factorization-systems/identity-modality.lagda.md @@ -8,7 +8,7 @@ module orthogonal-factorization-systems.identity-modality where ```agda open import foundation.equivalences -open import foundation.functions +open import foundation.function-types open import foundation.universe-levels open import orthogonal-factorization-systems.local-types diff --git a/src/orthogonal-factorization-systems/lifting-operations.lagda.md b/src/orthogonal-factorization-systems/lifting-operations.lagda.md index 3e2f41c74c..06ef2d040d 100644 --- a/src/orthogonal-factorization-systems/lifting-operations.lagda.md +++ b/src/orthogonal-factorization-systems/lifting-operations.lagda.md @@ -8,7 +8,7 @@ module orthogonal-factorization-systems.lifting-operations where ```agda open import foundation.dependent-pair-types -open import foundation.functions +open import foundation.function-types open import foundation.homotopies open import foundation.sections open import foundation.universe-levels diff --git a/src/orthogonal-factorization-systems/lifting-squares.lagda.md b/src/orthogonal-factorization-systems/lifting-squares.lagda.md index b906d4f801..4e993d6170 100644 --- a/src/orthogonal-factorization-systems/lifting-squares.lagda.md +++ b/src/orthogonal-factorization-systems/lifting-squares.lagda.md @@ -11,7 +11,7 @@ open import foundation.commuting-3-simplices-of-homotopies open import foundation.commuting-squares-of-maps open import foundation.commuting-triangles-of-homotopies open import foundation.dependent-pair-types -open import foundation.functions +open import foundation.function-types open import foundation.homotopies open import foundation.universe-levels diff --git a/src/orthogonal-factorization-systems/lifts-of-maps.lagda.md b/src/orthogonal-factorization-systems/lifts-of-maps.lagda.md index 0bab9bbfcd..4f74ef23f9 100644 --- a/src/orthogonal-factorization-systems/lifts-of-maps.lagda.md +++ b/src/orthogonal-factorization-systems/lifts-of-maps.lagda.md @@ -11,7 +11,7 @@ open import foundation.action-on-identifications-functions open import foundation.contractible-types open import foundation.dependent-pair-types open import foundation.equivalences -open import foundation.functions +open import foundation.function-types open import foundation.fundamental-theorem-of-identity-types open import foundation.homotopies open import foundation.identity-types diff --git a/src/orthogonal-factorization-systems/local-types.lagda.md b/src/orthogonal-factorization-systems/local-types.lagda.md index 64d0a065c9..07cd9d0a4e 100644 --- a/src/orthogonal-factorization-systems/local-types.lagda.md +++ b/src/orthogonal-factorization-systems/local-types.lagda.md @@ -8,24 +8,23 @@ module orthogonal-factorization-systems.local-types where ```agda open import foundation.action-on-identifications-functions +open import foundation.contractible-maps +open import foundation.contractible-types open import foundation.dependent-pair-types +open import foundation.empty-types open import foundation.equivalences -open import foundation.functions +open import foundation.function-extensionality +open import foundation.function-types +open import foundation.functoriality-dependent-function-types +open import foundation.identity-types +open import foundation.propositions +open import foundation.retractions +open import foundation.sections 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.empty-types -open import foundation-core.function-extensionality -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 ```
diff --git a/src/orthogonal-factorization-systems/modal-operators.lagda.md b/src/orthogonal-factorization-systems/modal-operators.lagda.md index 5af7c42d45..e432f239f2 100644 --- a/src/orthogonal-factorization-systems/modal-operators.lagda.md +++ b/src/orthogonal-factorization-systems/modal-operators.lagda.md @@ -9,7 +9,7 @@ module orthogonal-factorization-systems.modal-operators where ```agda open import foundation.dependent-pair-types open import foundation.equivalences -open import foundation.functions +open import foundation.function-types open import foundation.locally-small-types open import foundation.propositions open import foundation.small-types diff --git a/src/orthogonal-factorization-systems/null-types.lagda.md b/src/orthogonal-factorization-systems/null-types.lagda.md index 8c93de6112..175c63075e 100644 --- a/src/orthogonal-factorization-systems/null-types.lagda.md +++ b/src/orthogonal-factorization-systems/null-types.lagda.md @@ -9,7 +9,7 @@ module orthogonal-factorization-systems.null-types where ```agda open import foundation.constant-maps open import foundation.equivalences -open import foundation.functions +open import foundation.function-types open import foundation.propositions open import foundation.type-arithmetic-unit-type open import foundation.unit-type diff --git a/src/orthogonal-factorization-systems/pullback-hom.lagda.md b/src/orthogonal-factorization-systems/pullback-hom.lagda.md index bdaba988d7..195c3c9c35 100644 --- a/src/orthogonal-factorization-systems/pullback-hom.lagda.md +++ b/src/orthogonal-factorization-systems/pullback-hom.lagda.md @@ -12,7 +12,7 @@ open import foundation.dependent-pair-types open import foundation.equivalences open import foundation.fibered-maps open import foundation.function-extensionality -open import foundation.functions +open import foundation.function-types open import foundation.functoriality-dependent-pair-types open import foundation.homotopies open import foundation.morphisms-cospans diff --git a/src/orthogonal-factorization-systems/raise-modalities.lagda.md b/src/orthogonal-factorization-systems/raise-modalities.lagda.md index bd3b959bff..7301a98dde 100644 --- a/src/orthogonal-factorization-systems/raise-modalities.lagda.md +++ b/src/orthogonal-factorization-systems/raise-modalities.lagda.md @@ -7,7 +7,7 @@ module orthogonal-factorization-systems.raise-modalities where
Imports ```agda -open import foundation.functions +open import foundation.function-types open import foundation.raising-universe-levels open import foundation.universe-levels diff --git a/src/orthogonal-factorization-systems/sigma-closed-reflective-subuniverses.lagda.md b/src/orthogonal-factorization-systems/sigma-closed-reflective-subuniverses.lagda.md index a5b77f4c32..37006c6940 100644 --- a/src/orthogonal-factorization-systems/sigma-closed-reflective-subuniverses.lagda.md +++ b/src/orthogonal-factorization-systems/sigma-closed-reflective-subuniverses.lagda.md @@ -8,7 +8,7 @@ module orthogonal-factorization-systems.sigma-closed-reflective-subuniverses whe ```agda open import foundation.dependent-pair-types -open import foundation.functions +open import foundation.function-types open import foundation.propositions open import foundation.universe-levels diff --git a/src/orthogonal-factorization-systems/uniquely-eliminating-modalities.lagda.md b/src/orthogonal-factorization-systems/uniquely-eliminating-modalities.lagda.md index 97525e9b03..e503c2bd99 100644 --- a/src/orthogonal-factorization-systems/uniquely-eliminating-modalities.lagda.md +++ b/src/orthogonal-factorization-systems/uniquely-eliminating-modalities.lagda.md @@ -13,7 +13,7 @@ open import foundation.contractible-types open import foundation.dependent-pair-types open import foundation.equivalences open import foundation.function-extensionality -open import foundation.functions +open import foundation.function-types open import foundation.homotopies open import foundation.identity-types open import foundation.propositions diff --git a/src/orthogonal-factorization-systems/wide-function-classes.lagda.md b/src/orthogonal-factorization-systems/wide-function-classes.lagda.md index cfc1246cbb..f51bdd8fe1 100644 --- a/src/orthogonal-factorization-systems/wide-function-classes.lagda.md +++ b/src/orthogonal-factorization-systems/wide-function-classes.lagda.md @@ -9,7 +9,7 @@ module orthogonal-factorization-systems.wide-function-classes where ```agda open import foundation.cartesian-product-types open import foundation.dependent-pair-types -open import foundation.functions +open import foundation.function-types open import foundation.propositions open import foundation.universe-levels diff --git a/src/reflection/precategory-solver.lagda.md b/src/reflection/precategory-solver.lagda.md index 67a7949466..564fe24917 100644 --- a/src/reflection/precategory-solver.lagda.md +++ b/src/reflection/precategory-solver.lagda.md @@ -13,7 +13,7 @@ open import category-theory.precategories open import foundation.action-on-identifications-functions open import foundation.dependent-pair-types -open import foundation.functions +open import foundation.function-types open import foundation.identity-types open import foundation.unit-type open import foundation.universe-levels diff --git a/src/ring-theory/ideals-generated-by-subsets-rings.lagda.md b/src/ring-theory/ideals-generated-by-subsets-rings.lagda.md index 30a35b6112..e2f8fdf80b 100644 --- a/src/ring-theory/ideals-generated-by-subsets-rings.lagda.md +++ b/src/ring-theory/ideals-generated-by-subsets-rings.lagda.md @@ -11,7 +11,7 @@ open import foundation.action-on-identifications-functions open import foundation.cartesian-product-types open import foundation.dependent-pair-types open import foundation.fibers-of-maps -open import foundation.functions +open import foundation.function-types open import foundation.identity-types open import foundation.logical-equivalences open import foundation.powersets diff --git a/src/ring-theory/isomorphisms-rings.lagda.md b/src/ring-theory/isomorphisms-rings.lagda.md index fd84c61881..d3fff2e4e0 100644 --- a/src/ring-theory/isomorphisms-rings.lagda.md +++ b/src/ring-theory/isomorphisms-rings.lagda.md @@ -13,7 +13,7 @@ open import foundation.contractible-types open import foundation.dependent-pair-types open import foundation.equality-dependent-function-types open import foundation.equivalences -open import foundation.functions +open import foundation.function-types open import foundation.functoriality-dependent-pair-types open import foundation.fundamental-theorem-of-identity-types open import foundation.homotopies diff --git a/src/ring-theory/left-ideals-generated-by-subsets-rings.lagda.md b/src/ring-theory/left-ideals-generated-by-subsets-rings.lagda.md index f35a1d3890..a10777ae7b 100644 --- a/src/ring-theory/left-ideals-generated-by-subsets-rings.lagda.md +++ b/src/ring-theory/left-ideals-generated-by-subsets-rings.lagda.md @@ -11,7 +11,7 @@ open import foundation.action-on-identifications-functions open import foundation.cartesian-product-types open import foundation.dependent-pair-types open import foundation.fibers-of-maps -open import foundation.functions +open import foundation.function-types open import foundation.identity-types open import foundation.logical-equivalences open import foundation.powersets diff --git a/src/ring-theory/poset-of-ideals-rings.lagda.md b/src/ring-theory/poset-of-ideals-rings.lagda.md index c7c99542b2..9a1c673b15 100644 --- a/src/ring-theory/poset-of-ideals-rings.lagda.md +++ b/src/ring-theory/poset-of-ideals-rings.lagda.md @@ -8,7 +8,7 @@ module ring-theory.poset-of-ideals-rings where ```agda open import foundation.dependent-pair-types -open import foundation.functions +open import foundation.function-types open import foundation.identity-types open import foundation.powersets open import foundation.propositions diff --git a/src/ring-theory/poset-of-left-ideals-rings.lagda.md b/src/ring-theory/poset-of-left-ideals-rings.lagda.md index 1c0d94c6ca..265e7f8d91 100644 --- a/src/ring-theory/poset-of-left-ideals-rings.lagda.md +++ b/src/ring-theory/poset-of-left-ideals-rings.lagda.md @@ -8,7 +8,7 @@ module ring-theory.poset-of-left-ideals-rings where ```agda open import foundation.dependent-pair-types -open import foundation.functions +open import foundation.function-types open import foundation.identity-types open import foundation.powersets open import foundation.propositions diff --git a/src/ring-theory/poset-of-right-ideals-rings.lagda.md b/src/ring-theory/poset-of-right-ideals-rings.lagda.md index 0ce7a855ea..44e6e9ef33 100644 --- a/src/ring-theory/poset-of-right-ideals-rings.lagda.md +++ b/src/ring-theory/poset-of-right-ideals-rings.lagda.md @@ -8,7 +8,7 @@ module ring-theory.poset-of-right-ideals-rings where ```agda open import foundation.dependent-pair-types -open import foundation.functions +open import foundation.function-types open