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Make IsFunctor a seperate record

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Frederik Hanghøj Iversen 2018-01-30 16:23:36 +01:00
parent 52dea06df9
commit c87a6fb469
4 changed files with 88 additions and 48 deletions
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39
src/Cat/Categories/Cat.agda
View file

@ -21,6 +21,7 @@ eqpair : ∀ {ℓa ℓb} {A : Set ℓa} {B : Set ℓb} {a a' : A} {b b' : B}
eqpair eqa eqb i = eqa i , eqb i
open Functor
open IsFunctor
open Category
-- The category of categories
@ -36,11 +37,11 @@ module _ (ℓ ℓ' : Level) where
eq→ = refl
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))
((h ∘f (g ∘f f)) .isFunctor .ident)
(((h ∘f g) ∘f f) .isFunctor .ident)
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))
((h ∘f (g ∘f f)) .isFunctor .distrib) (((h ∘f g) ∘f f) .isFunctor .distrib)
assc : h ∘f (g ∘f f) ≡ (h ∘f g) ∘f f
assc = Functor≡ eq* eq→ eqI eqD
@ -59,12 +60,12 @@ module _ (ℓ ℓ' : Level) where
postulate
eqI-r : PathP (λ i → {c : ℂ .Object}
→ PathP (λ _ → Arrow 𝔻 (func* F c) (func* F c)) (func→ F (ℂ .𝟙)) (𝔻 .𝟙))
(ident (F ∘f identity)) (ident F)
((F ∘f identity) .isFunctor .ident) (F .isFunctor .ident)
eqD-r : PathP
(λ i →
{A B C : ℂ .Object} {f : ℂ .Arrow A B} {g : ℂ .Arrow B C} →
eq→ i (ℂ ._⊕_ g f) ≡ 𝔻 ._⊕_ (eq→ i g) (eq→ i f))
((F ∘f identity) .distrib) (distrib F)
((F ∘f identity) .isFunctor .distrib) (F .isFunctor .distrib)
ident-r : F ∘f identity ≡ F
ident-r = Functor≡ eq* eq→ eqI-r eqD-r
module _ where
@ -75,10 +76,10 @@ module _ (ℓ ℓ' : Level) where
(λ i → {x y : Object ℂ} → ℂ .Arrow x y → 𝔻 .Arrow (eq* i x) (eq* i y))
((identity ∘f F) .func→) (F .func→)
eqI : PathP (λ i → ∀ {A : ℂ .Object} → eq→ i (ℂ .𝟙 {A}) ≡ 𝔻 .𝟙 {eq* i A})
(ident (identity ∘f F)) (ident F)
((identity ∘f F) .isFunctor .ident) (F .isFunctor .ident)
eqD : PathP (λ i → {A B C : ℂ .Object} {f : ℂ .Arrow A B} {g : ℂ .Arrow B C}
→ eq→ i (ℂ ._⊕_ g f) ≡ 𝔻 ._⊕_ (eq→ i g) (eq→ i f))
(distrib (identity ∘f F)) (distrib F)
((identity ∘f F) .isFunctor .distrib) (F .isFunctor .distrib)
ident-l : identity ∘f F ≡ F
ident-l = Functor≡ eq* eq→ eqI eqD
@ -134,10 +135,10 @@ module _ {ℓ ℓ' : Level} where
}
proj₁ : Arrow Catt :product: ℂ
proj₁ = record { func* = fst ; func→ = fst ; ident = refl ; distrib = refl }
