Refactor Functor - only in module Functor

This commit is contained in:
Frederik Hanghøj Iversen 2018-02-06 14:24:34 +01:00
parent a27292dd53
commit 9349b37550
5 changed files with 133 additions and 72 deletions

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@ -14,15 +14,18 @@ module _ {c c' d d' : Level} { : Category c c'} {𝔻 : Cat
open Functor open Functor
module _ (F G : Functor 𝔻) where module _ (F G : Functor 𝔻) where
private
module F = Functor F
module G = Functor G
-- What do you call a non-natural tranformation? -- What do you call a non-natural tranformation?
Transformation : Set (c d') Transformation : Set (c d')
Transformation = (C : Object ) 𝔻 [ F .func* C , G .func* C ] Transformation = (C : Object ) 𝔻 [ F.func* C , G.func* C ]
Natural : Transformation Set (c (c' d')) Natural : Transformation Set (c (c' d'))
Natural θ Natural θ
= {A B : Object } = {A B : Object }
(f : [ A , B ]) (f : [ A , B ])
𝔻 [ θ B F .func→ f ] 𝔻 [ G .func→ f θ A ] 𝔻 [ θ B F.func→ f ] 𝔻 [ G.func→ f θ A ]
NaturalTransformation : Set (c c' d') NaturalTransformation : Set (c c' d')
NaturalTransformation = Σ Transformation Natural NaturalTransformation = Σ Transformation Natural
@ -30,13 +33,12 @@ module _ {c c' d d' : Level} { : Category c c'} {𝔻 : Cat
-- NaturalTranformation : Set (c ⊔ (c' ⊔ d')) -- NaturalTranformation : Set (c ⊔ (c' ⊔ d'))
-- NaturalTranformation = ∀ (θ : Transformation) {A B : .Object} → (f : .Arrow A B) → 𝔻 ._⊕_ (θ B) (F .func→ f) ≡ 𝔻 ._⊕_ (G .func→ f) (θ A) -- NaturalTranformation = ∀ (θ : Transformation) {A B : .Object} → (f : .Arrow A B) → 𝔻 ._⊕_ (θ B) (F .func→ f) ≡ 𝔻 ._⊕_ (G .func→ f) (θ A)
module _ {F G : Functor 𝔻} where NaturalTransformation≡ : {α β : NaturalTransformation}
NaturalTransformation≡ : {α β : NaturalTransformation F G}
(eq₁ : α .proj₁ β .proj₁) (eq₁ : α .proj₁ β .proj₁)
(eq₂ : PathP (eq₂ : PathP
(λ i {A B : Object } (f : [ A , B ]) (λ i {A B : Object } (f : [ A , B ])
𝔻 [ eq₁ i B F .func→ f ] 𝔻 [ eq₁ i B F.func→ f ]
𝔻 [ G .func→ f eq₁ i A ]) 𝔻 [ G.func→ f eq₁ i A ])
(α .proj₂) (β .proj₂)) (α .proj₂) (β .proj₂))
α β α β
NaturalTransformation≡ eq₁ eq₂ i = eq₁ i , eq₂ i NaturalTransformation≡ eq₁ eq₂ i = eq₁ i , eq₂ i
@ -52,7 +54,8 @@ module _ {c c' d d' : Level} { : Category c c'} {𝔻 : Cat
𝔻 [ F→ f 𝟙 𝔻 ] ≡⟨⟩ 𝔻 [ F→ f 𝟙 𝔻 ] ≡⟨⟩
𝔻 [ F→ f identityTrans F A ] 𝔻 [ F→ f identityTrans F A ]
where where
F→ = F .func→ module F = Functor F
F→ = F.func→
module 𝔻 = IsCategory (isCategory 𝔻) module 𝔻 = IsCategory (isCategory 𝔻)
identityNat : (F : Functor 𝔻) NaturalTransformation F F identityNat : (F : Functor 𝔻) NaturalTransformation F F
@ -60,20 +63,23 @@ module _ {c c' d d' : Level} { : Category c c'} {𝔻 : Cat
module _ {F G H : Functor 𝔻} where module _ {F G H : Functor 𝔻} where
private private
module F = Functor F