import foundation.identity-types open import foundation.powersets open import foundation.propositions diff --git a/src/ring-theory/powers-of-elements-rings.lagda.md b/src/ring-theory/powers-of-elements-rings.lagda.md index a637632921..b9804bc923 100644 --- a/src/ring-theory/powers-of-elements-rings.lagda.md +++ b/src/ring-theory/powers-of-elements-rings.lagda.md @@ -14,7 +14,7 @@ open import elementary-number-theory.parity-natural-numbers open import foundation.action-on-identifications-functions open import foundation.empty-types -open import foundation.functions +open import foundation.function-types open import foundation.identity-types open import foundation.universe-levels diff --git a/src/ring-theory/right-ideals-generated-by-subsets-rings.lagda.md b/src/ring-theory/right-ideals-generated-by-subsets-rings.lagda.md index acfcb530f2..8e268bd112 100644 --- a/src/ring-theory/right-ideals-generated-by-subsets-rings.lagda.md +++ b/src/ring-theory/right-ideals-generated-by-subsets-rings.lagda.md @@ -11,7 +11,7 @@ open import foundation.action-on-identifications-functions open import foundation.cartesian-product-types open import foundation.dependent-pair-types open import foundation.fibers-of-maps -open import foundation.functions +open import foundation.function-types open import foundation.identity-types open import foundation.logical-equivalences open import foundation.powersets diff --git a/src/ring-theory/sums-rings.lagda.md b/src/ring-theory/sums-rings.lagda.md index 41fa35876b..449cc19129 100644 --- a/src/ring-theory/sums-rings.lagda.md +++ b/src/ring-theory/sums-rings.lagda.md @@ -10,7 +10,7 @@ module ring-theory.sums-rings where open import elementary-number-theory.addition-natural-numbers open import elementary-number-theory.natural-numbers -open import foundation.functions +open import foundation.function-types open import foundation.homotopies open import foundation.identity-types open import foundation.universe-levels diff --git a/src/ring-theory/sums-semirings.lagda.md b/src/ring-theory/sums-semirings.lagda.md index 8aca5454f7..da71e01156 100644 --- a/src/ring-theory/sums-semirings.lagda.md +++ b/src/ring-theory/sums-semirings.lagda.md @@ -12,7 +12,7 @@ open import elementary-number-theory.natural-numbers open import foundation.action-on-identifications-functions open import foundation.coproduct-types -open import foundation.functions +open import foundation.function-types open import foundation.homotopies open import foundation.identity-types open import foundation.unit-type diff --git a/src/set-theory/countable-sets.lagda.md b/src/set-theory/countable-sets.lagda.md index abd1880ae6..fbb07cc526 100644 --- a/src/set-theory/countable-sets.lagda.md +++ b/src/set-theory/countable-sets.lagda.md @@ -20,7 +20,7 @@ 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.function-types open import foundation.functoriality-cartesian-product-types open import foundation.functoriality-coproduct-types open import foundation.maybe diff --git a/src/set-theory/cumulative-hierarchy.lagda.md b/src/set-theory/cumulative-hierarchy.lagda.md index f85e301bd0..f376b96293 100644 --- a/src/set-theory/cumulative-hierarchy.lagda.md +++ b/src/set-theory/cumulative-hierarchy.lagda.md @@ -17,7 +17,7 @@ open import foundation.coproduct-types open import foundation.dependent-pair-types open import foundation.empty-types open import foundation.existential-quantification -open import foundation.functions +open import foundation.function-types open import foundation.functoriality-propositional-truncation open import foundation.identity-types open import foundation.logical-equivalences diff --git a/src/species/cauchy-composition-species-of-types-in-subuniverses.lagda.md b/src/species/cauchy-composition-species-of-types-in-subuniverses.lagda.md index b4e6f4cca4..5cf28b5f1e 100644 --- a/src/species/cauchy-composition-species-of-types-in-subuniverses.lagda.md +++ b/src/species/cauchy-composition-species-of-types-in-subuniverses.lagda.md @@ -12,7 +12,7 @@ open import foundation.contractible-types open import foundation.dependent-pair-types open import foundation.equality-cartesian-product-types open import foundation.equivalences -open import foundation.functions +open import foundation.function-types open import foundation.functoriality-cartesian-product-types open import foundation.functoriality-dependent-function-types open import foundation.functoriality-dependent-pair-types diff --git a/src/species/cauchy-exponentials-species-of-types-in-subuniverses.lagda.md b/src/species/cauchy-exponentials-species-of-types-in-subuniverses.lagda.md index b30b59e134..599ba01bfb 100644 --- a/src/species/cauchy-exponentials-species-of-types-in-subuniverses.lagda.md +++ b/src/species/cauchy-exponentials-species-of-types-in-subuniverses.lagda.md @@ -12,7 +12,7 @@ open import foundation.coproduct-decompositions open import foundation.coproduct-types open import foundation.dependent-pair-types open import foundation.equivalences -open import foundation.functions +open import foundation.function-types open import foundation.functoriality-cartesian-product-types open import foundation.functoriality-dependent-function-types open import foundation.functoriality-dependent-pair-types diff --git a/src/species/cauchy-products-species-of-types-in-subuniverses.lagda.md b/src/species/cauchy-products-species-of-types-in-subuniverses.lagda.md index dc141a0271..50ee887c03 100644 --- a/src/species/cauchy-products-species-of-types-in-subuniverses.lagda.md +++ b/src/species/cauchy-products-species-of-types-in-subuniverses.lagda.md @@ -14,7 +14,7 @@ open import foundation.coproduct-types open import foundation.dependent-pair-types open import foundation.empty-types open import foundation.equivalences -open import foundation.functions +open import foundation.function-types open import foundation.functoriality-dependent-pair-types open import foundation.homotopies open import foundation.identity-types diff --git a/src/species/cauchy-series-species-of-types-in-subuniverses.lagda.md b/src/species/cauchy-series-species-of-types-in-subuniverses.lagda.md index 1ebcaedc43..d296ac285f 100644 --- a/src/species/cauchy-series-species-of-types-in-subuniverses.lagda.md +++ b/src/species/cauchy-series-species-of-types-in-subuniverses.lagda.md @@ -10,7 +10,7 @@ module species.cauchy-series-species-of-types-in-subuniverses where open import foundation.cartesian-product-types open import foundation.dependent-pair-types open import foundation.equivalences -open import foundation.functions +open import foundation.function-types open import foundation.functoriality-cartesian-product-types open import foundation.functoriality-dependent-pair-types open import foundation.functoriality-function-types diff --git a/src/species/composition-cauchy-series-species-of-types-in-subuniverses.lagda.md b/src/species/composition-cauchy-series-species-of-types-in-subuniverses.lagda.md index f23faa6197..b49689e733 100644 --- a/src/species/composition-cauchy-series-species-of-types-in-subuniverses.lagda.md +++ b/src/species/composition-cauchy-series-species-of-types-in-subuniverses.lagda.md @@ -8,7 +8,7 @@ module species.composition-cauchy-series-species-of-types-in-subuniverses where ```agda open import foundation.equivalences -open import foundation.functions +open import foundation.function-types open import foundation.subuniverses open import foundation.universe-levels diff --git a/src/species/coproducts-species-of-types-in-subuniverses.lagda.md b/src/species/coproducts-species-of-types-in-subuniverses.lagda.md index b8f6e56a69..5acba6e5cf 100644 --- a/src/species/coproducts-species-of-types-in-subuniverses.lagda.md +++ b/src/species/coproducts-species-of-types-in-subuniverses.lagda.md @@ -10,7 +10,7 @@ module species.coproducts-species-of-types-in-subuniverses where open import foundation.coproduct-types open import foundation.dependent-pair-types open import foundation.equivalences -open import foundation.functions +open import foundation.function-types open import foundation.homotopies open import foundation.identity-types open import foundation.subuniverses diff --git a/src/species/dirichlet-exponentials-species-of-types-in-subuniverses.lagda.md b/src/species/dirichlet-exponentials-species-of-types-in-subuniverses.lagda.md index 9f92d74733..a21c283bb6 100644 --- a/src/species/dirichlet-exponentials-species-of-types-in-subuniverses.lagda.md +++ b/src/species/dirichlet-exponentials-species-of-types-in-subuniverses.lagda.md @@ -10,7 +10,7 @@ module species.dirichlet-exponentials-species-of-types-in-subuniverses where open import foundation.cartesian-product-types open import foundation.dependent-pair-types open import foundation.equivalences -open import foundation.functions +open import foundation.function-types open import foundation.functoriality-cartesian-product-types open import foundation.functoriality-dependent-function-types open import foundation.functoriality-dependent-pair-types diff --git a/src/species/dirichlet-products-species-of-types-in-subuniverses.lagda.md b/src/species/dirichlet-products-species-of-types-in-subuniverses.lagda.md index 9f6121466a..4fedc029c4 100644 --- a/src/species/dirichlet-products-species-of-types-in-subuniverses.lagda.md +++ b/src/species/dirichlet-products-species-of-types-in-subuniverses.lagda.md @@ -11,7 +11,7 @@ open import foundation.cartesian-product-types open import foundation.contractible-types open import foundation.dependent-pair-types open import foundation.equivalences -open import foundation.functions +open import foundation.function-types open import foundation.functoriality-dependent-pair-types open import foundation.homotopies open import foundation.identity-types diff --git a/src/species/exponentials-cauchy-series-of-types-in-subuniverses.lagda.md b/src/species/exponentials-cauchy-series-of-types-in-subuniverses.lagda.md index 65f313ba03..cd9e392b7f 100644 --- a/src/species/exponentials-cauchy-series-of-types-in-subuniverses.lagda.md +++ b/src/species/exponentials-cauchy-series-of-types-in-subuniverses.lagda.md @@ -10,7 +10,7 @@ module species.exponentials-cauchy-series-of-types-in-subuniverses where open import foundation.cartesian-product-types open import foundation.dependent-pair-types open import foundation.equivalences -open import foundation.functions +open import foundation.function-types open import foundation.functoriality-dependent-pair-types open import foundation.subuniverses open import foundation.type-arithmetic-cartesian-product-types diff --git a/src/species/morphisms-finite-species.lagda.md b/src/species/morphisms-finite-species.lagda.md index 1a2df0a494..cad1eb0d55 100644 --- a/src/species/morphisms-finite-species.lagda.md +++ b/src/species/morphisms-finite-species.lagda.md @@ -11,7 +11,7 @@ open import foundation.contractible-types open import foundation.dependent-pair-types open import foundation.equality-dependent-function-types open import foundation.equivalences -open import foundation.functions +open import foundation.function-types open import foundation.fundamental-theorem-of-identity-types open import foundation.homotopies open import foundation.identity-types diff --git a/src/species/morphisms-species-of-types.lagda.md b/src/species/morphisms-species-of-types.lagda.md index 97538ab2b2..595c500054 100644 --- a/src/species/morphisms-species-of-types.lagda.md +++ b/src/species/morphisms-species-of-types.lagda.md @@ -11,7 +11,7 @@ open import foundation.contractible-types open import foundation.dependent-pair-types open import foundation.equality-dependent-function-types open import foundation.equivalences -open import foundation.functions +open import foundation.function-types open import foundation.fundamental-theorem-of-identity-types open import foundation.homotopies open import foundation.identity-types diff --git a/src/species/products-cauchy-series-species-of-types-in-subuniverses.lagda.md b/src/species/products-cauchy-series-species-of-types-in-subuniverses.lagda.md index 1f00a463ad..8cc4b98260 100644 --- a/src/species/products-cauchy-series-species-of-types-in-subuniverses.lagda.md +++ b/src/species/products-cauchy-series-species-of-types-in-subuniverses.lagda.md @@ -10,7 +10,7 @@ module species.products-cauchy-series-species-of-types-in-subuniverses where open import foundation.cartesian-product-types open import foundation.coproduct-types open import foundation.equivalences -open import foundation.functions +open import foundation.function-types open import foundation.functoriality-cartesian-product-types open import foundation.subuniverses open import foundation.universe-levels diff --git a/src/species/products-dirichlet-series-species-of-types-in-subuniverses.lagda.md b/src/species/products-dirichlet-series-species-of-types-in-subuniverses.lagda.md index 2a3e42fcbd..3b1b54e159 100644 --- a/src/species/products-dirichlet-series-species-of-types-in-subuniverses.lagda.md +++ b/src/species/products-dirichlet-series-species-of-types-in-subuniverses.lagda.md @@ -11,7 +11,7 @@ open import foundation.cartesian-product-types open import foundation.coproduct-types open import foundation.dependent-pair-types open import foundation.equivalences -open import foundation.functions +open import foundation.function-types open import foundation.functoriality-cartesian-product-types open import foundation.functoriality-dependent-pair-types open import foundation.functoriality-function-types diff --git a/src/species/small-cauchy-composition-species-of-finite-inhabited-types.lagda.md b/src/species/small-cauchy-composition-species-of-finite-inhabited-types.lagda.md index dcb1a63389..951521adbf 100644 --- a/src/species/small-cauchy-composition-species-of-finite-inhabited-types.lagda.md +++ b/src/species/small-cauchy-composition-species-of-finite-inhabited-types.lagda.md @@ -12,7 +12,7 @@ module species.small-cauchy-composition-species-of-finite-inhabited-types where open import foundation.contractible-types open import foundation.decidable-types open import foundation.equivalences -open import foundation.functions +open import foundation.function-types open import foundation.functoriality-cartesian-product-types open import foundation.functoriality-dependent-pair-types open import foundation.identity-types diff --git a/src/species/species-of-types-in-subuniverses.lagda.md b/src/species/species-of-types-in-subuniverses.lagda.md index 8bdc9166d5..8440ab4a5a 100644 --- a/src/species/species-of-types-in-subuniverses.lagda.md +++ b/src/species/species-of-types-in-subuniverses.lagda.md @@ -10,7 +10,7 @@ module species.species-of-types-in-subuniverses where open import foundation.cartesian-product-types open import foundation.dependent-pair-types open import foundation.equivalences -open import foundation.functions +open import foundation.function-types open import foundation.propositions open import foundation.subuniverses open import foundation.transport diff --git a/src/species/unit-cauchy-composition-species-of-types-in-subuniverses.lagda.md b/src/species/unit-cauchy-composition-species-of-types-in-subuniverses.lagda.md index f986c7f833..8721a6b7e3 100644 --- a/src/species/unit-cauchy-composition-species-of-types-in-subuniverses.lagda.md +++ b/src/species/unit-cauchy-composition-species-of-types-in-subuniverses.lagda.md @@ -9,7 +9,7 @@ module species.unit-cauchy-composition-species-of-types-in-subuniverses where ```agda open import foundation.contractible-types open import foundation.dependent-pair-types -open import foundation.functions +open import foundation.function-types open import foundation.subuniverses open import foundation.universe-levels diff --git a/src/structured-types/coherent-h-spaces.lagda.md b/src/structured-types/coherent-h-spaces.lagda.md index c1deef485e..790ade388f 100644 --- a/src/structured-types/coherent-h-spaces.lagda.md +++ b/src/structured-types/coherent-h-spaces.lagda.md @@ -12,7 +12,7 @@ open import foundation.action-on-identifications-functions open import foundation.dependent-pair-types open import foundation.equivalences open import foundation.function-extensionality -open import foundation.functions +open import foundation.function-types open import foundation.functoriality-dependent-pair-types open import foundation.identity-types open import foundation.type-arithmetic-dependent-pair-types diff --git a/src/structured-types/equivalences-types-equipped-with-endomorphisms.lagda.md b/src/structured-types/equivalences-types-equipped-with-endomorphisms.lagda.md index 1109f405ce..40e8d8d1f1 100644 --- a/src/structured-types/equivalences-types-equipped-with-endomorphisms.lagda.md +++ b/src/structured-types/equivalences-types-equipped-with-endomorphisms.lagda.md @@ -12,7 +12,7 @@ open import foundation.commuting-squares-of-maps open import foundation.contractible-types open import foundation.dependent-pair-types open import foundation.equivalences -open import foundation.functions +open import foundation.function-types open import foundation.functoriality-dependent-pair-types open import foundation.fundamental-theorem-of-identity-types open import foundation.homotopies diff --git a/src/structured-types/finite-multiplication-magmas.lagda.md b/src/structured-types/finite-multiplication-magmas.lagda.md index 16c9bd16c7..6c31a6bd9a 100644 --- a/src/structured-types/finite-multiplication-magmas.lagda.md +++ b/src/structured-types/finite-multiplication-magmas.lagda.md @@ -11,7 +11,7 @@ open import elementary-number-theory.natural-numbers open import foundation.coproduct-types open import foundation.equivalences -open import foundation.functions +open import foundation.function-types open import foundation.unit-type open import foundation.universe-levels diff --git a/src/structured-types/mere-equivalences-types-equipped-with-endomorphisms.lagda.md b/src/structured-types/mere-equivalences-types-equipped-with-endomorphisms.lagda.md index cb641112fa..17971592e6 100644 --- a/src/structured-types/mere-equivalences-types-equipped-with-endomorphisms.lagda.md +++ b/src/structured-types/mere-equivalences-types-equipped-with-endomorphisms.lagda.md @@ -10,7 +10,7 @@ module structured-types.mere-equivalences-types-equipped-with-endomorphisms wher open import foundation.contractible-types open import foundation.dependent-pair-types open import foundation.equivalences -open import foundation.functions +open import foundation.function-types open import foundation.fundamental-theorem-of-identity-types open import foundation.identity-types open import foundation.propositional-truncations diff --git a/src/structured-types/morphisms-coherent-h-spaces.lagda.md b/src/structured-types/morphisms-coherent-h-spaces.lagda.md index 77f8cf8293..a7f70b2878 100644 --- a/src/structured-types/morphisms-coherent-h-spaces.lagda.md +++ b/src/structured-types/morphisms-coherent-h-spaces.lagda.md @@ -10,7 +10,7 @@ module structured-types.morphisms-coherent-h-spaces where open import foundation.action-on-identifications-binary-functions open import foundation.action-on-identifications-functions open import foundation.dependent-pair-types -open import foundation.functions +open import foundation.function-types open import foundation.homotopies open import foundation.identity-types open import foundation.path-algebra diff --git a/src/structured-types/pointed-cartesian-product-types.lagda.md b/src/structured-types/pointed-cartesian-product-types.lagda.md index fc084f2b5c..42a6eea437 100644 --- a/src/structured-types/pointed-cartesian-product-types.lagda.md +++ b/src/structured-types/pointed-cartesian-product-types.lagda.md @@ -10,7 +10,7 @@ module structured-types.pointed-cartesian-product-types where open import foundation.cartesian-product-types open import foundation.dependent-pair-types open import foundation.equality-cartesian-product-types -open import foundation.functions +open import foundation.function-types open import foundation.homotopies open import foundation.identity-types open import foundation.universe-levels diff --git a/src/structured-types/pointed-dependent-functions.lagda.md b/src/structured-types/pointed-dependent-functions.lagda.md index 8d542de1cb..7b4563094d 100644 --- a/src/structured-types/pointed-dependent-functions.lagda.md +++ b/src/structured-types/pointed-dependent-functions.lagda.md @@ -9,7 +9,7 @@ module structured-types.pointed-dependent-functions where ```agda open import foundation.dependent-pair-types open import foundation.fibers-of-maps -open import foundation.functions +open import foundation.function-types open import foundation.identity-types open import foundation.universe-levels diff --git a/src/structured-types/pointed-equivalences.lagda.md b/src/structured-types/pointed-equivalences.lagda.md index b22fcb75de..f68e212527 100644 --- a/src/structured-types/pointed-equivalences.lagda.md +++ b/src/structured-types/pointed-equivalences.lagda.md @@ -15,7 +15,7 @@ open import foundation.dependent-pair-types open import foundation.equivalences open import foundation.fibers-of-maps open import foundation.function-extensionality -open import foundation.functions +open import foundation.function-types open import foundation.functoriality-dependent-pair-types open import foundation.homotopies open import foundation.identity-types diff --git a/src/structured-types/pointed-maps.lagda.md b/src/structured-types/pointed-maps.lagda.md index 81a8105f82..5be0aa5c63 100644 --- a/src/structured-types/pointed-maps.lagda.md +++ b/src/structured-types/pointed-maps.lagda.md @@ -11,7 +11,7 @@ open import foundation.action-on-identifications-dependent-functions open import foundation.action-on-identifications-functions open import foundation.constant-maps open import foundation.dependent-pair-types -open import foundation.functions +open import foundation.function-types open import foundation.identity-types open import foundation.transport open import foundation.universe-levels diff --git a/src/structured-types/pointed-types-equipped-with-automorphisms.lagda.md b/src/structured-types/pointed-types-equipped-with-automorphisms.lagda.md index c7d0f8ac73..bce925f1f9 100644 --- a/src/structured-types/pointed-types-equipped-with-automorphisms.lagda.md +++ b/src/structured-types/pointed-types-equipped-with-automorphisms.lagda.md @@ -13,7 +13,7 @@ open import foundation.cartesian-product-types open import foundation.contractible-types open import foundation.dependent-pair-types open import foundation.equivalences -open import foundation.functions +open import foundation.function-types open import foundation.functoriality-dependent-pair-types open import foundation.fundamental-theorem-of-identity-types open import foundation.homotopies diff --git a/src/structured-types/types-equipped-with-endomorphisms.lagda.md b/src/structured-types/types-equipped-with-endomorphisms.lagda.md index 58e1ad673f..7c1dd530be 100644 --- a/src/structured-types/types-equipped-with-endomorphisms.lagda.md +++ b/src/structured-types/types-equipped-with-endomorphisms.lagda.md @@ -9,7 +9,7 @@ module structured-types.types-equipped-with-endomorphisms where ```agda open import foundation.dependent-pair-types open import foundation.endomorphisms -open import foundation.functions +open import foundation.function-types open import foundation.unit-type open import foundation.universe-levels ``` diff --git a/src/synthetic-homotopy-theory/26-descent.lagda.md b/src/synthetic-homotopy-theory/26-descent.lagda.md index 33289f2936..912e5babc4 100644 --- a/src/synthetic-homotopy-theory/26-descent.lagda.md +++ b/src/synthetic-homotopy-theory/26-descent.lagda.md @@ -21,7 +21,7 @@ open import foundation.equivalence-extensionality open import foundation.equivalences open import foundation.fibers-of-maps open import foundation.function-extensionality -open import foundation.functions +open import foundation.function-types open import foundation.functoriality-dependent-function-types open import foundation.functoriality-dependent-pair-types open import foundation.functoriality-function-types diff --git a/src/synthetic-homotopy-theory/26-id-pushout.lagda.md b/src/synthetic-homotopy-theory/26-id-pushout.lagda.md index 0b33d3bd45..b179d5559f 100644 --- a/src/synthetic-homotopy-theory/26-id-pushout.lagda.md +++ b/src/synthetic-homotopy-theory/26-id-pushout.lagda.md @@ -15,7 +15,7 @@ open import foundation.dependent-pair-types open import foundation.equality-dependent-function-types open import foundation.equivalences open import foundation.function-extensionality -open import foundation.functions +open import foundation.function-types open import foundation.functoriality-dependent-function-types open import foundation.functoriality-dependent-pair-types open import foundation.fundamental-theorem-of-identity-types diff --git a/src/synthetic-homotopy-theory/27-sequences.lagda.md b/src/synthetic-homotopy-theory/27-sequences.lagda.md index