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Frederik Hanghøj Iversen 2018-01-21 19:24:13 +01:00
parent e1e8b60930 922570a5bd
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Makefile Normal file
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build: src/**.agda
agda src/Cat.agda

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module Cat where
import Cat.Categories.Sets
import Cat.Categories.Cat
import Cat.Categories.Rel
import Cat.Category.Pathy
import Cat.Category.Bij
import Cat.Category.Free
import Cat.Category.Properties
import Cat.Category
import Cat.Cubical
import Cat.Functor

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src/Cat/Categories/Cat.agda
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{-# OPTIONS --cubical #-}
{-# OPTIONS --cubical --allow-unsolved-metas #-}
module Category.Categories.Cat where
module Cat.Categories.Cat where
open import Agda.Primitive
open import Cubical
open import Function
open import Data.Product renaming (proj₁ to fst ; proj₂ to snd)
open import Category
open import Cat.Category
open import Cat.Functor
-- Tip from Andrea:
-- Use co-patterns - they help with showing more understandable types in goals.
lift-eq : ∀ {ℓ} {A B : Set ℓ} {a a' : A} {b b' : B} → a ≡ a' → b ≡ b' → (a , b) ≡ (a' , b')
fst (lift-eq a b i) = a i
snd (lift-eq a b i) = b i
eqpair : ∀ {ℓa ℓb} {A : Set ℓa} {B : Set ℓb} {a a' : A} {b b' : B}
→ a ≡ a' → b ≡ b' → (a , b) ≡ (a' , b')
eqpair eqa eqb i = eqa i , eqb i
open Functor
open Category
module _ {ℓ ℓ' : Level} {A B : Category ℓ ℓ'} where
lift-eq-functors : {f g : Functor A B}
→ (eq* : f .func* ≡ g .func*)
→ (eq→ : PathP (λ i → ∀ {x y} → A .Arrow x y → B .Arrow (eq* i x) (eq* i y))
(f .func→) (g .func→))
-- → (eq→ : Functor.func→ f ≡ {!!}) -- Functor.func→ g)
-- Use PathP
-- directly to show heterogeneous equalities by using previous
-- equalities (i.e. continuous paths) to create new continuous paths.
→ (eqI : PathP (λ i → ∀ {c : A .Object} → eq→ i (A .𝟙 {c}) ≡ B .𝟙 {eq* i c})
(ident f) (ident g))
→ (eqD : PathP (λ i → { c c' c'' : A .Object} {a : A .Arrow c c'} {a' : A .Arrow c' c''}
→ eq→ i (A ._⊕_ a' a) ≡ B ._⊕_ (eq→ i a') (eq→ i a))
(distrib f) (distrib g))
→ f ≡ g
lift-eq-functors eq* eq→ eqI eqD i = record { func* = eq* i ; func→ = eq→ i ; ident = eqI i ; distrib = eqD i }
-- The category of categories
module _ {ℓ ℓ' : Level} where
private
_⊛_ = functor-comp
module _ {A B C D : Category {ℓ} {ℓ'}} {f : Functor A B} {g : Functor B C} {h : Functor C D} where
assc : h ⊛ (g ⊛ f) ≡ (h ⊛ g) ⊛ f
assc = {!!}
module _ {A B C D : Category ℓ ℓ'} {f : Functor A B} {g : Functor B C} {h : Functor C D} where
eq* : func* (h ∘f (g ∘f f)) ≡ func* ((h ∘f g) ∘f f)
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 = {!!}
module _ {A B : Category {ℓ} {ℓ'}} where
lift-eq : (f g : Functor A B)
→ (eq* : Functor.func* f ≡ Functor.func* g)
-- TODO: Must transport here using the equality from above.
-- Reason:
-- func→ : Arrow A dom cod → Arrow B (func* dom) (func* cod)
-- func→₁ : Arrow A dom cod → Arrow B (func*₁ dom) (func*₁ cod)
-- In other words, func→ and func→₁ does not have the same type.
-- → Functor.func→ f ≡ Functor.func→ g
-- → Functor.ident f ≡ Functor.ident g
-- → Functor.distrib f ≡ Functor.distrib g
→ f ≡ g
lift-eq
(functor func* func→ idnt distrib)
(functor func*₁ func→₁ idnt₁ distrib₁)