module G = Functor G
module H = Functor H
_∘nt_ : Transformation G H Transformation F G Transformation F H _∘nt_ : Transformation G H Transformation F G Transformation F H
(θ ∘nt η) C = 𝔻 [ θ C η C ] (θ ∘nt η) C = 𝔻 [ θ C η C ]
NatComp _:⊕:_ : NaturalTransformation G H NaturalTransformation F G NaturalTransformation F H NatComp _:⊕:_ : NaturalTransformation G H NaturalTransformation F G NaturalTransformation F H
proj₁ ((θ , _) :⊕: (η , _)) = θ ∘nt η proj₁ ((θ , _) :⊕: (η , _)) = θ ∘nt η
proj₂ ((θ , θNat) :⊕: (η , ηNat)) {A} {B} f = begin proj₂ ((θ , θNat) :⊕: (η , ηNat)) {A} {B} f = begin
𝔻 [ (θ ∘nt η) B F .func→ f ] ≡⟨⟩ 𝔻 [ (θ ∘nt η) B F.func→ f ] ≡⟨⟩
𝔻 [ 𝔻 [ θ B η B ] F .func→ f ] ≡⟨ sym assoc 𝔻 [ 𝔻 [ θ B η B ] F.func→ f ] ≡⟨ sym assoc
𝔻 [ θ B 𝔻 [ η B F .func→ f ] ] ≡⟨ cong (λ φ 𝔻 [ θ B φ ]) (ηNat f) 𝔻 [ θ B 𝔻 [ η B F.func→ f ] ] ≡⟨ cong (λ φ 𝔻 [ θ B φ ]) (ηNat f)
𝔻 [ θ B 𝔻 [ G .func→ f η A ] ] ≡⟨ assoc 𝔻 [ θ B 𝔻 [ G.func→ f η A ] ] ≡⟨ assoc
𝔻 [ 𝔻 [ θ B G .func→ f ] η A ] ≡⟨ cong (λ φ 𝔻 [ φ η A ]) (θNat f) 𝔻 [ 𝔻 [ θ B G.func→ f ] η A ] ≡⟨ cong (λ φ 𝔻 [ φ η A ]) (θNat f)
𝔻 [ 𝔻 [ H .func→ f θ A ] η A ] ≡⟨ sym assoc 𝔻 [ 𝔻 [ H.func→ f θ A ] η A ] ≡⟨ sym assoc
𝔻 [ H .func→ f 𝔻 [ θ A η A ] ] ≡⟨⟩ 𝔻 [ H.func→ f 𝔻 [ θ A η A ] ] ≡⟨⟩
𝔻 [ H .func→ f (θ ∘nt η) A ] 𝔻 [ H.func→ f (θ ∘nt η) A ]
where where
open IsCategory (isCategory 𝔻) open IsCategory (isCategory 𝔻)

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@ -56,8 +56,10 @@ Representable {' = '} = Functor (Sets {'})
-- The "co-yoneda" embedding. -- The "co-yoneda" embedding.
representable : { '} { : Category '} Category.Object Representable representable : { '} { : Category '} Category.Object Representable
representable { = } A = record representable { = } A = record
{ func* = λ B [ A , B ] { raw = record
; func→ = [_∘_] { func* = λ B [ A , B ]
; func→ = [_∘_]
}
; isFunctor = record ; isFunctor = record
{ ident = funExt λ _ proj₂ ident { ident = funExt λ _ proj₂ ident
; distrib = funExt λ x sym assoc ; distrib = funExt λ x sym assoc
@ -73,8 +75,10 @@ Presheaf {' = '} = Functor (Opposite ) (Sets {'})
-- Alternate name: `yoneda` -- Alternate name: `yoneda`
presheaf : { ' : Level} { : Category '} Category.Object (Opposite ) Presheaf presheaf : { ' : Level} { : Category '} Category.Object (Opposite ) Presheaf
presheaf { = } B = record presheaf { = } B = record
{ func* = λ A [ A , B ] { raw = record
; func→ = λ f g [ g f ] { func* = λ A [ A , B ]
; func→ = λ f g [ g f ]
}
; isFunctor = record ; isFunctor = record
{ ident = funExt λ x proj₁ ident { ident = funExt λ x proj₁ ident
; distrib = funExt λ x assoc ; distrib = funExt λ x assoc

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@ -53,10 +53,6 @@ record RawCategory ( ' : Level) : Set (lsuc (' ⊔ )) where
-- (univalent). -- (univalent).
