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Improve and add a few path operators #746

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merged 4 commits into from
Apr 25, 2022
Merged

Improve and add a few path operators #746

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c6316a5
Change type symP for better goal inference.
Mar 27, 2022
f839c50
Add implicit args to substRefl in order to be able to specify A.
Mar 27, 2022
b4c4cf9
fromPathP⁻ and toPathP⁻
Mar 27, 2022
d1467f7
Adapt to changed symP.
Mar 27, 2022
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3 changes: 2 additions & 1 deletion Cubical/Categories/Category/Properties.agda
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Expand Up @@ -83,7 +83,8 @@ module _ {C : Category ℓ ℓ'} where
→ (f : C [ x , y ]) (g : C [ y , z ]) (g' : C [ y' , z ])
→ (r : PathP (λ i → C [ p i , z ]) g g')
→ f ⋆⟨ C ⟩ g ≡ seqP {p = p} f g'
lCatWhiskerP f g g' r = cong (λ v → f ⋆⟨ C ⟩ v) (sym (fromPathP (symP r)))
lCatWhiskerP {z = z} {p = p} f g g' r =
cong (λ v → f ⋆⟨ C ⟩ v) (sym (fromPathP (symP {A = λ i → C [ p (~ i) , z ]} r)))

rCatWhiskerP : {x y' y z : C .ob} {p : y' ≡ y}
→ (f' : C [ x , y' ]) (f : C [ x , y ]) (g : C [ y , z ])
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7 changes: 7 additions & 0 deletions Cubical/Foundations/Path.agda
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Expand Up @@ -22,6 +22,13 @@ cong′ : ∀ {B : Type ℓ'} (f : A → B) {x y : A} (p : x ≡ y)
cong′ f = cong f
{-# INLINE cong′ #-}

module _ {A : I → Type ℓ} {x : A i0} {y : A i1} where
toPathP⁻ : x ≡ transport⁻ (λ i → A i) y → PathP A x y
toPathP⁻ p = symP (toPathP (sym p))

fromPathP⁻ : PathP A x y → x ≡ transport⁻ (λ i → A i) y
fromPathP⁻ p = sym (fromPathP {A = λ i → A (~ i)} (symP p))

PathP≡Path : ∀ (P : I → Type ℓ) (p : P i0) (q : P i1) →
PathP P p q ≡ Path (P i1) (transport (λ i → P i) p) q
PathP≡Path P p q i = PathP (λ j → P (i ∨ j)) (transport-filler (λ j → P j) p i) q
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6 changes: 3 additions & 3 deletions Cubical/Foundations/Prelude.agda
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Expand Up @@ -50,8 +50,8 @@ sym : x ≡ y → y ≡ x
sym p i = p (~ i)
{-# INLINE sym #-}

symP : {A : I → Type ℓ} → {x : A i0} → {y : A i1} →
(p : PathP A x y) → PathP (λ i → A (~ i)) y x
symP : {A : I → Type ℓ} → {x : A i1} → {y : A i0} →
(p : PathP (λ i → A (~ i)) x y) → PathP A y x
symP p j = p (~ j)

cong : (f : (a : A) → B a) (p : x ≡ y) →
Expand Down Expand Up @@ -261,7 +261,7 @@ subst2 : ∀ {ℓ' ℓ''} {B : Type ℓ'} {z w : B} (C : A → B → Type ℓ'')
(p : x ≡ y) (q : z ≡ w) → C x z → C y w
subst2 B p q b = transport (λ i → B (p i) (q i)) b

substRefl : (px : B x) → subst B refl px ≡ px
substRefl : ∀ {B : A → Type ℓ} {x} → (px : B x) → subst B refl px ≡ px
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Why add these to the type?

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@anuyts anuyts Apr 19, 2022 •
edited
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I think I needed to explicitly pass the value for A and ℓ somewhere. The type of B defines a name for A and ℓ.

substRefl px = transportRefl px

subst-filler : (B : A → Type ℓ') (p : x ≡ y) (b : B x)
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