eq-func* = {!!}
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
idHere = identity {ℓ} {ℓ'} {A}
lem : (Functor.func* f) ∘ (Functor.func* idHere) ≡ Functor.func* f
module _ {A B : Category ℓ ℓ'} {f : Functor A B} where
lem : (func* f) ∘ (func* (identity {C = A})) ≡ func* f
lem = refl
ident-r : f ⊛ identity ≡ f
ident-r = lift-eq (f ⊛ identity) f refl
ident-l : identity ⊛ f ≡ f
ident-l = {!!}
-- lemmm : func→ {C = A} {D = B} (f ∘f identity) ≡ func→ f
lemmm : PathP
(λ i →
{x y : Object A} → Arrow A x y → Arrow B (func* f x) (func* f y))
(func→ (f ∘f identity)) (func→ f)
lemmm = refl
postulate lemz : PathP (λ i → {c : A .Object} → PathP (λ _ → Arrow B (func* f c) (func* f c)) (func→ f (A .𝟙)) (B .𝟙))
(ident (f ∘f identity)) (ident f)
-- lemz = {!!}
postulate ident-r : f ∘f identity ≡ f
-- ident-r = lift-eq-functors lem lemmm {!lemz!} {!!}
postulate ident-l : identity ∘f f ≡ f
-- ident-l = lift-eq-functors lem lemmm {!refl!} {!!}
CatCat : Category {lsuc (ℓ ⊔ ℓ')} {ℓ ⊔ ℓ'}
CatCat =
Cat : Category (lsuc (ℓ ⊔ ℓ')) (ℓ ⊔ ℓ')
Cat =
record
{ Object = Category {ℓ} {ℓ'}
{ Object = Category ℓ ℓ'
; Arrow = Functor
; 𝟙 = identity
; _⊕_ = functor-comp
; assoc = {!!}
; ident = ident-r , ident-l
; _⊕_ = _∘f_
-- What gives here? Why can I not name the variables directly?
; isCategory = record
{ assoc = λ {_ _ _ _ f g h} → assc {f = f} {g = g} {h = h}
; ident = ident-r , ident-l
}
}
module _ {ℓ : Level} (C D : Category ℓ ℓ) where
private
:Object: = C .Object × D .Object
:Arrow: : :Object: → :Object: → Set ℓ
:Arrow: (c , d) (c' , d') = Arrow C c c' × Arrow D d d'
:𝟙: : {o : :Object:} → :Arrow: o o
:𝟙: = C .𝟙 , D .𝟙
_:⊕:_ :
{a b c : :Object:} →
:Arrow: b c →
:Arrow: a b →
:Arrow: a c
_:⊕:_ = λ { (bc∈C , bc∈D) (ab∈C , ab∈D) → (C ._⊕_) bc∈C ab∈C , D ._⊕_ bc∈D ab∈D}
instance
:isCategory: : IsCategory :Object: :Arrow: :𝟙: _:⊕:_
:isCategory: = record
{ assoc = eqpair C.assoc D.assoc
; ident
= eqpair (fst C.ident) (fst D.ident)
, eqpair (snd C.ident) (snd D.ident)
}
where
open module C = IsCategory (C .isCategory)
open module D = IsCategory (D .isCategory)
:product: : Category ℓ ℓ
:product: = record
{ Object = :Object:
; Arrow = :Arrow:
; 𝟙 = :𝟙:
; _⊕_ = _:⊕:_
}
proj₁ : Arrow Cat :product: C
proj₁ = record { func* = fst ; func→ = fst ; ident = refl ; distrib = refl }
proj₂ : Arrow Cat :product: D
proj₂ = record { func* = snd ; func→ = snd ; ident = refl ; distrib = refl }
module _ {X : Object (Cat {ℓ} {ℓ})} (x₁ : Arrow Cat X C) (x₂ : Arrow Cat X D) where
open Functor
-- ident' : {c : Object X} → ((func→ x₁) {dom = c} (𝟙 X) , (func→ x₂) {dom = c} (𝟙 X)) ≡ 𝟙 (catProduct C D)
-- ident' {c = c} = lift-eq (ident x₁) (ident x₂)
x : Functor X :product:
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)
}
-- Need to "lift equality of functors"
-- If I want to do this like I do it for pairs it's gonna be a pain.
postulate isUniqL : (Cat ⊕ proj₁) x ≡ x₁
-- isUniqL = lift-eq-functors refl refl {!!} {!!}
postulate isUniqR : (Cat ⊕ proj₂) x ≡ x₂
-- isUniqR = lift-eq-functors refl refl {!!} {!!}
isUniq : (Cat ⊕ proj₁) x ≡ x₁ × (Cat ⊕ proj₂) x ≡ x₂
isUniq = isUniqL , isUniqR
uniq : ∃![ x ] ((Cat ⊕ proj₁) x ≡ x₁ × (Cat ⊕ proj₂) x ≡ x₂)
uniq = x , isUniq
instance
isProduct : IsProduct Cat proj₁ proj₂
isProduct = uniq
product : Product {ℂ = Cat} C D
product = record
{ obj = :product:
; proj₁ = proj₁
; proj₂ = proj₂
}

