fredefox/cat
1
0
Fork 0
Code Issues Pull requests Releases Wiki Activity

Use equality construction principle

Browse source 
Also update submodules
This commit is contained in:
Frederik Hanghøj Iversen 2018-01-30 22:41:18 +01:00
parent 255b0236f9
commit 86d3d7368e
7 changed files with 68 additions and 39 deletions
Show stats Download patch file Download diff file Expand all files Collapse all files

2
libs/agda-stdlib

@ -1 +1 @@
Subproject commit de23244a73d6dab55715fd5a107a5de805c55764
Subproject commit b5bfbc3c170b0bd0c9aaac1b4d4b3f9b06832bf6

2
libs/cubical

@ -1 +1 @@
Subproject commit a83f5f4c63e5dfd8143ac03163868c63a56802de
Subproject commit 1d6730c4999daa2b04e9dd39faa0791d0b5c3b48

18
src/Cat/Categories/Cat.agda
View file

@ -44,7 +44,7 @@ module _ (ℓ ℓ' : Level) where
((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
assc = Functor≡ eq* eq→ (IsFunctor≡ eqI eqD)
module _ {ℂ 𝔻 : Category ℓ ℓ'} {F : Functor ℂ 𝔻} where
module _ where
@ -58,16 +58,17 @@ module _ (ℓ ℓ' : Level) where
(func→ (F ∘f identity)) (func→ F)
eq→ = refl
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))
((F ∘f identity) .isFunctor .ident) (F .isFunctor .ident)
eqI-r
: (λ i → {c : ℂ .Object} → (λ _ → 𝔻 [ func* F c , func* F c ])
[ func→ F (ℂ .𝟙) ≡ 𝔻 .𝟙 ])
[(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) .isFunctor .distrib) (F .isFunctor .distrib)
ident-r : F ∘f identity ≡ F
ident-r = Functor≡ eq* eq→ eqI-r eqD-r
ident-r = Functor≡ eq* eq→ (IsFunctor≡ eqI-r eqD-r)
module _ where
private
postulate
@ -75,13 +76,14 @@ module _ (ℓ ℓ' : Level) where
eq→ : PathP
(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})
((identity ∘f F) .isFunctor .ident) (F .isFunctor .ident)
eqI : (λ i → ∀ {A : ℂ .Object} → eq→ i (ℂ .𝟙 {A}) ≡ 𝔻 .𝟙 {eq* i A})
[ ((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 ])
((identity ∘f F) .isFunctor .distrib) (F .isFunctor .distrib)
-- (λ z → eq* i z) (eq→ i)
ident-l : identity ∘f F ≡ F
ident-l = Functor≡ eq* eq→ eqI eqD
ident-l = Functor≡ eq* eq→ λ i → record { ident = eqI i ; distrib = eqD i }
Cat : Category (lsuc (ℓ ⊔ ℓ')) (ℓ ⊔ ℓ')
Cat =

2
src/Cat/Categories/Fun.agda
View file

@ -53,7 +53,7 @@ module _ {ℓc ℓc' ℓd ℓd' : Level} {ℂ : Category ℓc ℓc'} {𝔻 : Cat
𝔻 [ F→ f ∘ identityTrans F A ] ∎
where
F→ = F .func→
open module 𝔻 = IsCategory (𝔻 .isCategory)
module 𝔻 = IsCategory (𝔻 .isCategory)
identityNat : (F : Functor ℂ 𝔻) → NaturalTransformation F F
identityNat F = identityTrans F , identityNatural F

