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Make some names more explicit

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Frederik Hanghøj Iversen 2018-01-21 19:23:24 +01:00
parent 26d210dcc3
commit 922570a5bd
1 changed files with 32 additions and 14 deletions
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42
src/Cat/Categories/Cat.agda
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@ -24,9 +24,9 @@ open Functor
open Category open Category
module _ {ℓ ℓ' : Level} {A B : Category ℓ ℓ'} where module _ {ℓ ℓ' : Level} {A B : Category ℓ ℓ'} where
lift-eq-functors : {f g : Functor A B} lift-eq-functors : {f g : Functor A B}
→ (eq* : Functor.func* f ≡ Functor.func* g) → (eq* : f .func* ≡ g .func*)
→ (eq→ : PathP (λ i → ∀ {x y} → Arrow A x y → Arrow B (eq* i x) (eq* i y)) → (eq→ : PathP (λ i → ∀ {x y} → A .Arrow x y → B .Arrow (eq* i x) (eq* i y))
(func→ f) (func→ g)) (f .func→) (g .func→))
-- → (eq→ : Functor.func→ f ≡ {!!}) -- Functor.func→ g) -- → (eq→ : Functor.func→ f ≡ {!!}) -- Functor.func→ g)
-- Use PathP -- Use PathP
-- directly to show heterogeneous equalities by using previous -- directly to show heterogeneous equalities by using previous
@ -43,8 +43,25 @@ module _ {ℓ ℓ' : Level} {A B : Category ℓ ℓ'} where
module _ {ℓ ℓ' : Level} where module _ {ℓ ℓ' : Level} where
private private
module _ {A B C D : Category ℓ ℓ'} {f : Functor A B} {g : Functor B C} {h : Functor C D} where module _ {A B C D : Category ℓ ℓ'} {f : Functor A B} {g : Functor B C} {h : Functor C D} where
postulate assc : h ∘f (g ∘f f) ≡ (h ∘f g) ∘f f eq* : func* (h ∘f (g ∘f f)) ≡ func* ((h ∘f g) ∘f f)
-- assc = lift-eq-functors refl refl {!refl!} λ i j → {!!} eq* = refl
eq→ : PathP
(λ i → {x y : A .Object} → A .Arrow x y → D .Arrow (eq* i x) (eq* i y))
(func→ (h ∘f (g ∘f f))) (func→ ((h ∘f g) ∘f f))
eq→ = refl
id-l = (h ∘f (g ∘f f)) .ident -- = func→ (h ∘f (g ∘f f)) (𝟙 A) ≡ 𝟙 D
id-r = ((h ∘f g) ∘f f) .ident -- = func→ ((h ∘f g) ∘f f) (𝟙 A) ≡ 𝟙 D
postulate eqI : PathP
(λ i → ∀ {c : A .Object} → eq→ i (A .𝟙 {c}) ≡ D .𝟙 {eq* i c})
(ident ((h ∘f (g ∘f f))))
(ident ((h ∘f g) ∘f f))
postulate eqD : PathP (λ i → { c c' c'' : A .Object} {a : A .Arrow c c'} {a' : A .Arrow c' c''}
→ eq→ i (A ._⊕_ a' a) ≡ D ._⊕_ (eq→ i a') (eq→ i a))
(distrib (h ∘f (g ∘f f))) (distrib ((h ∘f g) ∘f f))
-- eqD = {!!}
assc : h ∘f (g ∘f f) ≡ (h ∘f g) ∘f f
assc = lift-eq-functors eq* eq→ eqI eqD
module _ {A B : Category ℓ ℓ'} {f : Functor A B} where module _ {A B : Category ℓ ℓ'} {f : Functor A B} where
lem : (func* f) ∘ (func* (identity {C = A})) ≡ func* f lem : (func* f) ∘ (func* (identity {C = A})) ≡ func* f
@ -71,9 +88,10 @@ module _ {ℓ ℓ' : Level} where
; 𝟙 = identity ; 𝟙 = identity
; _⊕_ = _∘f_ ; _⊕_ = _∘f_
-- What gives here? Why can I not name the variables directly? -- What gives here? Why can I not name the variables directly?
; isCategory = {!!} ; isCategory = record
-- ; assoc = λ {_ _ _ _ f g h} → assc {f = f} {g = g} {h = h} { assoc = λ {_ _ _ _ f g h} → assc {f = f} {g = g} {h = h}
-- ; ident = ident-r , ident-l ; ident = ident-r , ident-l
}
} }
module _ {ℓ : Level} (C D : Category ℓ ℓ) where module _ {ℓ : Level} (C D : Category ℓ ℓ) where
@ -132,11 +150,11 @@ module _ {ℓ : Level} (C D : Category ℓ ℓ) where
-- Need to "lift equality of functors" -- Need to "lift equality of functors"
-- If I want to do this like I do it for pairs it's gonna be a pain. -- If I want to do this like I do it for pairs it's gonna be a pain.
isUniqL : (Cat ⊕ proj₁) x ≡ x₁ postulate isUniqL : (Cat ⊕ proj₁) x ≡ x₁
isUniqL = lift-eq-functors refl refl {!!} {!!} -- isUniqL = lift-eq-functors refl refl {!!} {!!}
isUniqR : (Cat ⊕ proj₂) x ≡ x₂ postulate isUniqR : (Cat ⊕ proj₂) x ≡ x₂
isUniqR = lift-eq-functors refl refl {!!} {!!} -- isUniqR = lift-eq-functors refl refl {!!} {!!}
isUniq : (Cat ⊕ proj₁) x ≡ x₁ × (Cat ⊕ proj₂) x ≡ x₂ isUniq : (Cat ⊕ proj₁) x ≡ x₁ × (Cat ⊕ proj₂) x ≡ x₂
isUniq = isUniqL , isUniqR isUniq = isUniqL , isUniqR