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src/Cat/Categories/Rel.agda
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@ -154,12 +154,11 @@ module _ {A B C D : Set} {S : Subset (A × B)} {R : Subset (B × C)} {Q : Subset
≡ (Σ[ b ∈ B ] (a , b) ∈ S × (Σ[ c ∈ C ] (b , c) ∈ R × (c , d) ∈ Q))
is-assoc = equivToPath equi
Rel : Category
Rel : Category (lsuc lzero) (lsuc lzero)
Rel = record
{ Object = Set
; Arrow = λ S R → Subset (S × R)
; 𝟙 = λ {S} → Diag S
; _⊕_ = λ {A B C} S R → λ {( a , c ) → Σ[ b ∈ B ] ( (a , b) ∈ R × (b , c) ∈ S )}
; assoc = funExt is-assoc
; ident = funExt ident-l , funExt ident-r
; isCategory = record { assoc = funExt is-assoc ; ident = funExt ident-l , funExt ident-r }
}

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src/Cat/Categories/Sets.agda
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@ -9,44 +9,45 @@ open import Data.Product renaming (proj₁ to fst ; proj₂ to snd)
open import Cat.Category
open import Cat.Functor
open Category
-- Sets are built-in to Agda. The set of all small sets is called Set.
Fun : {ℓ : Level} → ( T U : Set ℓ ) → Set ℓ
Fun T U = T → U
Sets : {ℓ : Level} → Category {lsuc ℓ} {ℓ}
Sets : {ℓ : Level} → Category (lsuc ℓ) ℓ
Sets {ℓ} = record
{ Object = Set ℓ
; Arrow = λ T U → Fun {ℓ} T U
; 𝟙 = λ x → x
; _⊕_ = λ g f x → g ( f x )
; assoc = refl
; ident = funExt (λ x → refl) , funExt (λ x → refl)
; Arrow = λ T U → T → U
; 𝟙 = id
; _⊕_ = _∘′_
; isCategory = record { assoc = refl ; ident = funExt (λ _ → refl) , funExt (λ _ → refl) }
}
where
open import Function
Representable : {ℓ ℓ' : Level} → (ℂ : Category {ℓ} {ℓ'}) → Set (ℓ ⊔ lsuc ℓ')
-- Covariant Presheaf
Representable : {ℓ ℓ' : Level} → (ℂ : Category ℓ ℓ') → Set (ℓ ⊔ lsuc ℓ')
Representable {ℓ' = ℓ'} ℂ = Functor ℂ (Sets {ℓ'})
representable : {ℓ ℓ' : Level} → {ℂ : Category {ℓ} {ℓ'}} → Category.Object ℂ → Representable ℂ
-- The "co-yoneda" embedding.
representable : ∀ {ℓ ℓ'} {ℂ : Category ℓ ℓ'} → Category.Object ℂ → Representable ℂ
representable {ℂ = ℂ} A = record
{ func* = λ B → ℂ.Arrow A B
; func→ = λ f g → f ℂ.⊕ g
; ident = funExt λ _ → snd ℂ.ident
; distrib = funExt λ x → sym ℂ.assoc
{ func* = λ B → ℂ .Arrow A B
; func→ = ℂ ._⊕_
; ident = funExt λ _ → snd ident
; distrib = funExt λ x → sym assoc
}
where
open module ℂ = Category ℂ
open IsCategory (ℂ .isCategory)
Presheaf : {ℓ ℓ' : Level} → (ℂ : Category {ℓ} {ℓ'}) → Set (ℓ ⊔ lsuc ℓ')
-- Contravariant Presheaf
Presheaf : ∀ {ℓ ℓ'} (ℂ : Category ℓ ℓ') → Set (ℓ ⊔ lsuc ℓ')
Presheaf {ℓ' = ℓ'} ℂ = Functor (Opposite ℂ) (Sets {ℓ'})
presheaf : {ℓ ℓ' : Level} → {ℂ : Category {ℓ} {ℓ'}} → Category.Object (Opposite ℂ) → Presheaf ℂ
-- Alternate name: `yoneda`
presheaf : {ℓ ℓ' : Level} {ℂ : Category ℓ ℓ'} → Category.Object (Opposite ℂ) → Presheaf ℂ
presheaf {ℂ = ℂ} B = record
{ func* = λ A → ℂ.Arrow A B
; func→ = λ f g → g ℂ.⊕ f
; ident = funExt λ x → fst ℂ.ident
; distrib = funExt λ x → ℂ.assoc
{ func* = λ A → ℂ .Arrow A B
; func→ = λ f g → ℂ ._⊕_ g f
; ident = funExt λ x → fst ident
; distrib = funExt λ x → assoc
}
where
open module ℂ = Category ℂ
open IsCategory (ℂ .isCategory)

