144 lines
4.3 KiB
Agda
144 lines
4.3 KiB
Agda
{-# OPTIONS --cubical #-}
|
||
module Cat.Category.Functor where
|
||
|
||
open import Agda.Primitive
|
||
open import Cubical
|
||
open import Function
|
||
|
||
open import Cat.Category
|
||
|
||
open Category hiding (_∘_ ; raw)
|
||
|
||
module _ {ℓc ℓc' ℓd ℓd'}
|
||
(ℂ : Category ℓc ℓc')
|
||
(𝔻 : Category ℓd ℓd')
|
||
where
|
||
|
||
private
|
||
ℓ = ℓc ⊔ ℓc' ⊔ ℓd ⊔ ℓd'
|
||
𝓤 = Set ℓ
|
||
|
||
record RawFunctor : 𝓤 where
|
||
field
|
||
func* : Object ℂ → Object 𝔻
|
||
func→ : ∀ {A B} → ℂ [ A , B ] → 𝔻 [ func* A , func* B ]
|
||
|
||
record IsFunctor (F : RawFunctor) : 𝓤 where
|
||
open RawFunctor F
|
||
field
|
||
ident : {c : Object ℂ} → func→ (𝟙 ℂ {c}) ≡ 𝟙 𝔻 {func* c}
|
||
distrib : {A B C : Object ℂ} {f : ℂ [ A , B ]} {g : ℂ [ B , C ]}
|
||
→ func→ (ℂ [ g ∘ f ]) ≡ 𝔻 [ func→ g ∘ func→ f ]
|
||
|
||
record Functor : Set (ℓc ⊔ ℓc' ⊔ ℓd ⊔ ℓd') where
|
||
field
|
||
raw : RawFunctor
|
||
{{isFunctor}} : IsFunctor raw
|
||
|
||
private
|
||
module R = RawFunctor raw
|
||
|
||
func* : Object ℂ → Object 𝔻
|
||
func* = R.func*
|
||
|
||
func→ : ∀ {A B} → ℂ [ A , B ] → 𝔻 [ func* A , func* B ]
|
||
func→ = R.func→
|
||
|
||
open IsFunctor
|
||
open Functor
|
||
|
||
module _
|
||
{ℓa ℓb : Level}
|
||
{ℂ 𝔻 : Category ℓa ℓb}
|
||
{F : RawFunctor ℂ 𝔻}
|
||
where
|
||
private
|
||
module 𝔻 = IsCategory (isCategory 𝔻)
|
||
|
||
propIsFunctor : isProp (IsFunctor _ _ F)
|
||
propIsFunctor isF0 isF1 i = record
|
||
{ ident = 𝔻.arrowIsSet _ _ isF0.ident isF1.ident i
|
||
; distrib = 𝔻.arrowIsSet _ _ isF0.distrib isF1.distrib i
|
||
}
|
||
where
|
||
module isF0 = IsFunctor isF0
|
||
module isF1 = IsFunctor isF1
|
||
|
||
-- Alternate version of above where `F` is indexed by an interval
|
||
module _
|
||
{ℓa ℓb : Level}
|
||
{ℂ 𝔻 : Category ℓa ℓb}
|
||
{F : I → RawFunctor ℂ 𝔻}
|
||
where
|
||
private
|
||
module 𝔻 = IsCategory (isCategory 𝔻)
|
||
IsProp' : {ℓ : Level} (A : I → Set ℓ) → Set ℓ
|
||
IsProp' A = (a0 : A i0) (a1 : A i1) → A [ a0 ≡ a1 ]
|
||
|
||
IsFunctorIsProp' : IsProp' λ i → IsFunctor _ _ (F i)
|
||
IsFunctorIsProp' isF0 isF1 = lemPropF {B = IsFunctor ℂ 𝔻}
|
||
(\ F → propIsFunctor {F = F}) (\ i → F i)
|
||
where
|
||
open import Cubical.NType.Properties using (lemPropF)
|
||
|
||
module _ {ℓ ℓ' : Level} {ℂ 𝔻 : Category ℓ ℓ'} where
|
||
Functor≡ : {F G : Functor ℂ 𝔻}
|
||
→ (eq* : func* F ≡ func* G)
|
||
→ (eq→ : (λ i → ∀ {x y} → ℂ [ x , y ] → 𝔻 [ eq* i x , eq* i y ])
|
||
[ func→ F ≡ func→ G ])
|
||
→ F ≡ G
|
||
Functor≡ {F} {G} eq* eq→ i = record
|
||
{ raw = eqR i
|
||
; isFunctor = eqIsF i
|
||
}
|
||
where
|
||
eqR : raw F ≡ raw G
|
||
eqR i = record { func* = eq* i ; func→ = eq→ i }
|
||
eqIsF : (λ i → IsFunctor ℂ 𝔻 (eqR i)) [ isFunctor F ≡ isFunctor G ]
|
||
eqIsF = IsFunctorIsProp' (isFunctor F) (isFunctor G)
|
||
|
||
module _ {ℓ ℓ' : Level} {A B C : Category ℓ ℓ'} (F : Functor B C) (G : Functor A B) where
|
||
private
|
||
F* = func* F
|
||
F→ = func→ F
|
||
G* = func* G
|
||
G→ = func→ G
|
||
module _ {a0 a1 a2 : Object A} {α0 : A [ a0 , a1 ]} {α1 : A [ a1 , a2 ]} where
|
||
|
||
dist : (F→ ∘ G→) (A [ α1 ∘ α0 ]) ≡ C [ (F→ ∘ G→) α1 ∘ (F→ ∘ G→) α0 ]
|
||
dist = begin
|
||
(F→ ∘ G→) (A [ α1 ∘ α0 ]) ≡⟨ refl ⟩
|
||
F→ (G→ (A [ α1 ∘ α0 ])) ≡⟨ cong F→ (G .isFunctor .distrib)⟩
|
||
F→ (B [ G→ α1 ∘ G→ α0 ]) ≡⟨ F .isFunctor .distrib ⟩
|
||
C [ (F→ ∘ G→) α1 ∘ (F→ ∘ G→) α0 ] ∎
|
||
|
||
_∘fr_ : RawFunctor A C
|
||
RawFunctor.func* _∘fr_ = F* ∘ G*
|
||
RawFunctor.func→ _∘fr_ = F→ ∘ G→
|
||
instance
|
||
isFunctor' : IsFunctor A C _∘fr_
|
||
isFunctor' = record
|
||
{ ident = begin
|
||
(F→ ∘ G→) (𝟙 A) ≡⟨ refl ⟩
|
||
F→ (G→ (𝟙 A)) ≡⟨ cong F→ (G .isFunctor .ident)⟩
|
||
F→ (𝟙 B) ≡⟨ F .isFunctor .ident ⟩
|
||
𝟙 C ∎
|
||
; distrib = dist
|
||
}
|
||
|
||
_∘f_ : Functor A C
|
||
raw _∘f_ = _∘fr_
|
||
|
||
-- The identity functor
|
||
identity : ∀ {ℓ ℓ'} → {C : Category ℓ ℓ'} → Functor C C
|
||
identity = record
|
||
{ raw = record
|
||
{ func* = λ x → x
|
||
; func→ = λ x → x
|
||
}
|
||
; isFunctor = record
|
||
{ ident = refl
|
||
; distrib = refl
|
||
}
|
||
}
|