adb67fcd59..0667134622 100644 --- a/src/synthetic-homotopy-theory/27-sequences.lagda.md +++ b/src/synthetic-homotopy-theory/27-sequences.lagda.md @@ -13,7 +13,7 @@ open import foundation.contractible-types open import foundation.dependent-pair-types open import foundation.equality-dependent-function-types open import foundation.equivalences -open import foundation.functions +open import foundation.function-types open import foundation.fundamental-theorem-of-identity-types open import foundation.homotopies open import foundation.identity-types diff --git a/src/synthetic-homotopy-theory/cavallos-trick.lagda.md b/src/synthetic-homotopy-theory/cavallos-trick.lagda.md index 017aea853a..ff5efc2233 100644 --- a/src/synthetic-homotopy-theory/cavallos-trick.lagda.md +++ b/src/synthetic-homotopy-theory/cavallos-trick.lagda.md @@ -9,7 +9,7 @@ module synthetic-homotopy-theory.cavallos-trick where ```agda open import foundation.action-on-identifications-functions open import foundation.dependent-pair-types -open import foundation.functions +open import foundation.function-types open import foundation.homotopies open import foundation.identity-types open import foundation.sections diff --git a/src/synthetic-homotopy-theory/cocones-under-spans.lagda.md b/src/synthetic-homotopy-theory/cocones-under-spans.lagda.md index 0729c77a43..ac8d493fc8 100644 --- a/src/synthetic-homotopy-theory/cocones-under-spans.lagda.md +++ b/src/synthetic-homotopy-theory/cocones-under-spans.lagda.md @@ -14,7 +14,7 @@ open import foundation.dependent-pair-types open import foundation.equality-dependent-pair-types open import foundation.equivalences open import foundation.function-extensionality -open import foundation.functions +open import foundation.function-types open import foundation.functoriality-dependent-pair-types open import foundation.fundamental-theorem-of-identity-types open import foundation.homotopies diff --git a/src/synthetic-homotopy-theory/dependent-cocones-under-spans.lagda.md b/src/synthetic-homotopy-theory/dependent-cocones-under-spans.lagda.md index b9c595e2e5..04dfbd0a6c 100644 --- a/src/synthetic-homotopy-theory/dependent-cocones-under-spans.lagda.md +++ b/src/synthetic-homotopy-theory/dependent-cocones-under-spans.lagda.md @@ -10,10 +10,10 @@ module synthetic-homotopy-theory.dependent-cocones-under-spans where open import foundation.action-on-identifications-dependent-functions open import foundation.action-on-identifications-functions open import foundation.contractible-types +open import foundation.dependent-identifications open import foundation.dependent-pair-types -open import foundation.dependent-paths open import foundation.equivalences -open import foundation.functions +open import foundation.function-types open import foundation.functoriality-dependent-pair-types open import foundation.fundamental-theorem-of-identity-types open import foundation.homotopies @@ -45,7 +45,7 @@ functions and a family of dependent identifications ```text - (s : S) → path-over P (H s) (i' (f s)) (j' (g s)). + (s : S) → dependent-identification P (H s) (i' (f s)) (j' (g s)). ``` ## Definitions @@ -65,7 +65,7 @@ module _ Σ ( (b : B) → P (vertical-map-cocone f g c b)) ( λ hB → (s : S) → - path-over P + dependent-identification P ( coherence-square-cocone f g c s) ( hA (f s)) ( hB (g s)))) @@ -84,7 +84,7 @@ module _ coherence-square-dependent-cocone : (s : S) → - path-over P + dependent-identification P ( coherence-square-cocone f g c s) ( horizontal-map-dependent-cocone (f s)) ( vertical-map-dependent-cocone (g s)) @@ -194,7 +194,7 @@ module _ ( vertical-map-dependent-cocone f g c P d , refl-htpy) ( is-contr-equiv ( Σ ( (s : S) → - path-over P + dependent-identification P ( coherence-square-cocone f g c s) ( horizontal-map-dependent-cocone f g c P d (f s)) ( vertical-map-dependent-cocone f g c P d (g s))) diff --git a/src/synthetic-homotopy-theory/dependent-universal-property-pushouts.lagda.md b/src/synthetic-homotopy-theory/dependent-universal-property-pushouts.lagda.md index 5c460b9b40..7ff94b9f67 100644 --- a/src/synthetic-homotopy-theory/dependent-universal-property-pushouts.lagda.md +++ b/src/synthetic-homotopy-theory/dependent-universal-property-pushouts.lagda.md @@ -15,7 +15,7 @@ open import foundation.equality-dependent-pair-types open import foundation.equivalences open import foundation.fibers-of-maps open import foundation.function-extensionality -open import foundation.functions +open import foundation.function-types open import foundation.functoriality-dependent-pair-types open import foundation.homotopies open import foundation.identity-types @@ -107,7 +107,7 @@ htpy-eq-dependent-cocone-map f g c ind-c {P} h h' p = ( dependent-cocone-map f g c P h') ( p)) ( λ s → - map-compute-path-over-eq-value h h' + map-compute-dependent-identification-eq-value h h' ( coherence-square-cocone f g c s) ( horizontal-htpy-eq-dependent-cocone f g c P ( dependent-cocone-map f g c P h) diff --git a/src/synthetic-homotopy-theory/descent-circle.lagda.md b/src/synthetic-homotopy-theory/descent-circle.lagda.md index 56e1e0f311..aa53af00af 100644 --- a/src/synthetic-homotopy-theory/descent-circle.lagda.md +++ b/src/synthetic-homotopy-theory/descent-circle.lagda.md @@ -14,13 +14,13 @@ open import foundation.commuting-squares-of-maps open import foundation.commuting-triangles-of-maps open import foundation.contractible-maps open import foundation.contractible-types +open import foundation.dependent-identifications open import foundation.dependent-pair-types -open import foundation.dependent-paths open import foundation.equivalence-extensionality open import foundation.equivalences open import foundation.fibers-of-maps open import foundation.function-extensionality -open import foundation.functions +open import foundation.function-types open import foundation.functoriality-dependent-pair-types open import foundation.functoriality-function-types open import foundation.fundamental-theorem-of-identity-types @@ -243,17 +243,20 @@ module _ α : type-descent-data-circle P ≃ Q (base-free-loop l) α = pr1 αH - map-compute-path-over-loop-circle : + map-compute-dependent-identification-loop-circle : ( x y : type-descent-data-circle P) → - ( map-equiv (aut-descent-data-circle P) x ＝ y) → - ( path-over Q (loop-free-loop l) (map-equiv α x) (map-equiv α y)) - map-compute-path-over-loop-circle x y q = + map-equiv (aut-descent-data-circle P) x ＝ y → + dependent-identification Q + ( loop-free-loop l) + ( map-equiv α x) + ( map-equiv α y) + map-compute-dependent-identification-loop-circle x y q = inv (pr2 αH x) ∙ (ap (map-equiv α) q) - is-equiv-map-compute-path-over-loop-circle : + is-equiv-map-compute-dependent-identification-loop-circle : ( x y : type-descent-data-circle P) → - is-equiv (map-compute-path-over-loop-circle x y) - is-equiv-map-compute-path-over-loop-circle x y = + is-equiv (map-compute-dependent-identification-loop-circle x y) + is-equiv-map-compute-dependent-identification-loop-circle x y = fundamental-theorem-id ( is-contr-equiv' ( fib (map-equiv α) (tr Q (loop-free-loop l) (map-equiv α x))) @@ -261,17 +264,20 @@ module _ ( is-contr-map-is-equiv ( is-equiv-map-equiv α) ( tr Q (loop-free-loop l) (map-equiv α x)))) - ( map-compute-path-over-loop-circle x) + ( map-compute-dependent-identification-loop-circle x) ( y) - compute-path-over-loop-circle : + compute-dependent-identification-loop-circle : ( x y : type-descent-data-circle P) → ( map-equiv (aut-descent-data-circle P) x ＝ y) ≃ - ( path-over Q (loop-free-loop l) (map-equiv α x) (map-equiv α y)) - pr1 (compute-path-over-loop-circle x y) = - map-compute-path-over-loop-circle x y - pr2 (compute-path-over-loop-circle x y) = - is-equiv-map-compute-path-over-loop-circle x y + ( dependent-identification Q + ( loop-free-loop l) + ( map-equiv α x) + ( map-equiv α y)) + pr1 (compute-dependent-identification-loop-circle x y) = + map-compute-dependent-identification-loop-circle x y + pr2 (compute-dependent-identification-loop-circle x y) = + is-equiv-map-compute-dependent-identification-loop-circle x y ``` ```agda @@ -293,7 +299,7 @@ module _ ( s (base-free-loop l)) pr2 (ev-fixpoint-descent-data-circle s) = map-inv-is-equiv - ( is-equiv-map-compute-path-over-loop-circle + ( is-equiv-map-compute-dependent-identification-loop-circle ( l) ( Q) ( P) @@ -310,10 +316,10 @@ module _ fixpoint-descent-data-circle l P ≃ free-dependent-loop l Q equiv-fixpoint-descent-data-circle-free-dependent-loop = equiv-Σ - ( λ x → path-over Q (loop-free-loop l) x x) + ( λ x → dependent-identification Q (loop-free-loop l) x x) ( α) ( λ x → - compute-path-over-loop-circle l Q P αH x x) + compute-dependent-identification-loop-circle l Q P αH x x) comparison-fixpoint-descent-data-circle : fixpoint-descent-data-circle l P → free-dependent-loop l Q @@ -336,7 +342,7 @@ module _ ( ( horizontal-concat-Id² ( refl {x = ap (tr Q (loop-free-loop l)) (inv issec-inv-α)}) ( issec-map-inv-is-equiv - ( is-equiv-map-compute-path-over-loop-circle + ( is-equiv-map-compute-dependent-identification-loop-circle ( l) ( Q) ( P) diff --git a/src/synthetic-homotopy-theory/infinite-cyclic-types.lagda.md b/src/synthetic-homotopy-theory/infinite-cyclic-types.lagda.md index 3d96304176..6baee03601 100644 --- a/src/synthetic-homotopy-theory/infinite-cyclic-types.lagda.md +++ b/src/synthetic-homotopy-theory/infinite-cyclic-types.lagda.md @@ -16,7 +16,7 @@ open import foundation.contractible-maps open import foundation.contractible-types open import foundation.dependent-pair-types open import foundation.equivalences -open import foundation.functions +open import foundation.function-types open import foundation.functoriality-dependent-function-types open import foundation.functoriality-dependent-pair-types open import foundation.homotopies diff --git a/src/synthetic-homotopy-theory/joins-of-types.lagda.md b/src/synthetic-homotopy-theory/joins-of-types.lagda.md index f763b58e16..268d0d3bf1 100644 --- a/src/synthetic-homotopy-theory/joins-of-types.lagda.md +++ b/src/synthetic-homotopy-theory/joins-of-types.lagda.md @@ -13,7 +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.function-types open import foundation.identity-types open import foundation.propositions open import foundation.type-arithmetic-cartesian-product-types diff --git a/src/synthetic-homotopy-theory/multiplication-circle.lagda.md b/src/synthetic-homotopy-theory/multiplication-circle.lagda.md index 651552c4c0..508171fba4 100644 --- a/src/synthetic-homotopy-theory/multiplication-circle.lagda.md +++ b/src/synthetic-homotopy-theory/multiplication-circle.lagda.md @@ -10,7 +10,7 @@ module synthetic-homotopy-theory.multiplication-circle where open import foundation.action-on-identifications-functions open import foundation.dependent-pair-types open import foundation.function-extensionality -open import foundation.functions +open import foundation.function-types open import foundation.homotopies open import foundation.identity-types open import foundation.transport @@ -41,12 +41,20 @@ htpy-id-id-Π-𝕊¹ : Π-𝕊¹ ( eq-value id id) ( loop-𝕊¹) - ( map-compute-path-over-eq-value-id-id loop-𝕊¹ loop-𝕊¹ loop-𝕊¹ refl) + ( map-compute-dependent-identification-eq-value-id-id + ( loop-𝕊¹) + ( loop-𝕊¹) + ( loop-𝕊¹) + ( refl)) htpy-id-id-Π-𝕊¹ = apply-dependent-universal-property-𝕊¹ ( eq-value id id) ( loop-𝕊¹) - ( map-compute-path-over-eq-value-id-id loop-𝕊¹ loop-𝕊¹ loop-𝕊¹ refl) + ( map-compute-dependent-identification-eq-value-id-id + ( loop-𝕊¹) + ( loop-𝕊¹) + ( loop-𝕊¹) + ( refl)) htpy-id-id-𝕊¹ : (x : 𝕊¹) → Id x x htpy-id-id-𝕊¹ = pr1 htpy-id-id-Π-𝕊¹ @@ -61,11 +69,11 @@ htpy-id-id-base-𝕊¹ = pr1 (pr2 htpy-id-id-Π-𝕊¹) Mul-Π-𝕊¹ : 𝕊¹ → UU lzero Mul-Π-𝕊¹ x = 𝕊¹-Pointed-Type →∗ (pair 𝕊¹ x) -path-over-Mul-Π-𝕊¹ : +dependent-identification-Mul-Π-𝕊¹ : {x : 𝕊¹} (p : Id base-𝕊¹ x) (q : Mul-Π-𝕊¹ base-𝕊¹) (r : Mul-Π-𝕊¹ x) → (H : pr1 q ~ pr1 r) → Id (pr2 q ∙ p) (H base-𝕊¹ ∙ pr2 r) → Id (tr Mul-Π-𝕊¹ p q) r -path-over-Mul-Π-𝕊¹ {x} refl q r H u = +dependent-identification-Mul-Π-𝕊¹ {x} refl q r H u = eq-htpy-pointed-map ( 𝕊¹-Pointed-Type) ( 𝕊¹-Pointed-Type) @@ -76,7 +84,7 @@ path-over-Mul-Π-𝕊¹ {x} refl q r H u = eq-id-id-𝕊¹-Pointed-Type : Id (tr Mul-Π-𝕊¹ loop-𝕊¹ id-pointed-map) id-pointed-map eq-id-id-𝕊¹-Pointed-Type = - path-over-Mul-Π-𝕊¹ loop-𝕊¹ + dependent-identification-Mul-Π-𝕊¹ loop-𝕊¹ ( id-pointed-map) ( id-pointed-map) ( htpy-id-id-𝕊¹) diff --git a/src/synthetic-homotopy-theory/pullback-property-pushouts.lagda.md