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src/Cat/Category.agda
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@ -4,139 +4,119 @@ module Cat.Category where
open import Agda.Primitive
open import Data.Unit.Base
open import Data.Product renaming (proj₁ to fst ; proj₂ to snd)
open import Data.Product renaming
( proj₁ to fst
; proj₂ to snd
; ∃! to ∃!≈
)
open import Data.Empty
open import Function
open import Cubical
postulate undefined : {ℓ : Level} → {A : Set ℓ} → A
∃! : ∀ {a b} {A : Set a}
→ (A → Set b) → Set (a ⊔ b)
∃! = ∃!≈ _≡_
record Category {ℓ ℓ'} : Set (lsuc (ℓ' ⊔ ℓ)) where
constructor category
∃!-syntax : ∀ {a b} {A : Set a} → (A → Set b) → Set (a ⊔ b)
∃!-syntax = ∃
syntax ∃!-syntax (λ x → B) = ∃![ x ] B
record IsCategory {ℓ ℓ' : Level}
(Object : Set ℓ)
(Arrow : Object → Object → Set ℓ')
(𝟙 : {o : Object} → Arrow o o)
(_⊕_ : { a b c : Object } → Arrow b c → Arrow a b → Arrow a c)
: Set (lsuc (ℓ' ⊔ ℓ)) where
field
assoc : {A B C D : Object} { f : Arrow A B } { g : Arrow B C } { h : Arrow C D }
→ h ⊕ (g ⊕ f) ≡ (h ⊕ g) ⊕ f
ident : {A B : Object} {f : Arrow A B}
→ f ⊕ 𝟙 ≡ f × 𝟙 ⊕ f ≡ f
-- open IsCategory public
record Category (ℓ ℓ' : Level) : Set (lsuc (ℓ' ⊔ ℓ)) where
-- adding no-eta-equality can speed up type-checking.
no-eta-equality
field
Object : Set ℓ
Arrow : Object → Object → Set ℓ'
𝟙 : {o : Object} → Arrow o o
_⊕_ : { a b c : Object } → Arrow b c → Arrow a b → Arrow a c
assoc : { A B C D : Object } { f : Arrow A B } { g : Arrow B C } { h : Arrow C D }
→ h ⊕ (g ⊕ f) ≡ (h ⊕ g) ⊕ f
ident : { A B : Object } { f : Arrow A B }
→ f ⊕ 𝟙 ≡ f × 𝟙 ⊕ f ≡ f
{{isCategory}} : IsCategory Object Arrow 𝟙 _⊕_
infixl 45 _⊕_
domain : { a b : Object } → Arrow a b → Object
domain {a = a} _ = a
codomain : { a b : Object } → Arrow a b → Object
codomain {b = b} _ = b
open Category public
open Category
module _ {ℓ ℓ' : Level} {ℂ : Category {ℓ} {ℓ'}} { A B : Object ℂ } where
private
open module ℂ = Category ℂ
_+_ = ℂ._⊕_
module _ {ℓ ℓ' : Level} {ℂ : Category ℓ ℓ'} where
module _ { A B : ℂ .Object } where
Isomorphism : (f : ℂ .Arrow A B) → Set ℓ'
Isomorphism f = Σ[ g ∈ ℂ .Arrow B A ] ℂ ._⊕_ g f ≡ ℂ .𝟙 × ℂ ._⊕_ f g ≡ ℂ .𝟙
Isomorphism : (f : ℂ.Arrow A B) → Set ℓ'
Isomorphism f = Σ[ g ∈ ℂ.Arrow B A ] g + f ≡ ℂ.𝟙 × f + g ≡ ℂ.𝟙
Epimorphism : {X : ℂ .Object } → (f : ℂ .Arrow A B) → Set ℓ'
Epimorphism {X} f = ( g₀ g₁ : ℂ .Arrow B X ) → ℂ ._⊕_ g₀ f ≡ ℂ ._⊕_ g₁ f → g₀ ≡ g₁
Epimorphism : {X : ℂ.Object } → (f : ℂ.Arrow A B) → Set ℓ'
Epimorphism {X} f = ( g₀ g₁ : ℂ.Arrow B X ) → g₀ + f ≡ g₁ + f → g₀ ≡ g₁
Monomorphism : {X : ℂ .Object} → (f : ℂ .Arrow A B) → Set ℓ'
Monomorphism {X} f = ( g₀ g₁ : ℂ .Arrow X A ) → ℂ ._⊕_ f g₀ ≡ ℂ ._⊕_ f g₁ → g₀ ≡ g₁
Monomorphism : {X : ℂ.Object} → (f : ℂ.Arrow A B) → Set ℓ'
Monomorphism {X} f = ( g₀ g₁ : ℂ.Arrow X A ) → f + g₀ ≡ f + g₁ → g₀ ≡ g₁
-- Isomorphism of objects
_≅_ : (A B : Object ℂ) → Set ℓ'
_≅_ A B = Σ[ f ∈ ℂ .Arrow A B ] (Isomorphism f)
iso-is-epi : ∀ {X} (f : ℂ.Arrow A B) → Isomorphism f → Epimorphism {X = X} f
-- Idea: Pre-compose with f- on both sides of the equality of eq to get
-- g₀ + f + f- ≡ g₁ + f + f-
-- which by left-inv reduces to the goal.
iso-is-epi f (f- , left-inv , right-inv) g₀ g₁ eq =
trans (sym (fst ℂ.ident))
( trans (cong (_+_ g₀) (sym right-inv))
( trans ℂ.assoc
( trans (cong (λ x → x + f-) eq)
( trans (sym ℂ.assoc)
( trans (cong (_+_ g₁) right-inv) (fst ℂ.ident))
)
)
)
)
module _ {ℓ ℓ' : Level} (ℂ : Category ℓ ℓ') {A B obj : Object ℂ} where
IsProduct : (π₁ : Arrow ℂ obj A) (π₂ : Arrow ℂ obj B) → Set (ℓ ⊔ ℓ')
IsProduct π₁ π₂
= ∀ {X : ℂ .Object} (x₁ : ℂ .Arrow X A) (x₂ : ℂ .Arrow X B)
→ ∃![ x ] (ℂ ._⊕_ π₁ x ≡ x₁ × ℂ ._⊕_ π₂ x ≡ x₂)
iso-is-mono : ∀ {X} (f : ℂ.Arrow A B ) → Isomorphism f → Monomorphism {X = X} f
-- For the next goal we do something similar: Post-compose with f- and use