proj₁ = record { func* = fst ; func→ = fst ; isFunctor = record { ident = refl ; distrib = refl } }
proj₂ : Arrow Catt :product: 𝔻
proj₂ = record { func* = snd ; func→ = snd ; ident = refl ; distrib = refl }
proj₂ = record { func* = snd ; func→ = snd ; isFunctor = record { ident = refl ; distrib = refl } }
module _ {X : Object Catt} (x₁ : Arrow Catt X ℂ) (x₂ : Arrow Catt X 𝔻) where
open Functor
@ -149,9 +150,14 @@ module _ {ℓ ℓ' : Level} where
x = record
{ func* = λ x → (func* x₁) x , (func* x₂) x
; func→ = λ x → func→ x₁ x , func→ x₂ x
; ident = lift-eq (ident x₁) (ident x₂)
; distrib = lift-eq (distrib x₁) (distrib x₂)
; isFunctor = record
{ ident = lift-eq x₁.ident x₂.ident
; distrib = lift-eq x₁.distrib x₂.distrib
}
}
where
open module x₁ = IsFunctor (x₁ .isFunctor)
open module x₂ = IsFunctor (x₂ .isFunctor)
-- Need to "lift equality of functors"
-- If I want to do this like I do it for pairs it's gonna be a pain.
@ -260,10 +266,12 @@ module _ (ℓ : Level) where
:func→: {c} {c} (identityNat F , ℂ .𝟙) ≡⟨⟩
(identityTrans F C 𝔻⊕ F .func→ (ℂ .𝟙)) ≡⟨⟩
𝔻 .𝟙 𝔻⊕ F .func→ (ℂ .𝟙) ≡⟨ proj₂ 𝔻.ident ⟩
F .func→ (ℂ .𝟙) ≡⟨ F .ident ⟩
F .func→ (ℂ .𝟙) ≡⟨ F.ident ⟩
𝔻 .𝟙 ∎
where
open module 𝔻 = IsCategory (𝔻 .isCategory)
open module F = IsFunctor (F .isFunctor)
module _ {F×A G×B H×C : Functor ℂ 𝔻 × ℂ .Object} where
F = F×A .proj₁
A = F×A .proj₂
@ -302,7 +310,7 @@ module _ (ℓ : Level) where
≡ (η C 𝔻⊕ G .func→ g) 𝔻⊕ (θ B 𝔻⊕ F .func→ f)
:distrib: = begin
(ηθ C) 𝔻⊕ F .func→ (g ℂ⊕ f) ≡⟨ ηθNat (g ℂ⊕ f) ⟩
H .func→ (g ℂ⊕ f) 𝔻⊕ (ηθ A) ≡⟨ cong (λ φ → φ 𝔻⊕ ηθ A) (H .distrib) ⟩
H .func→ (g ℂ⊕ f) 𝔻⊕ (ηθ A) ≡⟨ cong (λ φ → φ 𝔻⊕ ηθ A) (H.distrib) ⟩
(H .func→ g 𝔻⊕ H .func→ f) 𝔻⊕ (ηθ A) ≡⟨ sym assoc ⟩
H .func→ g 𝔻⊕ (H .func→ f 𝔻⊕ (ηθ A)) ≡⟨⟩
H .func→ g 𝔻⊕ (H .func→ f 𝔻⊕ (ηθ A)) ≡⟨ cong (λ φ → H .func→ g 𝔻⊕ φ) assoc ⟩
@ -314,14 +322,17 @@ module _ (ℓ : Level) where
(η C 𝔻⊕ G .func→ g) 𝔻⊕ (θ B 𝔻⊕ F .func→ f) ∎
where
open IsCategory (𝔻 .isCategory)
open module H = IsFunctor (H .isFunctor)
:eval: : Functor ((:obj: ×p ℂ) .Product.obj) 𝔻
:eval: = record
{ func* = :func*:
; func→ = λ {dom} {cod} → :func→: {dom} {cod}
; ident = λ {o} → :ident: {o}
; isFunctor = record
{ ident = λ {o} → :ident: {o}
; distrib = λ {f u n k y} → :distrib: {f} {u} {n} {k} {y}
}
}
module _ (𝔸 : Category ℓ ℓ) (F : Functor ((𝔸 ×p ℂ) .Product.obj) 𝔻) where
open HasProducts (hasProducts {ℓ} {ℓ}) using (parallelProduct)