b/src/synthetic-homotopy-theory/pullback-property-pushouts.lagda.md index 69ccb47e66..56ac8f968a 100644 --- a/src/synthetic-homotopy-theory/pullback-property-pushouts.lagda.md +++ b/src/synthetic-homotopy-theory/pullback-property-pushouts.lagda.md @@ -9,7 +9,7 @@ module synthetic-homotopy-theory.pullback-property-pushouts where ```agda open import foundation.cones-over-cospans open import foundation.dependent-pair-types -open import foundation.functions +open import foundation.function-types open import foundation.functoriality-function-types open import foundation.pullbacks open import foundation.universe-levels diff --git a/src/synthetic-homotopy-theory/pushouts.lagda.md b/src/synthetic-homotopy-theory/pushouts.lagda.md index 744c34d276..3a878b719f 100644 --- a/src/synthetic-homotopy-theory/pushouts.lagda.md +++ b/src/synthetic-homotopy-theory/pushouts.lagda.md @@ -10,7 +10,7 @@ module synthetic-homotopy-theory.pushouts where open import foundation.action-on-identifications-functions open import foundation.dependent-pair-types open import foundation.equivalences -open import foundation.functions +open import foundation.function-types open import foundation.homotopies open import foundation.identity-types open import foundation.universe-levels diff --git a/src/synthetic-homotopy-theory/suspensions-of-types.lagda.md b/src/synthetic-homotopy-theory/suspensions-of-types.lagda.md index b12ff6716e..d34dece95c 100644 --- a/src/synthetic-homotopy-theory/suspensions-of-types.lagda.md +++ b/src/synthetic-homotopy-theory/suspensions-of-types.lagda.md @@ -13,7 +13,7 @@ open import foundation.contractible-types open import foundation.dependent-pair-types open import foundation.equivalences open import foundation.function-extensionality -open import foundation.functions +open import foundation.function-types open import foundation.functoriality-dependent-pair-types open import foundation.homotopies open import foundation.identity-systems diff --git a/src/synthetic-homotopy-theory/universal-cover-circle.lagda.md b/src/synthetic-homotopy-theory/universal-cover-circle.lagda.md index 3393f39c54..ea61a40d83 100644 --- a/src/synthetic-homotopy-theory/universal-cover-circle.lagda.md +++ b/src/synthetic-homotopy-theory/universal-cover-circle.lagda.md @@ -15,12 +15,12 @@ open import foundation.action-on-identifications-functions open import foundation.cartesian-product-types open import foundation.contractible-types open import foundation.coproduct-types +open import foundation.dependent-identifications open import foundation.dependent-pair-types -open import foundation.dependent-paths open import foundation.equality-dependent-pair-types open import foundation.equivalences open import foundation.function-extensionality -open import foundation.functions +open import foundation.function-types open import foundation.functoriality-dependent-function-types open import foundation.functoriality-dependent-pair-types open import foundation.fundamental-theorem-of-identity-types @@ -286,7 +286,7 @@ square-tr-contraction-total-space c refl f e e' H h y = ( h (map-equiv e' (map-equiv f y))))) ∙ ( apd h (H y)) -path-over-contraction-total-space' : +dependent-identification-contraction-total-space' : {l1 l2 l3 l4 : Level} {A : UU l1} {B : A → UU l2} (c : Σ A B) → {x x' : A} (p : Id x x') → {F : UU l3} {F' : UU l4} (f : F ≃ F') ( e : F ≃ B x) (e' : F' ≃ B x') @@ -294,7 +294,8 @@ path-over-contraction-total-space' : (h : (y : F) → Id c (pair x (map-equiv e y))) → (h' : (y' : F') → Id c (pair x' (map-equiv e' y'))) → UU (l1 ⊔ l2 ⊔ l3) -path-over-contraction-total-space' c {x} {x'} p {F} {F'} f e e' H h h' = +dependent-identification-contraction-total-space' + c {x} {x'} p {F} {F'} f e e' H h h' = ( map-Π ( λ y → concat' c (segment-Σ p f e e' H y)) h) ~ ( precomp-Π @@ -302,18 +303,19 @@ path-over-contraction-total-space' c {x} {x'} p {F} {F'} f e e' H h h' = ( λ y' → Id c (pair x' (map-equiv e' y'))) ( h')) -map-path-over-contraction-total-space' : +map-dependent-identification-contraction-total-space' : { l1 l2 l3 l4 : Level} {A : UU l1} {B : A → UU l2} (c : Σ A B) → { x x' : A} (p : Id x x') → { F : UU l3} {F' : UU l4} (f : F ≃ F') ( e : F ≃ B x) (e' : F' ≃ B x') ( H : ((map-equiv e') ∘ (map-equiv f)) ~ ((tr B p) ∘ (map-equiv e))) → ( h : contraction-total-space' c x e) → ( h' : contraction-total-space' c x' e') → - ( path-over-contraction-total-space' c p f e e' H h h') → - ( path-over (contraction-total-space c) p + ( dependent-identification-contraction-total-space' c p f e e' H h h') → + ( dependent-identification (contraction-total-space c) p ( map-inv-equiv (equiv-contraction-total-space c x e) h) ( map-inv-equiv (equiv-contraction-total-space c x' e') h')) -map-path-over-contraction-total-space' c {x} {.x} refl f e e' H h h' α = +map-dependent-identification-contraction-total-space' + c {x} {.x} refl f e e' H h h' α = map-inv-equiv ( equiv-ap ( ( equiv-tr-contraction-total-space' c refl f e e' H) ∘e @@ -342,18 +344,19 @@ map-path-over-contraction-total-space' c {x} {.x} refl f e e' H h h' α = ( equiv-precomp-Π e' (λ y' → Id c (pair x y')))) ( h')))))) -equiv-path-over-contraction-total-space' : +equiv-dependent-identification-contraction-total-space' : { l1 l2 l3 l4 : Level} {A : UU l1} {B : A → UU l2} (c : Σ A B) → { x x' : A} (p : Id x x') → { F : UU l3} {F' : UU l4} (f : F ≃ F') ( e : F ≃ B x) (e' : F' ≃ B x') ( H : ((map-equiv e') ∘ (map-equiv f)) ~ ((tr B p) ∘ (map-equiv e))) → ( h : contraction-total-space' c x e) → ( h' : contraction-total-space' c x' e') → - ( path-over (contraction-total-space c) p + ( dependent-identification (contraction-total-space c) p ( map-inv-equiv (equiv-contraction-total-space c x e) h) ( map-inv-equiv (equiv-contraction-total-space c x' e') h')) ≃ - ( path-over-contraction-total-space' c p f e e' H h h') -equiv-path-over-contraction-total-space' c {x} {.x} refl f e e' H h h' = + ( dependent-identification-contraction-total-space' c p f e e' H h h') +equiv-dependent-identification-contraction-total-space' + c {x} {.x} refl f e e' H h h' = ( inv-equiv ( equiv-con-inv-htpy h ( segment-Σ refl f e e' H) @@ -401,7 +404,7 @@ pr1 (center-total-fundamental-cover-circle l dup-circle) = base-free-loop l pr2 (center-total-fundamental-cover-circle l dup-circle) = map-equiv ( compute-fiber-fundamental-cover-circle l dup-circle) zero-ℤ -path-over-loop-contraction-total-fundamental-cover-circle : +dependent-identification-loop-contraction-total-fundamental-cover-circle : { l1 : Level} {X : UU l1} (l : free-loop X) → ( dup-circle : {l2 : Level} → dependent-universal-property-circle l2 l) → ( h : @@ -410,7 +413,7 @@ path-over-loop-contraction-total-fundamental-cover-circle : ( base-free-loop l) ( compute-fiber-fundamental-cover-circle l dup-circle)) → ( p : - path-over-contraction-total-space' + dependent-identification-contraction-total-space' ( center-total-fundamental-cover-circle l dup-circle) ( loop-free-loop l) ( equiv-succ-ℤ) @@ -419,7 +422,7 @@ path-over-loop-contraction-total-fundamental-cover-circle : ( compute-tr-fundamental-cover-circle l dup-circle) ( h) ( h)) → - path-over + dependent-identification ( contraction-total-space ( center-total-fundamental-cover-circle l dup-circle)) ( pr2 l) @@ -435,8 +438,9 @@ path-over-loop-contraction-total-fundamental-cover-circle : ( base-free-loop l) ( compute-fiber-fundamental-cover-circle l dup-circle)) ( h)) -path-over-loop-contraction-total-fundamental-cover-circle l dup-circle h p = - map-path-over-contraction-total-space' +dependent-identification-loop-contraction-total-fundamental-cover-circle + l dup-circle h p = + map-dependent-identification-contraction-total-space' ( center-total-fundamental-cover-circle l dup-circle) ( loop-free-loop l) ( equiv-succ-ℤ) @@ -456,7 +460,7 @@ contraction-total-fundamental-cover-circle-data : ( base-free-loop l) ( compute-fiber-fundamental-cover-circle l dup-circle)) → ( p : - path-over-contraction-total-space' + dependent-identification-contraction-total-space' ( center-total-fundamental-cover-circle l dup-circle) ( loop-free-loop l) ( equiv-succ-ℤ) @@ -481,7 +485,7 @@ contraction-total-fundamental-cover-circle-data ( base-free-loop l) ( compute-fiber-fundamental-cover-circle l dup-circle)) ( h)) - ( path-over-loop-contraction-total-fundamental-cover-circle + ( dependent-identification-loop-contraction-total-fundamental-cover-circle l dup-circle h p)) x y @@ -494,7 +498,7 @@ is-contr-total-fundamental-cover-circle-data : ( base-free-loop l) ( compute-fiber-fundamental-cover-circle l dup-circle)) → ( p : - path-over-contraction-total-space' + dependent-identification-contraction-total-space' ( center-total-fundamental-cover-circle l dup-circle) ( loop-free-loop l) ( equiv-succ-ℤ) diff --git a/src/synthetic-homotopy-theory/universal-property-circle.lagda.md b/src/synthetic-homotopy-theory/universal-property-circle.lagda.md index a95866262a..6de359fe45 100644 --- a/src/synthetic-homotopy-theory/universal-property-circle.lagda.md +++ b/src/synthetic-homotopy-theory/universal-property-circle.lagda.md @@ -16,7 +16,7 @@ open import foundation.dependent-pair-types open import foundation.equivalences open import foundation.fibers-of-maps open import foundation.function-extensionality -open import foundation.functions +open import foundation.function-types open import foundation.functoriality-dependent-pair-types open import foundation.homotopies open import foundation.identity-types @@ -124,7 +124,7 @@ module _ ( free-dependent-loop α (λ x → Id (f x) (g x))) pr1 (free-dependent-loop-htpy {l2} {P} {f} {g} (p , q)) = p pr2 (free-dependent-loop-htpy {l2} {P} {f} {g} (p , q)) = - map-compute-path-over-eq-value f g (loop-free-loop α) p p q + map-compute-dependent-identification-eq-value f g (loop-free-loop α) p p q isretr-ind-circle : ( ind-circle : {l : Level} → induction-principle-circle l α) diff --git a/src/synthetic-homotopy-theory/universal-property-pushouts.lagda.md b/src/synthetic-homotopy-theory/universal-property-pushouts.lagda.md index 927c584dc1..b332141c72 100644 --- a/src/synthetic-homotopy-theory/universal-property-pushouts.lagda.md +++ b/src/synthetic-homotopy-theory/universal-property-pushouts.lagda.md @@ -15,7 +15,7 @@ open import foundation.equality-dependent-pair-types open import foundation.equivalences open import foundation.fibers-of-maps open import foundation.function-extensionality -open import foundation.functions +open import foundation.function-types open import foundation.functoriality-dependent-pair-types open import foundation.homotopies open import foundation.identity-types diff --git a/src/trees/bases-directed-trees.lagda.md b/src/trees/bases-directed-trees.lagda.md index ceff9bbd43..fa49199117 100644 --- a/src/trees/bases-directed-trees.lagda.md +++ b/src/trees/bases-directed-trees.lagda.md @@ -15,7 +15,7 @@ open import foundation.coproduct-types open import foundation.dependent-pair-types open import foundation.empty-types open import foundation.equivalences -open import foundation.functions +open import foundation.function-types open import foundation.functoriality-dependent-pair-types open import foundation.identity-types open import foundation.logical-equivalences diff --git a/src/trees/bases-enriched-directed-trees.lagda.md b/src/trees/bases-enriched-directed-trees.lagda.md index 2f885e1988..989ca4c759 100644 --- a/src/trees/bases-enriched-directed-trees.lagda.md +++ b/src/trees/bases-enriched-directed-trees.lagda.md @@ -11,7 +11,7 @@ open import foundation.contractible-types open import foundation.coproduct-types open import foundation.dependent-pair-types open import foundation.equivalences -open import foundation.functions +open import foundation.function-types open import foundation.functoriality-coproduct-types open import foundation.functoriality-dependent-pair-types open import foundation.identity-types diff --git a/src/trees/combinator-directed-trees.lagda.md b/src/trees/combinator-directed-trees.lagda.md index 493ecb3b2c..f44bcafac5 100644 --- a/src/trees/combinator-directed-trees.lagda.md +++ b/src/trees/combinator-directed-trees.lagda.md @@ -15,7 +15,7 @@ open import foundation.dependent-pair-types open import foundation.empty-types open import foundation.equality-dependent-pair-types open import foundation.equivalences -open import foundation.functions +open import foundation.function-types open import foundation.functoriality-dependent-pair-types open