-- right-inv to get the goal.
iso-is-mono f (f- , (left-inv , right-inv)) g₀ g₁ eq =
trans (sym (snd ℂ.ident))
( trans (cong (λ x → x + g₀) (sym left-inv))
( trans (sym ℂ.assoc)
( trans (cong (_+_ f-) eq)
( trans ℂ.assoc
( trans (cong (λ x → x + g₁) left-inv) (snd ℂ.ident)
)
)
)
)
)
-- Tip from Andrea; Consider this style for efficiency:
-- record IsProduct {ℓ ℓ' : Level} (ℂ : Category {ℓ} {ℓ'})
-- {A B obj : Object ℂ} (π₁ : Arrow ℂ obj A) (π₂ : Arrow ℂ obj B) : Set (ℓ ⊔ ℓ') where
-- field
-- isProduct : ∀ {X : ℂ .Object} (x₁ : ℂ .Arrow X A) (x₂ : ℂ .Arrow X B)
-- → ∃![ x ] (ℂ ._⊕_ π₁ x ≡ x₁ × ℂ. _⊕_ π₂ x ≡ x₂)
iso-is-epi-mono : ∀ {X} (f : ℂ.Arrow A B ) → Isomorphism f → Epimorphism {X = X} f × Monomorphism {X = X} f
iso-is-epi-mono f iso = iso-is-epi f iso , iso-is-mono f iso
record Product {ℓ ℓ' : Level} {ℂ : Category ℓ ℓ'} (A B : ℂ .Object) : Set (ℓ ⊔ ℓ') where
no-eta-equality
field
obj : ℂ .Object
proj₁ : ℂ .Arrow obj A
proj₂ : ℂ .Arrow obj B
{{isProduct}} : IsProduct ℂ proj₁ proj₂
{-
epi-mono-is-not-iso : ∀ {ℓ ℓ'} → ¬ ((ℂ : Category {ℓ} {ℓ'}) {A B X : Object ℂ} (f : Arrow ℂ A B ) → Epimorphism {ℂ = ℂ} {X = X} f → Monomorphism {ℂ = ℂ} {X = X} f → Isomorphism {ℂ = ℂ} f)
epi-mono-is-not-iso f =
let k = f {!!} {!!} {!!} {!!}
in {!!}
-}
module _ {ℓ ℓ' : Level} (ℂ : Category ℓ ℓ') where
Opposite : Category ℓ ℓ'
Opposite =
record
{ Object = ℂ .Object
; Arrow = flip (ℂ .Arrow)
; 𝟙 = ℂ .𝟙
; _⊕_ = flip (ℂ ._⊕_)
; isCategory = record { assoc = sym assoc ; ident = swap ident }
}
where
open IsCategory (ℂ .isCategory)
-- Isomorphism of objects
_≅_ : { ℓ ℓ' : Level } → { ℂ : Category {ℓ} {ℓ'} } → ( A B : Object ℂ ) → Set ℓ'
_≅_ {ℂ = ℂ} A B = Σ[ f ∈ ℂ.Arrow A B ] (Isomorphism {ℂ = ℂ} f)
where
open module ℂ = Category ℂ
-- A consequence of no-eta-equality; `Opposite-is-involution` is no longer
-- definitional - i.e.; you must match on the fields:
--
-- Opposite-is-involution : ∀ {ℓ ℓ'} → {C : Category {ℓ} {ℓ'}} → Opposite (Opposite C) ≡ C
-- Object (Opposite-is-involution {C = C} i) = Object C
-- Arrow (Opposite-is-involution i) = {!!}
-- 𝟙 (Opposite-is-involution i) = {!!}
-- _⊕_ (Opposite-is-involution i) = {!!}
-- assoc (Opposite-is-involution i) = {!!}
-- ident (Opposite-is-involution i) = {!!}
Product : {ℓ : Level} → ( C D : Category {ℓ} {ℓ} ) → Category {ℓ} {ℓ}
Product C D =
record
{ Object = C.Object × D.Object
; Arrow = λ { (c , d) (c' , d') →
let carr = C.Arrow c c'
darr = D.Arrow d d'
in carr × darr}
; 𝟙 = C.𝟙 , D.𝟙
; _⊕_ = λ { (bc∈C , bc∈D) (ab∈C , ab∈D) → bc∈C C.⊕ ab∈C , bc∈D D.⊕ ab∈D}
; assoc = eqpair C.assoc D.assoc
; ident =
let (Cl , Cr) = C.ident
(Dl , Dr) = D.ident
in eqpair Cl Dl , eqpair Cr Dr
}
where
open module C = Category C
open module D = Category D
-- Two pairs are equal if their components are equal.
eqpair : {ℓ : Level} → { A : Set ℓ } → { B : Set ℓ } → { a a' : A } → { b b' : B } → a ≡ a' → b ≡ b' → (a , b) ≡ (a' , b')
eqpair {a = a} {b = b} eqa eqb = subst eqa (subst eqb (refl {x = (a , b)}))
Opposite : ∀ {ℓ ℓ'} → Category {ℓ} {ℓ'} → Category {ℓ} {ℓ'}
Opposite ℂ =
record
{ Object = ℂ.Object
; Arrow = λ A B → ℂ.Arrow B A
; 𝟙 = ℂ.𝟙
; _⊕_ = λ g f → f ℂ.⊕ g
; assoc = sym ℂ.assoc
; ident = swap ℂ.ident
}
where
open module ℂ = Category ℂ
Hom : {ℓ ℓ' : Level} → (ℂ : Category {ℓ} {ℓ'}) → (A B : Object ℂ) → Set ℓ'
Hom : {ℓ ℓ' : Level} → (ℂ : Category ℓ ℓ') → (A B : Object ℂ) → Set ℓ'
Hom ℂ A B = Arrow ℂ A B
module _ {ℓ ℓ' : Level} {ℂ : Category {ℓ} {ℓ'}} where
private
Obj = Object ℂ
Arr = Arrow ℂ
_+_ = _⊕_ ℂ
HomFromArrow : (A : Obj) → {B B' : Obj} → (g : Arr B B')
module _ {ℓ ℓ' : Level} {ℂ : Category ℓ ℓ'} where
HomFromArrow : (A : ℂ .Object) → {B B' : ℂ .Object} → (g : ℂ .Arrow B B')
→ Hom ℂ A B → Hom ℂ A B'
HomFromArrow _A g = λ f → g + f
HomFromArrow _A = _⊕_ ℂ