8
src/Cat/Categories/Sets.agda
View file

@ -50,9 +50,11 @@ representable : ∀ {ℓ ℓ'} {ℂ : Category ℓ ℓ'} → Category.Object ℂ
representable {ℂ = ℂ} A = record
{ func* = λ B → ℂ .Arrow A B
; func→ = ℂ ._⊕_
; ident = funExt λ _ → snd ident
; isFunctor = record
{ ident = funExt λ _ → snd ident
; distrib = funExt λ x → sym assoc
}
}
where
open IsCategory (ℂ .isCategory)
@ -65,8 +67,10 @@ presheaf : {ℓ ℓ' : Level} {ℂ : Category ℓ ℓ'} → Category.Object (Opp
presheaf {ℂ = ℂ} B = record
{ func* = λ A → ℂ .Arrow A B
; func→ = λ f g → ℂ ._⊕_ g f
; ident = funExt λ x → fst ident
; isFunctor = record
{ ident = funExt λ x → fst ident
; distrib = funExt λ x → assoc
}
}
where
open IsCategory (ℂ .isCategory)

4
src/Cat/Category/Properties.agda
View file

@ -87,6 +87,8 @@ module _ {ℓ : Level} {ℂ : Category ℓ ℓ} where
yoneda = record
{ func* = prshf
; func→ = :func→:
; ident = :ident:
; isFunctor = record
{ ident = :ident:
; distrib = {!!}
}
}