import foundation.homotopies open import foundation.identity-types diff --git a/src/trees/combinator-enriched-directed-trees.lagda.md b/src/trees/combinator-enriched-directed-trees.lagda.md index 2a3462fc84..bca174548f 100644 --- a/src/trees/combinator-enriched-directed-trees.lagda.md +++ b/src/trees/combinator-enriched-directed-trees.lagda.md @@ -9,7 +9,7 @@ module trees.combinator-enriched-directed-trees where ```agda open import foundation.dependent-pair-types open import foundation.equivalences -open import foundation.functions +open import foundation.function-types open import foundation.homotopies open import foundation.identity-types open import foundation.maybe diff --git a/src/trees/directed-trees.lagda.md b/src/trees/directed-trees.lagda.md index 663f64483e..8c7810e671 100644 --- a/src/trees/directed-trees.lagda.md +++ b/src/trees/directed-trees.lagda.md @@ -20,7 +20,7 @@ open import foundation.dependent-pair-types open import foundation.empty-types open import foundation.equality-dependent-pair-types open import foundation.equivalences -open import foundation.functions +open import foundation.function-types open import foundation.functoriality-coproduct-types open import foundation.identity-types open import foundation.isolated-points diff --git a/src/trees/equivalences-directed-trees.lagda.md b/src/trees/equivalences-directed-trees.lagda.md index f7e65b6dc6..bab333224b 100644 --- a/src/trees/equivalences-directed-trees.lagda.md +++ b/src/trees/equivalences-directed-trees.lagda.md @@ -14,7 +14,7 @@ open import foundation.coproduct-types open import foundation.dependent-pair-types open import foundation.empty-types open import foundation.equivalences -open import foundation.functions +open import foundation.function-types open import foundation.functoriality-dependent-pair-types open import foundation.homotopies open import foundation.identity-types diff --git a/src/trees/equivalences-enriched-directed-trees.lagda.md b/src/trees/equivalences-enriched-directed-trees.lagda.md index 1733ad63d0..e24e0e65d7 100644 --- a/src/trees/equivalences-enriched-directed-trees.lagda.md +++ b/src/trees/equivalences-enriched-directed-trees.lagda.md @@ -15,7 +15,7 @@ open import foundation.equality-dependent-function-types open import foundation.equivalence-extensionality open import foundation.equivalences open import foundation.function-extensionality -open import foundation.functions +open import foundation.function-types open import foundation.functoriality-dependent-pair-types open import foundation.homotopies open import foundation.identity-types diff --git a/src/trees/extensional-w-types.lagda.md b/src/trees/extensional-w-types.lagda.md index e0cce31c58..1f227e745a 100644 --- a/src/trees/extensional-w-types.lagda.md +++ b/src/trees/extensional-w-types.lagda.md @@ -13,7 +13,7 @@ open import foundation.dependent-pair-types open import foundation.equality-dependent-function-types open import foundation.equivalence-extensionality open import foundation.equivalences -open import foundation.functions +open import foundation.function-types open import foundation.functoriality-dependent-function-types open import foundation.functoriality-dependent-pair-types open import foundation.fundamental-theorem-of-identity-types @@ -22,6 +22,7 @@ open import foundation.identity-types open import foundation.propositional-truncations open import foundation.propositions open import foundation.slice +open import foundation.transport open import foundation.type-arithmetic-dependent-pair-types open import foundation.univalent-type-families open import foundation.universe-levels diff --git a/src/trees/functoriality-combinator-directed-trees.lagda.md b/src/trees/functoriality-combinator-directed-trees.lagda.md index 0e110287b8..b181907762 100644 --- a/src/trees/functoriality-combinator-directed-trees.lagda.md +++ b/src/trees/functoriality-combinator-directed-trees.lagda.md @@ -11,7 +11,7 @@ open import foundation.action-on-identifications-functions open import foundation.binary-transport open import foundation.contractible-types open import foundation.dependent-pair-types -open import foundation.functions +open import foundation.function-types open import foundation.homotopies open import foundation.identity-types open import foundation.transport diff --git a/src/trees/functoriality-w-types.lagda.md b/src/trees/functoriality-w-types.lagda.md index c9f3d846bd..3f65f6dd88 100644 --- a/src/trees/functoriality-w-types.lagda.md +++ b/src/trees/functoriality-w-types.lagda.md @@ -14,7 +14,7 @@ open import foundation.dependent-pair-types open import foundation.embeddings open import foundation.equivalences open import foundation.fibers-of-maps -open import foundation.functions +open import foundation.function-types open import foundation.functoriality-dependent-function-types open import foundation.functoriality-dependent-pair-types open import foundation.identity-types diff --git a/src/trees/induction-w-types.lagda.md b/src/trees/induction-w-types.lagda.md index ed1e506905..bf924b6e13 100644 --- a/src/trees/induction-w-types.lagda.md +++ b/src/trees/induction-w-types.lagda.md @@ -12,7 +12,7 @@ open import elementary-number-theory.natural-numbers open import foundation.action-on-identifications-functions open import foundation.dependent-pair-types open import foundation.function-extensionality -open import foundation.functions +open import foundation.function-types open import foundation.identity-types open import foundation.negation open import foundation.transport diff --git a/src/trees/morphisms-algebras-polynomial-endofunctors.lagda.md b/src/trees/morphisms-algebras-polynomial-endofunctors.lagda.md index b5bd19de80..38ad1fe9f1 100644 --- a/src/trees/morphisms-algebras-polynomial-endofunctors.lagda.md +++ b/src/trees/morphisms-algebras-polynomial-endofunctors.lagda.md @@ -11,7 +11,7 @@ open import foundation.action-on-identifications-functions open import foundation.contractible-types open import foundation.dependent-pair-types open import foundation.equivalences -open import foundation.functions +open import foundation.function-types open import foundation.functoriality-dependent-pair-types open import foundation.fundamental-theorem-of-identity-types open import foundation.homotopies diff --git a/src/trees/morphisms-directed-trees.lagda.md b/src/trees/morphisms-directed-trees.lagda.md index a795b49f63..f4cb1d987a 100644 --- a/src/trees/morphisms-directed-trees.lagda.md +++ b/src/trees/morphisms-directed-trees.lagda.md @@ -12,7 +12,7 @@ open import foundation.contractible-types open import foundation.dependent-pair-types open import foundation.equality-dependent-pair-types open import foundation.equivalences -open import foundation.functions +open import foundation.function-types open import foundation.functoriality-dependent-pair-types open import foundation.homotopies open import foundation.identity-types diff --git a/src/trees/multisets.lagda.md b/src/trees/multisets.lagda.md index 79e4cb0fd1..de99986bde 100644 --- a/src/trees/multisets.lagda.md +++ b/src/trees/multisets.lagda.md @@ -9,7 +9,7 @@ module trees.multisets where ```agda open import foundation.dependent-pair-types open import foundation.empty-types -open import foundation.functions +open import foundation.function-types open import foundation.universe-levels open import trees.elementhood-relation-w-types diff --git a/src/trees/planar-binary-trees.lagda.md b/src/trees/planar-binary-trees.lagda.md index a717bc8137..c9ad78c9ae 100644 --- a/src/trees/planar-binary-trees.lagda.md +++ b/src/trees/planar-binary-trees.lagda.md @@ -9,7 +9,7 @@ module trees.planar-binary-trees where ```agda open import foundation.booleans open import foundation.empty-types -open import foundation.functions +open import foundation.function-types open import foundation.universe-levels open import trees.w-types diff --git a/src/trees/polynomial-endofunctors.lagda.md b/src/trees/polynomial-endofunctors.lagda.md index 0251bd976b..9c837c62df 100644 --- a/src/trees/polynomial-endofunctors.lagda.md +++ b/src/trees/polynomial-endofunctors.lagda.md @@ -10,7 +10,7 @@ module trees.polynomial-endofunctors where open import foundation.contractible-types open import foundation.dependent-pair-types open import foundation.equivalences -open import foundation.functions +open import foundation.function-types open import foundation.functoriality-dependent-pair-types open import foundation.fundamental-theorem-of-identity-types open import foundation.homotopies diff --git a/src/trees/rooted-morphisms-directed-trees.lagda.md b/src/trees/rooted-morphisms-directed-trees.lagda.md index 1a87beaeab..2b46217d61 100644 --- a/src/trees/rooted-morphisms-directed-trees.lagda.md +++ b/src/trees/rooted-morphisms-directed-trees.lagda.md @@ -12,7 +12,7 @@ open import foundation.binary-transport open import foundation.contractible-types open import foundation.dependent-pair-types open import foundation.equivalences -open import foundation.functions +open import foundation.function-types open import foundation.functoriality-dependent-pair-types open import foundation.homotopies open import foundation.identity-types diff --git a/src/trees/rooted-morphisms-enriched-directed-trees.lagda.md b/src/trees/rooted-morphisms-enriched-directed-trees.lagda.md index cadd6a88c4..1836a44f6d 100644 --- a/src/trees/rooted-morphisms-enriched-directed-trees.lagda.md +++ b/src/trees/rooted-morphisms-enriched-directed-trees.lagda.md @@ -8,7 +8,7 @@ module trees.rooted-morphisms-enriched-directed-trees where ```agda open import foundation.dependent-pair-types -open import foundation.functions +open import foundation.function-types open import foundation.homotopies open import foundation.identity-types open import foundation.propositions diff --git a/src/trees/small-multisets.lagda.md b/src/trees/small-multisets.lagda.md index e44f59f124..ff3d51f090 100644 --- a/src/trees/small-multisets.lagda.md +++ b/src/trees/small-multisets.lagda.md @@ -11,7 +11,7 @@ open import foundation.action-on-identifications-functions open import foundation.dependent-pair-types open import foundation.empty-types open import foundation.equivalences -open import foundation.functions +open import foundation.function-types open import foundation.functoriality-dependent-pair-types open import foundation.homotopies open import foundation.identity-types diff --git a/src/trees/underlying-trees-elements-coalgebras-polynomial-endofunctors.lagda.md b/src/trees/underlying-trees-elements-coalgebras-polynomial-endofunctors.lagda.md index 02e6f1df15..68696ff7f5 100644 --- a/src/trees/underlying-trees-elements-coalgebras-polynomial-endofunctors.lagda.md +++ b/src/trees/underlying-trees-elements-coalgebras-polynomial-endofunctors.lagda.md @@ -15,7 +15,7 @@ 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.function-types open import foundation.functoriality-dependent-pair-types open import foundation.fundamental-theorem-of-identity-types open import foundation.homotopies @@ -23,6 +23,7 @@ open import foundation.identity-types open import foundation.isolated-points open import foundation.negation open import foundation.propositions +open import foundation.transport open import foundation.type-arithmetic-empty-type open import foundation.universe-levels diff --git a/src/trees/underlying-trees-of-elements-of-w-types.lagda.md b/src/trees/underlying-trees-of-elements-of-w-types.lagda.md index 963e04ba74..04fb19fc69 100644 --- a/src/trees/underlying-trees-of-elements-of-w-types.lagda.md +++ b/src/trees/underlying-trees-of-elements-of-w-types.lagda.md @@ -13,12 +13,13 @@ 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.function-types open import foundation.homotopies open import foundation.identity-types open import foundation.isolated-points open import foundation.negation open import foundation.propositions +open import foundation.transport open import foundation.universe-levels open import graph-theory.directed-graphs diff --git a/src/trees/w-type-of-natural-numbers.lagda.md b/src/trees/w-type-of-natural-numbers.lagda.md index e5fdbf759e..96c9d8b180 100644 --- a/src/trees/w-type-of-natural-numbers.lagda.md +++ b/src/trees/w-type-of-natural-numbers.lagda.md @@ -15,7 +15,7 @@ open import foundation.contractible-types open import foundation.dependent-pair-types open import foundation.equivalences open import foundation.function-extensionality -open import