7
src/Cat/Category/Bij.agda
View file

@ -1,6 +1,9 @@
{-# OPTIONS --cubical #-}
{-# OPTIONS --cubical --allow-unsolved-metas #-}
module Cat.Category.Bij where
open import Cubical.PathPrelude hiding ( Id )
open import Cubical.FromStdLib
module _ {A : Set} {a : A} {P : A → Set} where
Q : A → Set
@ -20,7 +23,7 @@ module _ {A : Set} {a : A} {P : A → Set} where
w x = {!!}
vw-bij : (a : P a) → (w ∘ v) a ≡ a
vw-bij a = refl
vw-bij a = {!!}
-- tubij a with (t ∘ u) a
-- ... | q = {!!}

22
src/Cat/Category/Free.agda
View file

@ -1,21 +1,20 @@
module Category.Free where
module Cat.Category.Free where
open import Agda.Primitive
open import Cubical.PathPrelude hiding (Path)
open import Data.Product
open import Category as C
open import Cat.Category as C
module _ {ℓ ℓ' : Level} (ℂ : Category {ℓ} {ℓ'}) where
module _ {ℓ ℓ' : Level} (ℂ : Category ℓ ℓ') where
private
open module ℂ = Category ℂ
Obj = ℂ.Object
Path : ( a b : Obj ) → Set
Path a b = undefined
postulate emptyPath : (o : Obj) → Path o o
postulate concatenate : {a b c : Obj} → Path b c → Path a b → Path a c
postulate
Path : ( a b : Obj ) → Set ℓ'
emptyPath : (o : Obj) → Path o o
concatenate : {a b c : Obj} → Path b c → Path a b → Path a c
private
module _ {A B C D : Obj} {r : Path A B} {q : Path B C} {p : Path C D} where
@ -27,12 +26,11 @@ module _ {ℓ ℓ' : Level} (ℂ : Category {ℓ} {ℓ'}) where
ident-r : concatenate {A} {A} {B} p (emptyPath A) ≡ p
ident-l : concatenate {A} {B} {B} (emptyPath B) p ≡ p
Free : Category
Free : Category ℓ ℓ'
Free = record
{ Object = Obj
; Arrow = Path
; 𝟙 = λ {o} → emptyPath o
; _⊕_ = λ {a b c} → concatenate {a} {b} {c}
; assoc = p-assoc
; ident = ident-r , ident-l
; isCategory = record { assoc = p-assoc ; ident = ident-r , ident-l }
}