record IsCategory {a b : Level} ( : RawCategory a b) : Set (lsuc (a b)) where record IsCategory {a b : Level} ( : RawCategory a b) : Set (lsuc (a b)) where
open RawCategory open RawCategory
-- (Object : Set )
-- (Arrow : Object → Object → Set ')
-- (𝟙 : {o : Object} → Arrow o o)
-- (_∘_ : { a b c : Object } → Arrow b c → Arrow a b → Arrow a c)
field field
assoc : {A B C D : Object} { f : Arrow A B } { g : Arrow B C } { h : Arrow C D } assoc : {A B C D : Object} { f : Arrow A B } { g : Arrow B C } { h : Arrow C D }
h (g f) (h g) f h (g f) (h g) f
@ -100,9 +96,9 @@ module _ {a} {b} { : RawCategory a b} where
( x.arrowIsSet (fst x.ident) (fst y.ident) i ( x.arrowIsSet (fst x.ident) (fst y.ident) i
, x.arrowIsSet (snd x.ident) (snd y.ident) i , x.arrowIsSet (snd x.ident) (snd y.ident) i
) )
; arrowIsSet = λ p q ; arrowIsSet = λ p q
let let
golden : x.arrowIsSet p q y.arrowIsSet p q golden : x.arrowIsSet p q y.arrowIsSet p q
golden = {!!} golden = {!!}
in in
golden i golden i

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@ -6,61 +6,110 @@ open import Function
open import Cat.Category open import Cat.Category
open Category hiding (_∘_) open Category hiding (_∘_ ; raw)
module _ {c c' d d'} ( : Category c c') (𝔻 : Category d d') where module _ {c c' d d'}
record IsFunctor ( : Category c c')
(func* : Object Object 𝔻) (𝔻 : Category d d')
(func→ : {A B : Object } [ A , B ] 𝔻 [ func* A , func* B ]) where
: Set (c c' d d') where
private
= c c' d d'
𝓤 = Set
record RawFunctor : 𝓤 where
field
func* : Object Object 𝔻
func→ : {A B} [ A , B ] 𝔻 [ func* A , func* B ]
record IsFunctor (F : RawFunctor) : 𝓤 where
open RawFunctor F
field field
ident : {c : Object } func→ (𝟙 {c}) 𝟙 𝔻 {func* c} ident : {c : Object } func→ (𝟙 {c}) 𝟙 𝔻 {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 : {A B C : Object } {f : [ A , B ]} {g : [ B , C ]} distrib : {A B C : Object } {f : [ A , B ]} {g : [ B , C ]}
func→ ( [ g f ]) 𝔻 [ func→ g func→ f ] func→ ( [ g f ]) 𝔻 [ func→ g func→ f ]
record Functor : Set (c c' d d') where record Functor : Set (c c' d d') where
field field
func* : Object Object 𝔻 raw : RawFunctor
func→ : {A B} [ A , B ] 𝔻 [ func* A , func* B ] {{isFunctor}} : IsFunctor raw
{{isFunctor}} : IsFunctor func* func→
private
module R = RawFunctor raw
func* : Object Object 𝔻
func* = R.func*
func→ : {A B} [ A , B ] 𝔻 [ func* A , func* B ]
func→ = R.func→
open IsFunctor open IsFunctor
open Functor open Functor
-- TODO: Is `IsFunctor` a proposition?
module _
{a b : Level}
{ 𝔻 : Category a b}
{F : RawFunctor 𝔻}
where
private
module 𝔻 = IsCategory (isCategory 𝔻)
-- isProp : Set
-- isProp = (x y : A) → x ≡ y
IsFunctorIsProp : isProp (IsFunctor _ _ F)
IsFunctorIsProp 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 a path in
module _
{a b : Level}
{ 𝔻 : Category a b}
{F : I RawFunctor 𝔻}
where
private
module 𝔻 = IsCategory (isCategory 𝔻)
IsProp' : { : Level} (A : I Set ) Set
IsProp' A = (a0 : A i0) (a1 : A i1) A [ a0 a1 ]
postulate IsFunctorIsProp' : IsProp' λ i IsFunctor _ _ (F i)
-- IsFunctorIsProp' 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
module _ { ' : Level} { 𝔻 : Category '} where module _ { ' : Level} { 𝔻 : Category '} where
IsFunctor≡
: {func* : Object Object 𝔻}
{func→ : {A B : Object } [ A , B ] 𝔻 [ func* A , func* B ]}
{F G : IsFunctor 𝔻 func* func→}
(eqI
: (λ i {A} func→ (𝟙 {A}) 𝟙 𝔻 {func* A})
[ F .ident G .ident ])
(eqD :
(λ i {A B C} {f : [ A , B ]} {g : [ B , C ]}
func→ ( [ g f ]) 𝔻 [ func→ g func→ f ])