foundation.functions +open import foundation.function-types open import foundation.homotopies open import foundation.identity-types open import foundation.unit-type diff --git a/src/trees/w-types.lagda.md b/src/trees/w-types.lagda.md index 6d1c8709d3..b33ac63d96 100644 --- a/src/trees/w-types.lagda.md +++ b/src/trees/w-types.lagda.md @@ -13,7 +13,7 @@ open import foundation.dependent-pair-types open import foundation.empty-types open import foundation.equivalences open import foundation.function-extensionality -open import foundation.functions +open import foundation.function-types open import foundation.fundamental-theorem-of-identity-types open import foundation.homotopies open import foundation.identity-types diff --git a/src/type-theories/dependent-type-theories.lagda.md b/src/type-theories/dependent-type-theories.lagda.md index 23b0da3069..9fb662749e 100644 --- a/src/type-theories/dependent-type-theories.lagda.md +++ b/src/type-theories/dependent-type-theories.lagda.md @@ -13,7 +13,7 @@ open import foundation.action-on-identifications-binary-functions open import foundation.action-on-identifications-functions open import foundation.dependent-pair-types open import foundation.equivalences -open import foundation.functions +open import foundation.function-types open import foundation.homotopies open import foundation.identity-types open import foundation.propositions diff --git a/src/type-theories/fibered-dependent-type-theories.lagda.md b/src/type-theories/fibered-dependent-type-theories.lagda.md index d34c5a3265..fef7eb8f07 100644 --- a/src/type-theories/fibered-dependent-type-theories.lagda.md +++ b/src/type-theories/fibered-dependent-type-theories.lagda.md @@ -11,7 +11,7 @@ module type-theories.fibered-dependent-type-theories where ```agda open import foundation.action-on-identifications-functions open import foundation.dependent-pair-types -open import foundation.functions +open import foundation.function-types open import foundation.identity-types open import foundation.transport open import foundation.universe-levels diff --git a/src/type-theories/simple-type-theories.lagda.md b/src/type-theories/simple-type-theories.lagda.md index 0e9cde7a32..227dc224ff 100644 --- a/src/type-theories/simple-type-theories.lagda.md +++ b/src/type-theories/simple-type-theories.lagda.md @@ -10,7 +10,7 @@ module type-theories.simple-type-theories where ```agda open import foundation.action-on-identifications-functions -open import foundation.functions +open import foundation.function-types open import foundation.homotopies open import foundation.identity-types open import foundation.transport diff --git a/src/type-theories/unityped-type-theories.lagda.md b/src/type-theories/unityped-type-theories.lagda.md index a810e2c9b1..fb04422101 100644 --- a/src/type-theories/unityped-type-theories.lagda.md +++ b/src/type-theories/unityped-type-theories.lagda.md @@ -12,7 +12,7 @@ module type-theories.unityped-type-theories where open import elementary-number-theory.addition-natural-numbers open import elementary-number-theory.natural-numbers -open import foundation.functions +open import foundation.function-types open import foundation.homotopies open import foundation.identity-types open import foundation.sets diff --git a/src/univalent-combinatorics/2-element-decidable-subtypes.lagda.md b/src/univalent-combinatorics/2-element-decidable-subtypes.lagda.md index 5a398b7696..f526ae39d7 100644 --- a/src/univalent-combinatorics/2-element-decidable-subtypes.lagda.md +++ b/src/univalent-combinatorics/2-element-decidable-subtypes.lagda.md @@ -22,7 +22,7 @@ open import foundation.embeddings open import foundation.equality-dependent-pair-types open import foundation.equivalences open import foundation.function-extensionality -open import foundation.functions +open import foundation.function-types open import foundation.functoriality-coproduct-types open import foundation.functoriality-dependent-pair-types open import foundation.functoriality-propositional-truncation diff --git a/src/univalent-combinatorics/2-element-subtypes.lagda.md b/src/univalent-combinatorics/2-element-subtypes.lagda.md index ba83d06dea..9abd297688 100644 --- a/src/univalent-combinatorics/2-element-subtypes.lagda.md +++ b/src/univalent-combinatorics/2-element-subtypes.lagda.md @@ -13,7 +13,7 @@ open import foundation.coproduct-types open import foundation.dependent-pair-types open import foundation.embeddings open import foundation.equivalences -open import foundation.functions +open import foundation.function-types open import foundation.functoriality-coproduct-types open import foundation.identity-types open import foundation.injective-maps diff --git a/src/univalent-combinatorics/2-element-types.lagda.md b/src/univalent-combinatorics/2-element-types.lagda.md index a2ca47cef6..589672f091 100644 --- a/src/univalent-combinatorics/2-element-types.lagda.md +++ b/src/univalent-combinatorics/2-element-types.lagda.md @@ -22,7 +22,7 @@ open import foundation.double-negation open import foundation.empty-types open import foundation.equivalence-extensionality open import foundation.fibers-of-maps -open import foundation.functions +open import foundation.function-types open import foundation.functoriality-coproduct-types open import foundation.functoriality-dependent-pair-types open import foundation.fundamental-theorem-of-identity-types diff --git a/src/univalent-combinatorics/binomial-types.lagda.md b/src/univalent-combinatorics/binomial-types.lagda.md index af2e4fd141..503e6c7b75 100644 --- a/src/univalent-combinatorics/binomial-types.lagda.md +++ b/src/univalent-combinatorics/binomial-types.lagda.md @@ -23,7 +23,7 @@ open import foundation.empty-types open import foundation.equivalences open import foundation.equivalences-maybe open import foundation.fibers-of-maps -open import foundation.functions +open import foundation.function-types open import foundation.functoriality-coproduct-types open import foundation.functoriality-dependent-pair-types open import foundation.functoriality-function-types diff --git a/src/univalent-combinatorics/cartesian-product-types.lagda.md b/src/univalent-combinatorics/cartesian-product-types.lagda.md index e7ebb5d476..92fa5af670 100644 --- a/src/univalent-combinatorics/cartesian-product-types.lagda.md +++ b/src/univalent-combinatorics/cartesian-product-types.lagda.md @@ -15,7 +15,7 @@ open import foundation.contractible-types open import foundation.decidable-equality open import foundation.dependent-pair-types open import foundation.equivalences -open import foundation.functions +open import foundation.function-types open import foundation.functoriality-cartesian-product-types open import foundation.functoriality-coproduct-types open import foundation.functoriality-dependent-pair-types diff --git a/src/univalent-combinatorics/coproduct-types.lagda.md b/src/univalent-combinatorics/coproduct-types.lagda.md index 072fd6342a..97aca66740 100644 --- a/src/univalent-combinatorics/coproduct-types.lagda.md +++ b/src/univalent-combinatorics/coproduct-types.lagda.md @@ -14,7 +14,7 @@ open import foundation.action-on-identifications-functions open import foundation.coproduct-types open import foundation.dependent-pair-types open import foundation.equivalences -open import foundation.functions +open import foundation.function-types open import foundation.functoriality-coproduct-types open import foundation.functoriality-propositional-truncation open import foundation.homotopies diff --git a/src/univalent-combinatorics/counting-decidable-subtypes.lagda.md b/src/univalent-combinatorics/counting-decidable-subtypes.lagda.md index fd000177c2..c7c3c8a88a 100644 --- a/src/univalent-combinatorics/counting-decidable-subtypes.lagda.md +++ b/src/univalent-combinatorics/counting-decidable-subtypes.lagda.md @@ -19,7 +19,7 @@ open import foundation.dependent-pair-types open import foundation.embeddings open import foundation.equivalences open import foundation.fibers-of-maps -open import foundation.functions +open import foundation.function-types open import foundation.functoriality-coproduct-types open import foundation.functoriality-dependent-pair-types open import foundation.identity-types diff --git a/src/univalent-combinatorics/counting-dependent-pair-types.lagda.md b/src/univalent-combinatorics/counting-dependent-pair-types.lagda.md index 206511c466..8958b28837 100644 --- a/src/univalent-combinatorics/counting-dependent-pair-types.lagda.md +++ b/src/univalent-combinatorics/counting-dependent-pair-types.lagda.md @@ -21,7 +21,7 @@ open import foundation.dependent-pair-types open import foundation.equality-dependent-pair-types open import foundation.equivalences open import foundation.fibers-of-maps -open import foundation.functions +open import foundation.function-types open import foundation.functoriality-coproduct-types open import foundation.functoriality-dependent-pair-types open import foundation.identity-types diff --git a/src/univalent-combinatorics/counting.lagda.md b/src/univalent-combinatorics/counting.lagda.md index 00b540f1c7..af8ee73e13 100644 --- a/src/univalent-combinatorics/counting.lagda.md +++ b/src/univalent-combinatorics/counting.lagda.md @@ -16,7 +16,7 @@ open import foundation.decidable-types open import foundation.dependent-pair-types open import foundation.empty-types open import foundation.equivalences -open import foundation.functions +open import foundation.function-types open import foundation.homotopies open import foundation.identity-types open import foundation.injective-maps diff --git a/src/univalent-combinatorics/cyclic-types.lagda.md b/src/univalent-combinatorics/cyclic-types.lagda.md index 5e52cd7589..d8e2f41a8f 100644 --- a/src/univalent-combinatorics/cyclic-types.lagda.md +++ b/src/univalent-combinatorics/cyclic-types.lagda.md @@ -21,7 +21,7 @@ open import foundation.contractible-types open import foundation.coproduct-types open import foundation.dependent-pair-types open import foundation.equivalences -open import foundation.functions +open import foundation.function-types open import foundation.functoriality-dependent-pair-types open import foundation.fundamental-theorem-of-identity-types open import foundation.homotopies diff --git a/src/univalent-combinatorics/decidable-dependent-function-types.lagda.md b/src/univalent-combinatorics/decidable-dependent-function-types.lagda.md index cc325e9130..5f6cfff00c 100644 --- a/src/univalent-combinatorics/decidable-dependent-function-types.lagda.md +++ b/src/univalent-combinatorics/decidable-dependent-function-types.lagda.md @@ -16,7 +16,7 @@ open import foundation.decidable-propositions open import foundation.decidable-types open import foundation.empty-types open import foundation.equivalences -open import foundation.functions +open import foundation.function-types open import foundation.propositional-truncations open import foundation.propositions open import foundation.unit-type diff --git a/src/univalent-combinatorics/decidable-dependent-pair-types.lagda.md b/src/univalent-combinatorics/decidable-dependent-pair-types.lagda.md index bde5d9c310..9e48dbc5b3 100644 --- a/src/univalent-combinatorics/decidable-dependent-pair-types.lagda.md +++ b/src/univalent-combinatorics/decidable-dependent-pair-types.lagda.md @@ -15,7 +15,7 @@ open import foundation.decidable-types open import foundation.dependent-pair-types open import foundation.empty-types open import foundation.equivalences -open import foundation.functions +open import foundation.function-types open import foundation.unit-type open import foundation.universe-levels diff --git a/src/univalent-combinatorics/decidable-equivalence-relations.lagda.md b/src/univalent-combinatorics/decidable-equivalence-relations.lagda.md index 1b7e9c68cf..f2d7d5fb87 100644 --- a/src/univalent-combinatorics/decidable-equivalence-relations.lagda.md +++ b/src/univalent-combinatorics/decidable-equivalence-relations.lagda.md @@ -17,7 +17,7 @@ open import foundation.decidable-relations open import foundation.decidable-types open import foundation.equivalence-relations open import foundation.equivalences -open import foundation.functions +open import foundation.function-types open import foundation.functoriality-cartesian-product-types open import foundation.functoriality-dependent-pair-types open import foundation.propositional-truncations diff --git a/src/univalent-combinatorics/decidable-subtypes.lagda.md