3
src/Cat/Category/Pathy.agda
View file

@ -1,7 +1,8 @@
{-# OPTIONS --cubical #-}
module Category.Pathy where
module Cat.Category.Pathy where
open import Level
open import Cubical.PathPrelude
{-

50
src/Cat/Category/Properties.agda
View file

@ -2,22 +2,64 @@
module Cat.Category.Properties where
open import Agda.Primitive
open import Data.Product
open import Cubical.PathPrelude
open import Cat.Category
open import Cat.Functor
open import Cat.Categories.Sets
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
open IsCategory (isCategory)
iso-is-epi : Isomorphism {ℂ = ℂ} f → Epimorphism {ℂ = ℂ} {X = X} f
iso-is-epi (f- , left-inv , right-inv) g₀ g₁ eq =
begin
g₀ ≡⟨ sym (proj₁ ident) ⟩
g₀ ⊕ 𝟙 ≡⟨ cong (_⊕_ g₀) (sym right-inv) ⟩
g₀ ⊕ (f ⊕ f-) ≡⟨ assoc ⟩
(g₀ ⊕ f) ⊕ f- ≡⟨ cong (λ φ → φ ⊕ f-) eq ⟩
(g₁ ⊕ f) ⊕ f- ≡⟨ sym assoc ⟩
g₁ ⊕ (f ⊕ f-) ≡⟨ cong (_⊕_ g₁) right-inv ⟩
g₁ ⊕ 𝟙 ≡⟨ proj₁ ident ⟩
g₁ ∎
iso-is-mono : Isomorphism {ℂ = ℂ} f → Monomorphism {ℂ = ℂ} {X = X} f
iso-is-mono (f- , (left-inv , right-inv)) g₀ g₁ eq =
begin
g₀ ≡⟨ sym (proj₂ ident) ⟩
𝟙 ⊕ g₀ ≡⟨ cong (λ φ → φ ⊕ g₀) (sym left-inv) ⟩
(f- ⊕ f) ⊕ g₀ ≡⟨ sym assoc ⟩
f- ⊕ (f ⊕ g₀) ≡⟨ cong (_⊕_ f-) eq ⟩
f- ⊕ (f ⊕ g₁) ≡⟨ assoc ⟩
(f- ⊕ f) ⊕ g₁ ≡⟨ cong (λ φ → φ ⊕ g₁) left-inv ⟩
𝟙 ⊕ g₁ ≡⟨ proj₂ ident ⟩
g₁ ∎
iso-is-epi-mono : Isomorphism {ℂ = ℂ} f → Epimorphism {ℂ = ℂ} {X = X} f × Monomorphism {ℂ = ℂ} {X = X} f
iso-is-epi-mono iso = iso-is-epi iso , iso-is-mono iso
{-
epi-mono-is-not-iso : ∀ {ℓ ℓ'} → ¬ ((ℂ : Category {ℓ} {ℓ'}) {A B X : Object ℂ} (f : Arrow ℂ A B ) → Epimorphism {ℂ = ℂ} {X = X} f → Monomorphism {ℂ = ℂ} {X = X} f → Isomorphism {ℂ = ℂ} f)
epi-mono-is-not-iso f =
let k = f {!!} {!!} {!!} {!!}
in {!!}
-}
module _ {ℓa ℓa' ℓb ℓb'} where
Exponential : Category {ℓa} {ℓa'} → Category {ℓb} {ℓb'} → Category {{!!}} {{!!}}
Exponential : Category ℓa ℓa' → Category ℓb ℓb' → Category {!!} {!!}
Exponential A B = record
{ Object = {!!}
; Arrow = {!!}
; 𝟙 = {!!}
; _⊕_ = {!!}
; assoc = {!!}
; ident = {!!}
; isCategory = {!!}
}
_⇑_ = Exponential
yoneda : ∀ {ℓ ℓ'} → {ℂ : Category {ℓ} {ℓ'}} → Functor ℂ (Sets ⇑ (Opposite ℂ))
yoneda : ∀ {ℓ ℓ'} → {ℂ : Category ℓ ℓ'} → Functor ℂ (Sets ⇑ (Opposite ℂ))
yoneda = {!!}