33
src/Cat/Category/Properties.agda
View file

@ -9,6 +9,8 @@ open import Cubical
open import Cat.Category
open import Cat.Functor
open import Cat.Categories.Sets
open import Cat.Equality
open Equality.Data.Product
module _ {ℓ ℓ' : Level} {ℂ : Category ℓ ℓ'} { A B : ℂ .Category.Object } {X : ℂ .Category.Object} (f : ℂ .Category.Arrow A B) where
open Category ℂ
@ -58,6 +60,7 @@ module _ {ℓ : Level} {ℂ : Category ℓ ℓ} where
private
Catℓ = Cat ℓ ℓ
prshf = presheaf {ℂ = ℂ}
module ℂ = IsCategory (ℂ .isCategory)
-- Exp : Set (lsuc (lsuc ℓ))
-- Exp = Exponential (Cat (lsuc ℓ) ℓ)
@ -70,22 +73,24 @@ module _ {ℓ : Level} {ℂ : Category ℓ ℓ} where
module _ {A B : ℂ .Object} (f : ℂ .Arrow A B) where
:func→: : NaturalTransformation (prshf A) (prshf B)
:func→: = (λ C x → ℂ [ f ∘ x ]) , λ f₁ → funExt λ x → lem
where
lem = (ℂ .isCategory) .IsCategory.assoc
:func→: = (λ C x → ℂ [ f ∘ x ]) , λ f₁ → funExt λ _ → ℂ.assoc
module _ {c : ℂ .Object} where
eqTrans : (:func→: (ℂ .𝟙 {c})) .proj₁ ≡ (Fun .𝟙 {o = prshf c}) .proj₁
eqTrans = funExt λ x → funExt λ x → ℂ .isCategory .IsCategory.ident .proj₂
eqNat
: PathP (λ i → {A B : ℂ .Object} (f : Opposite ℂ .Arrow A B)
→ Sets [ eqTrans i B ∘ prshf c .Functor.func→ f ]
≡ Sets [ prshf c .Functor.func→ f ∘ eqTrans i A ])
((:func→: (ℂ .𝟙 {c})) .proj₂) ((Fun .𝟙 {o = prshf c}) .proj₂)
eqNat = λ i f i' x₁ → {!ℂ ._⊕_ ? ?!}
-- eqNat i f = {!!}
-- Sets ._⊕_ (eq₁ i B) (prshf A .func→ f) ≡ Sets ._⊕_ (prshf B .func→ f) (eq₁ i A)
eqTrans : (λ _ → Transformation (prshf c) (prshf c))
[ (λ _ x → ℂ [ ℂ .𝟙 ∘ x ]) ≡ identityTrans (prshf c) ]
eqTrans = funExt λ x → funExt λ x → ℂ.ident .proj₂
eqNat : (λ i → Natural (prshf c) (prshf c) (eqTrans i))
[(λ _ → funExt (λ _ → ℂ.assoc)) ≡ identityNatural (prshf c)]
eqNat = {!!}
-- eqNat = λ {A} {B} i ℂ[B,A] i' ℂ[A,c] →
-- let
-- k : ℂ [ {!!} , {!!} ]
-- k = ℂ[A,c]
-- in {!ℂ [ ? ∘ ? ]!}
:ident: : (:func→: (ℂ .𝟙 {c})) ≡ (Fun .𝟙 {o = prshf c})
:ident: = NaturalTransformation≡ eqTrans eqNat
:ident: = Σ≡ eqTrans eqNat
yoneda : Functor ℂ (Fun {ℂ = Opposite ℂ} {𝔻 = Sets {ℓ}})
yoneda = record

21
src/Cat/Equality.agda Normal file
View file

@ -0,0 +1,21 @@
-- Defines equality-principles for data-types from the standard library.
module Cat.Equality where
open import Level
open import Cubical
-- _[_≡_] = PathP
module Equality where
module Data where
module Product where
open import Data.Product public
module _ {ℓa ℓb : Level} {A : Set ℓa} {B : A → Set ℓb} {a b : Σ A B}
(proj₁≡ : (λ _ → A) [ proj₁ a ≡ proj₁ b ])
(proj₂≡ : (λ i → B (proj₁≡ i)) [ proj₂ a ≡ proj₂ b ]) where
Σ≡ : a ≡ b
proj₁ (Σ≡ i) = proj₁≡ i
proj₂ (Σ≡ i) = proj₂≡ i

29
src/Cat/Functor.agda
View file

@ -32,27 +32,28 @@ open Functor
module _ {ℓ ℓ' : Level} {ℂ 𝔻 : Category ℓ ℓ'} where
-- 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 }
IsFunctor≡
: {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
: (λ i → ∀ {A} → func→ (ℂ .𝟙 {A}) ≡ 𝔻 .𝟙 {func* A})
[ F .ident ≡ G .ident ])
→ (eqD :
(λ i → ∀ {A B C} {f : ℂ [ A , B ]} {g : ℂ [ B , C ]}
→ func→ (ℂ [ g ∘ f ]) ≡ 𝔻 [ func→ g ∘ func→ f ])
[ F .distrib ≡ G .distrib ])
→ (λ _ → IsFunctor ℂ 𝔻 (λ i → func* i) func→) [ 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 [ x , y ] → 𝔻 [ 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})
(F .isFunctor .ident) (G .isFunctor .ident))
→ (eqD : PathP (λ i → {A B C : ℂ .Object} {f : ℂ [ A , B ]} {g : ℂ [ B , C ]}
→ eq→ i (ℂ [ g ∘ f ]) ≡ 𝔻 [ eq→ i g ∘ eq→ i f ])
(F .isFunctor .distrib) (G .isFunctor .distrib))
→ (eqIsFunctor : (λ i → IsFunctor ℂ 𝔻 (eq* i) (eq→ i)) [ F .isFunctor ≡ G .isFunctor ])
→ F ≡ G
Functor≡ eq* eq→ eqI eqD i = record { func* = eq* i ; func→ = eq→ i ; isFunctor = record { ident = eqI i ; distrib = eqD i } }
Functor≡ eq* eq→ eqIsFunctor i = record { func* = eq* i ; func→ = eq→ i ; isFunctor = eqIsFunctor i }
module _ {ℓ ℓ' : Level} {A B C : Category ℓ ℓ'} (F : Functor B C) (G : Functor A B) where
private