b/src/univalent-combinatorics/decidable-subtypes.lagda.md index 8456de10e7..90a64f18aa 100644 --- a/src/univalent-combinatorics/decidable-subtypes.lagda.md +++ b/src/univalent-combinatorics/decidable-subtypes.lagda.md @@ -15,7 +15,7 @@ open import foundation.coproduct-types open import foundation.decidable-equality open import foundation.decidable-propositions open import foundation.embeddings -open import foundation.functions +open import foundation.function-types open import foundation.identity-types open import foundation.injective-maps open import foundation.propositional-truncations diff --git a/src/univalent-combinatorics/dependent-function-types.lagda.md b/src/univalent-combinatorics/dependent-function-types.lagda.md index c62633bc9c..74056f51ef 100644 --- a/src/univalent-combinatorics/dependent-function-types.lagda.md +++ b/src/univalent-combinatorics/dependent-function-types.lagda.md @@ -12,7 +12,7 @@ open import elementary-number-theory.natural-numbers open import foundation.coproduct-types open import foundation.dependent-pair-types open import foundation.equivalences -open import foundation.functions +open import foundation.function-types open import foundation.functoriality-dependent-function-types open import foundation.homotopies open import foundation.propositional-truncations diff --git a/src/univalent-combinatorics/dependent-pair-types.lagda.md b/src/univalent-combinatorics/dependent-pair-types.lagda.md index 1159c5922d..8c7dab5853 100644 --- a/src/univalent-combinatorics/dependent-pair-types.lagda.md +++ b/src/univalent-combinatorics/dependent-pair-types.lagda.md @@ -15,7 +15,7 @@ open import foundation.decidable-types open import foundation.empty-types open import foundation.equality-dependent-pair-types open import foundation.equivalences -open import foundation.functions +open import foundation.function-types open import foundation.functoriality-dependent-pair-types open import foundation.functoriality-propositional-truncation open import foundation.identity-types diff --git a/src/univalent-combinatorics/distributivity-of-set-truncation-over-finite-products.lagda.md b/src/univalent-combinatorics/distributivity-of-set-truncation-over-finite-products.lagda.md index e9548f4abc..142916afbb 100644 --- a/src/univalent-combinatorics/distributivity-of-set-truncation-over-finite-products.lagda.md +++ b/src/univalent-combinatorics/distributivity-of-set-truncation-over-finite-products.lagda.md @@ -16,7 +16,7 @@ open import foundation.dependent-pair-types open import foundation.empty-types open import foundation.equivalences open import foundation.function-extensionality -open import foundation.functions +open import foundation.function-types open import foundation.functoriality-cartesian-product-types open import foundation.functoriality-dependent-function-types open import foundation.functoriality-dependent-pair-types diff --git a/src/univalent-combinatorics/equivalences-cubes.lagda.md b/src/univalent-combinatorics/equivalences-cubes.lagda.md index 7898a11e20..89f968dd7a 100644 --- a/src/univalent-combinatorics/equivalences-cubes.lagda.md +++ b/src/univalent-combinatorics/equivalences-cubes.lagda.md @@ -14,7 +14,7 @@ open import foundation.dependent-pair-types open import foundation.equality-dependent-function-types open import foundation.equivalence-extensionality open import foundation.equivalences -open import foundation.functions +open import foundation.function-types open import foundation.fundamental-theorem-of-identity-types open import foundation.homotopies open import foundation.identity-types diff --git a/src/univalent-combinatorics/finite-choice.lagda.md b/src/univalent-combinatorics/finite-choice.lagda.md index 3d3657c62c..083f1e2af2 100644 --- a/src/univalent-combinatorics/finite-choice.lagda.md +++ b/src/univalent-combinatorics/finite-choice.lagda.md @@ -17,7 +17,7 @@ open import foundation.empty-types open import foundation.equivalences open import foundation.fiber-inclusions open import foundation.fibers-of-maps -open import foundation.functions +open import foundation.function-types open import foundation.functoriality-dependent-function-types open import foundation.functoriality-dependent-pair-types open import foundation.functoriality-propositional-truncation diff --git a/src/univalent-combinatorics/finite-types.lagda.md b/src/univalent-combinatorics/finite-types.lagda.md index e5532d5d53..2791564474 100644 --- a/src/univalent-combinatorics/finite-types.lagda.md +++ b/src/univalent-combinatorics/finite-types.lagda.md @@ -19,7 +19,7 @@ open import foundation.decidable-types open import foundation.dependent-pair-types open import foundation.empty-types open import foundation.equivalences -open import foundation.functions +open import foundation.function-types open import foundation.functoriality-coproduct-types open import foundation.functoriality-dependent-pair-types open import foundation.functoriality-propositional-truncation diff --git a/src/univalent-combinatorics/finitely-presented-types.lagda.md b/src/univalent-combinatorics/finitely-presented-types.lagda.md index 32ea58758f..e1e689bca5 100644 --- a/src/univalent-combinatorics/finitely-presented-types.lagda.md +++ b/src/univalent-combinatorics/finitely-presented-types.lagda.md @@ -12,7 +12,7 @@ open import elementary-number-theory.natural-numbers open import foundation.dependent-pair-types open import foundation.equivalences open import foundation.fibers-of-maps -open import foundation.functions +open import foundation.function-types open import foundation.propositional-truncations open import foundation.propositions open import foundation.set-presented-types diff --git a/src/univalent-combinatorics/inhabited-finite-types.lagda.md b/src/univalent-combinatorics/inhabited-finite-types.lagda.md index aa88cfe405..01f01c91af 100644 --- a/src/univalent-combinatorics/inhabited-finite-types.lagda.md +++ b/src/univalent-combinatorics/inhabited-finite-types.lagda.md @@ -10,7 +10,7 @@ module univalent-combinatorics.inhabited-finite-types where open import elementary-number-theory.natural-numbers open import foundation.equivalences -open import foundation.functions +open import foundation.function-types open import foundation.functoriality-dependent-function-types open import foundation.identity-types open import foundation.inhabited-types diff --git a/src/univalent-combinatorics/necklaces.lagda.md b/src/univalent-combinatorics/necklaces.lagda.md index c09263b6e5..895042674e 100644 --- a/src/univalent-combinatorics/necklaces.lagda.md +++ b/src/univalent-combinatorics/necklaces.lagda.md @@ -12,7 +12,7 @@ open import elementary-number-theory.natural-numbers open import foundation.dependent-pair-types open import foundation.equivalences open import foundation.function-extensionality -open import foundation.functions +open import foundation.function-types open import foundation.homotopies open import foundation.identity-types open import foundation.structure-identity-principle diff --git a/src/univalent-combinatorics/orientations-complete-undirected-graph.lagda.md b/src/univalent-combinatorics/orientations-complete-undirected-graph.lagda.md index 918d2c7aaa..86c83b6578 100644 --- a/src/univalent-combinatorics/orientations-complete-undirected-graph.lagda.md +++ b/src/univalent-combinatorics/orientations-complete-undirected-graph.lagda.md @@ -34,7 +34,7 @@ open import foundation.equivalence-extensionality open import foundation.equivalence-relations open import foundation.equivalences open import foundation.function-extensionality -open import foundation.functions +open import foundation.function-types open import foundation.functoriality-dependent-pair-types open import foundation.functoriality-propositional-truncation open import foundation.homotopies diff --git a/src/univalent-combinatorics/partitions.lagda.md b/src/univalent-combinatorics/partitions.lagda.md index 9f9df8a6d8..e3ebbded00 100644 --- a/src/univalent-combinatorics/partitions.lagda.md +++ b/src/univalent-combinatorics/partitions.lagda.md @@ -14,7 +14,7 @@ open import foundation.equality-cartesian-product-types open import foundation.equivalence-extensionality open import foundation.equivalence-relations open import foundation.equivalences -open import foundation.functions +open import foundation.function-types open import foundation.functoriality-dependent-pair-types open import foundation.homotopies open import foundation.identity-types diff --git a/src/univalent-combinatorics/pi-finite-types.lagda.md b/src/univalent-combinatorics/pi-finite-types.lagda.md index 9a54c3cbf7..f9c6d055ef 100644 --- a/src/univalent-combinatorics/pi-finite-types.lagda.md +++ b/src/univalent-combinatorics/pi-finite-types.lagda.md @@ -27,7 +27,7 @@ open import foundation.equivalences open import foundation.fiber-inclusions open import foundation.fibers-of-maps open import foundation.function-extensionality -open import foundation.functions +open import foundation.function-types open import foundation.functoriality-coproduct-types open import foundation.functoriality-dependent-function-types open import foundation.functoriality-dependent-pair-types diff --git a/src/univalent-combinatorics/pigeonhole-principle.lagda.md b/src/univalent-combinatorics/pigeonhole-principle.lagda.md index 93da23f9af..fcba23c7b8 100644 --- a/src/univalent-combinatorics/pigeonhole-principle.lagda.md +++ b/src/univalent-combinatorics/pigeonhole-principle.lagda.md @@ -16,7 +16,7 @@ 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.function-types open import foundation.homotopies open import foundation.identity-types open import foundation.injective-maps diff --git a/src/univalent-combinatorics/sequences-finite-types.lagda.md b/src/univalent-combinatorics/sequences-finite-types.lagda.md index 33035d317f..dbe3669be3 100644 --- a/src/univalent-combinatorics/sequences-finite-types.lagda.md +++ b/src/univalent-combinatorics/sequences-finite-types.lagda.md @@ -16,7 +16,7 @@ open import foundation.action-on-identifications-functions open import foundation.cartesian-product-types open import foundation.dependent-pair-types open import foundation.equivalences -open import foundation.functions +open import foundation.function-types open import foundation.identity-types open import foundation.injective-maps open import foundation.negation diff --git a/src/univalent-combinatorics/set-quotients-of-index-two.lagda.md b/src/univalent-combinatorics/set-quotients-of-index-two.lagda.md index b7690e6030..ed82ecd316 100644 --- a/src/univalent-combinatorics/set-quotients-of-index-two.lagda.md +++ b/src/univalent-combinatorics/set-quotients-of-index-two.lagda.md @@ -18,7 +18,7 @@ open import foundation.embeddings open import foundation.empty-types open import foundation.equivalence-relations open import foundation.equivalences -open import foundation.functions +open import foundation.function-types open import foundation.functoriality-set-quotients open import foundation.identity-types open import foundation.injective-maps diff --git a/src/univalent-combinatorics/standard-finite-types.lagda.md b/src/univalent-combinatorics/standard-finite-types.lagda.md index e4953d76c8..91f86384f6 100644 --- a/src/univalent-combinatorics/standard-finite-types.lagda.md +++ b/src/univalent-combinatorics/standard-finite-types.lagda.md @@ -21,7 +21,7 @@ open import foundation.empty-types open import foundation.equality-coproduct-types open import foundation.equivalences open import foundation.equivalences-maybe -open import foundation.functions +open import foundation.function-types open import foundation.homotopies open import foundation.identity-types open import foundation.injective-maps diff --git a/src/univalent-combinatorics/universal-property-standard-finite-types.lagda.md b/src/univalent-combinatorics/universal-property-standard-finite-types.lagda.md index 42ab18988b..29a39526bd 100644 --- a/src/univalent-combinatorics/universal-property-standard-finite-types.lagda.md +++ b/src/univalent-combinatorics/universal-property-standard-finite-types.lagda.md @@ -13,7 +13,7 @@ open import foundation.cartesian-product-types open import foundation.contractible-types open import foundation.coproduct-types open import foundation.equivalences -open import foundation.functions +open import foundation.function-types open import foundation.functoriality-cartesian-product-types open import foundation.unit-type open import foundation.universal-property-empty-type