[ F .distrib G .distrib ])
(λ _ IsFunctor 𝔻 (λ i func* i) func→) [ F G ]
IsFunctor≡ eqI eqD i = record { ident = eqI i ; distrib = eqD i }
Functor≡ : {F G : Functor 𝔻} Functor≡ : {F G : Functor 𝔻}
(eq* : F .func* G .func*) (eq* : func* F func* G)
(eq→ : (λ i {x y} [ x , y ] 𝔻 [ eq* i x , eq* i y ]) (eq→ : (λ i {x y} [ x , y ] 𝔻 [ eq* i x , eq* i y ])
[ F .func→ G .func→ ]) [ func→ F func→ G ])
-- → (eqIsF : PathP (λ i → IsFunctor 𝔻 (eq* i) (eq→ i)) (F .isFunctor) (G .isFunctor))
(eqIsFunctor : (λ i IsFunctor 𝔻 (eq* i) (eq→ i)) [ F .isFunctor G .isFunctor ])
F G F G
Functor≡ eq* eq→ eqIsFunctor i = record { func* = eq* i ; func→ = eq→ i ; isFunctor = eqIsFunctor i } Functor≡ {F} {G} eq* eq→ i = record
{ raw = eqR i
; isFunctor = f i
}
where
eqR : raw F raw G
eqR i = record { func* = eq* i ; func→ = eq→ i }
postulate T : isSet (IsFunctor _ _ (raw F))
f : (λ i IsFunctor 𝔻 (eqR i)) [ isFunctor F isFunctor G ]
f = IsFunctorIsProp' (isFunctor F) (isFunctor G)
module _ { ' : Level} {A B C : Category '} (F : Functor B C) (G : Functor A B) where module _ { ' : Level} {A B C : Category '} (F : Functor B C) (G : Functor A B) where
private private
F* = F .func* F* = func* F
F→ = F .func→ F→ = func→ F
G* = G .func* G* = func* G
G→ = G .func→ G→ = func→ G
module _ {a0 a1 a2 : Object A} {α0 : A [ a0 , a1 ]} {α1 : A [ a1 , a2 ]} where 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 : (F→ G→) (A [ α1 α0 ]) C [ (F→ G→) α1 (F→ G→) α0 ]
@ -70,12 +119,12 @@ module _ { ' : Level} {A B C : Category '} (F : Functor B C) (G : F
F→ (B [ G→ α1 G→ α0 ]) ≡⟨ F .isFunctor .distrib F→ (B [ G→ α1 G→ α0 ]) ≡⟨ F .isFunctor .distrib
C [ (F→ G→) α1 (F→ G→) α0 ] C [ (F→ G→) α1 (F→ G→) α0 ]
_∘f_ : Functor A C _∘fr_ : RawFunctor A C
_∘f_ = RawFunctor.func* _∘fr_ = F* G*
record RawFunctor.func→ _∘fr_ = F→ G→
{ func* = F* G* instance
; func→ = F→ G→ isFunctor' : IsFunctor A C _∘fr_
; isFunctor = record isFunctor' = record
{ ident = begin { ident = begin
(F→ G→) (𝟙 A) ≡⟨ refl (F→ G→) (𝟙 A) ≡⟨ refl
F→ (G→ (𝟙 A)) ≡⟨ cong F→ (G .isFunctor .ident) F→ (G→ (𝟙 A)) ≡⟨ cong F→ (G .isFunctor .ident)
@ -83,13 +132,17 @@ module _ { ' : Level} {A B C : Category '} (F : Functor B C) (G : F
𝟙 C 𝟙 C
; distrib = dist ; distrib = dist
} }
}
_∘f_ : Functor A C
raw _∘f_ = _∘fr_
-- The identity functor -- The identity functor
identity : { '} {C : Category '} Functor C C identity : { '} {C : Category '} Functor C C
identity = record identity = record
{ func* = λ x x { raw = record
; func→ = λ x x { func* = λ x x
; func→ = λ x x
}
; isFunctor = record ; isFunctor = record
{ ident = refl { ident = refl
; distrib = refl ; distrib = refl

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@ -25,8 +25,10 @@ module _ {a b : Level} where
T : Functor (Opposite ) (Fam a b) T : Functor (Opposite ) (Fam a b)
-- Empty context -- Empty context
[] : Terminal [] : Terminal
private
module T = Functor T
Type : (Γ : Object ) Set a Type : (Γ : Object ) Set a
Type Γ = proj₁ (T .func* Γ) Type Γ = proj₁ (T.func* Γ)
module _ {Γ : Object } {A : Type Γ} where module _ {Γ : Object } {A : Type Γ} where
@ -35,7 +37,7 @@ module _ {a b : Level} where
(λ f (λ f
{x : proj₁ (func* T B)} {x : proj₁ (func* T B)}
proj₂ (func* T B) x proj₂ (func* T A) (f x)) proj₂ (func* T B) x proj₂ (func* T A) (f x))
k = T .func→ γ k = T.func→ γ
k₁ : proj₁ (func* T B) proj₁ (func* T A) k₁ : proj₁ (func* T B) proj₁ (func* T A)
k₁ = proj₁ k k₁ = proj₁ k
k₂ : ({x : proj₁ (func* T B)} k₂ : ({x : proj₁ (func* T B)}