13
src/Cat/Cubical.agda
View file

@ -1,3 +1,4 @@
{-# OPTIONS --allow-unsolved-metas #-}
module Cat.Cubical where
open import Agda.Primitive
@ -9,8 +10,13 @@ open import Data.Empty
open import Cat.Category
-- See chapter 1 for a discussion on how presheaf categories are CwF's.
-- See section 6.8 in Huber's thesis for details on how to implement the
-- categorical version of CTT
module _ {ℓ ℓ' : Level} (Ns : Set ℓ) where
-- Σ is the "namespace"
-- Ns is the "namespace"
ℓo = (lsuc lzero ⊔ ℓ)
FiniteDecidableSubset : Set ℓ
@ -36,13 +42,12 @@ module _ {ℓ ℓ' : Level} (Ns : Set ℓ) where
Mor = Σ themap rules
-- The category of names and substitutions
ℂ : Category -- {ℓo} {lsuc lzero ⊔ ℓo}
ℂ : Category ℓ ℓ -- ℓo (lsuc lzero ⊔ ℓo)
ℂ = record
-- { Object = FiniteDecidableSubset
{ Object = Ns → Bool
; Arrow = Mor
; 𝟙 = {!!}
; _⊕_ = {!!}
; assoc = {!!}
; ident = {!!}
; isCategory = ?
}

50
src/Cat/Functor.agda
View file

@ -6,7 +6,7 @@ open import Function
open import Cat.Category
record Functor {ℓc ℓc' ℓd ℓd'} (C : Category {ℓc} {ℓc'}) (D : Category {ℓd} {ℓd'})
record Functor {ℓc ℓc' ℓd ℓd'} (C : Category ℓc ℓc') (D : Category ℓd ℓd')
: Set (ℓc ⊔ ℓc' ⊔ ℓd ⊔ ℓd') where
private
open module C = Category C
@ -21,43 +21,41 @@ record Functor {ℓc ℓc' ℓd ℓd'} (C : Category {ℓc} {ℓc'}) (D : Catego
distrib : { c c' c'' : C.Object} {a : C.Arrow c c'} {a' : C.Arrow c' c''}
→ func→ (a' C.⊕ a) ≡ func→ a' D.⊕ func→ a
module _ {ℓ ℓ' : Level} {A B C : Category {ℓ} {ℓ'}} (F : Functor B C) (G : Functor A B) where
module _ {ℓ ℓ' : Level} {A B C : Category ℓ ℓ'} (F : Functor B C) (G : Functor A B) where
open Functor
open Category
private
open module F = Functor F
open module G = Functor G
open module A = Category A
open module B = Category B
open module C = Category C
F* = F .func*
F→ = F .func→
G* = G .func*
G→ = G .func→
_A⊕_ = A ._⊕_
_B⊕_ = B ._⊕_
_C⊕_ = C ._⊕_
module _ {a0 a1 a2 : A .Object} {α0 : A .Arrow a0 a1} {α1 : A .Arrow a1 a2} where
F* = F.func*
F→ = F.func→
G* = G.func*
G→ = G.func→
module _ {a0 a1 a2 : A.Object} {α0 : A.Arrow a0 a1} {α1 : A.Arrow a1 a2} where
dist : (F→ ∘ G→) (α1 A.⊕ α0) ≡ (F→ ∘ G→) α1 C.⊕ (F→ ∘ G→) α0
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 C.⊕ (F→ ∘ G→) α0 ∎
(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 C⊕ (F→ ∘ G→) α0 ∎
functor-comp : Functor A C
functor-comp =
_∘f_ : Functor A C
_∘f_ =
record
{ func* = F* ∘ G*
; func→ = F→ ∘ G→
; ident = begin
(F→ ∘ G→) (A.𝟙) ≡⟨ refl ⟩
F→ (G→ (A.𝟙)) ≡⟨ cong F→ G.ident ⟩
F→ (B.𝟙) ≡⟨ F.ident ⟩
C.𝟙 ∎
(F→ ∘ G→) (A .𝟙) ≡⟨ refl ⟩
F→ (G→ (A .𝟙)) ≡⟨ cong F→ (G .ident)⟩
F→ (B .𝟙) ≡⟨ F .ident ⟩
C .𝟙 ∎
; distrib = dist
}
-- The identity functor
identity : {ℓ ℓ' : Level} → {C : Category {ℓ} {ℓ'}} → Functor C C
-- Identity = record { F* = λ x → x ; F→ = λ x → x ; ident = refl ; distrib = refl }
identity : ∀ {ℓ ℓ'} → {C : Category ℓ ℓ'} → Functor C C
identity = record
{ func* = λ x → x
; func→ = λ x → x