57
src/Cat/Functor.agda
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@ -6,36 +6,55 @@ open import Function
open import Cat.Category
record Functor {ℓc ℓc' ℓd ℓd'} (C : Category ℓc ℓc') (D : Category ℓd ℓd')
open Category
module _ {ℓc ℓc' ℓd ℓd'} (ℂ : Category ℓc ℓc') (𝔻 : Category ℓd ℓd') where
record IsFunctor
(func* : ℂ .Object → 𝔻 .Object)
(func→ : {A B : ℂ .Object} → ℂ .Arrow A B → 𝔻 .Arrow (func* A) (func* B))
: Set (ℓc ⊔ ℓc' ⊔ ℓd ⊔ ℓd') where
open Category
field
func* : C .Object → D .Object
func→ : {dom cod : C .Object} → C .Arrow dom cod → D .Arrow (func* dom) (func* cod)
ident : { c : C .Object } → func→ (C .𝟙 {c}) ≡ D .𝟙 {func* c}
ident : { c : ℂ .Object } → func→ (ℂ .𝟙 {c}) ≡ 𝔻 .𝟙 {func* c}
-- TODO: Avoid use of ugly explicit arguments somehow.
-- This guy managed to do it:
-- https://github.com/copumpkin/categories/blob/master/Categories/Functor/Core.agda
distrib : { c c' c'' : C .Object} {a : C .Arrow c c'} {a' : C .Arrow c' c''}
→ func→ (C ._⊕_ a' a) ≡ D ._⊕_ (func→ a') (func→ a)
distrib : {A B C : ℂ .Object} {f : ℂ .Arrow A B} {g : ℂ .Arrow B C}
→ func→ (ℂ ._⊕_ g f) ≡ 𝔻 ._⊕_ (func→ g) (func→ f)
record Functor : Set (ℓc ⊔ ℓc' ⊔ ℓd ⊔ ℓd') where
field
func* : ℂ .Object → 𝔻 .Object
func→ : {A B : ℂ .Object} → ℂ .Arrow A B → 𝔻 .Arrow (func* A) (func* B)
{{isFunctor}} : IsFunctor func* func→
open IsFunctor
open Functor
open Category
module _ {ℓ ℓ' : Level} {ℂ 𝔻 : Category ℓ ℓ'} where
private
_ℂ⊕_ = ℂ ._⊕_
-- IsFunctor≡ : ∀ {A B : ℂ .Object} {func* : ℂ .Object → 𝔻 .Object} {func→ : {A B : ℂ .Object} → ℂ .Arrow A B → 𝔻 .Arrow (func* A) (func* B)} {F G : IsFunctor ℂ 𝔻 func* func→}
-- → (eqI : PathP (λ i → ∀ {A : ℂ .Object} → func→ (ℂ .𝟙 {A}) ≡ 𝔻 .𝟙 {func* A})
-- (F .ident) (G .ident))
-- → (eqD : PathP (λ i → {A B C : ℂ .Object} {f : ℂ .Arrow A B} {g : ℂ .Arrow B C}
-- → func→ (ℂ ._⊕_ g f) ≡ 𝔻 ._⊕_ (func→ g) (func→ f))
-- (F .distrib) (G .distrib))
-- → F ≡ G
-- IsFunctor≡ eqI eqD i = record { ident = eqI i ; distrib = eqD i }
Functor≡ : {F G : Functor ℂ 𝔻}
→ (eq* : F .func* ≡ G .func*)
→ (eq→ : PathP (λ i → ∀ {x y} → ℂ .Arrow x y → 𝔻 .Arrow (eq* i x) (eq* i y))
(F .func→) (G .func→))
-- → (eqIsF : PathP (λ i → IsFunctor ℂ 𝔻 (eq* i) (eq→ i)) (F .isFunctor) (G .isFunctor))
→ (eqI : PathP (λ i → ∀ {A : ℂ .Object} → eq→ i (ℂ .𝟙 {A}) ≡ 𝔻 .𝟙 {eq* i A})
(ident F) (ident G))
(F .isFunctor .ident) (G .isFunctor .ident))
→ (eqD : PathP (λ i → {A B C : ℂ .Object} {f : ℂ .Arrow A B} {g : ℂ .Arrow B C}
→ eq→ i (ℂ ._⊕_ g f) ≡ 𝔻 ._⊕_ (eq→ i g) (eq→ i f))
(distrib F) (distrib G))
(F .isFunctor .distrib) (G .isFunctor .distrib))
→ F ≡ G
Functor≡ eq* eq→ eqI eqD i = record { func* = eq* i ; func→ = eq→ i ; ident = eqI i ; distrib = eqD i }
Functor≡ eq* eq→ eqI eqD i = record { func* = eq* i ; func→ = eq→ i ; isFunctor = record { ident = eqI i ; distrib = eqD i } }
module _ {ℓ ℓ' : Level} {A B C : Category ℓ ℓ'} (F : Functor B C) (G : Functor A B) where
private
@ -51,8 +70,8 @@ module _ {ℓ ℓ' : Level} {A B C : Category ℓ ℓ'} (F : Functor B C) (G : F
dist : (F→ ∘ G→) (α1 A⊕ α0) ≡ (F→ ∘ G→) α1 C⊕ (F→ ∘ G→) α0
dist = begin
(F→ ∘ G→) (α1 A⊕ α0) ≡⟨ refl ⟩
F→ (G→ (α1 A⊕ α0)) ≡⟨ cong F→ (G .distrib)⟩
F→ ((G→ α1) B⊕ (G→ α0)) ≡⟨ F .distrib ⟩
F→ (G→ (α1 A⊕ α0)) ≡⟨ cong F→ (G .isFunctor .distrib)⟩
F→ ((G→ α1) B⊕ (G→ α0)) ≡⟨ F .isFunctor .distrib ⟩
(F→ ∘ G→) α1 C⊕ (F→ ∘ G→) α0 ∎
_∘f_ : Functor A C
@ -60,19 +79,23 @@ module _ {ℓ ℓ' : Level} {A B C : Category ℓ ℓ'} (F : Functor B C) (G : F
record
{ func* = F* ∘ G*
; func→ = F→ ∘ G→
; ident = begin
; isFunctor = record
{ ident = begin
(F→ ∘ G→) (A .𝟙) ≡⟨ refl ⟩
F→ (G→ (A .𝟙)) ≡⟨ cong F→ (G .ident)⟩
F→ (B .𝟙) ≡⟨ F .ident ⟩
F→ (G→ (A .𝟙)) ≡⟨ cong F→ (G .isFunctor .ident)⟩
F→ (B .𝟙) ≡⟨ F .isFunctor .ident ⟩
C .𝟙 ∎
; distrib = dist
}
}
-- The identity functor
identity : ∀ {ℓ ℓ'} → {C : Category ℓ ℓ'} → Functor C C
identity = record
{ func* = λ x → x
; func→ = λ x → x
; ident = refl
; isFunctor = record
{ ident = refl
; distrib = refl
}
}