cat/src/Cat/Categories/Cat.agda

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{-# OPTIONS --cubical --allow-unsolved-metas #-}
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 Cat.Category
open import Cat.Functor
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-- Tip from Andrea:
-- Use co-patterns - they help with showing more understandable types in goals.
lift-eq : {} {A B : Set } {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
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eqpair : {a b} {A : Set a} {B : Set b} {a a' : A} {b b' : B}
a a' b b' (a , b) (a' , b')
eqpair eqa eqb i = eqa i , eqb i
open Functor
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open IsFunctor
open Category hiding (_∘_)
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-- The category of categories
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module _ ( ' : Level) where
private
module _ {A B C D : Category '} {f : Functor A B} {g : Functor B C} {h : Functor C D} where
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private
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
postulate eqI : PathP
(λ i {c : A .Object} eq→ i (A .𝟙 {c}) D .𝟙 {eq* i c})
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((h ∘f (g ∘f f)) .isFunctor .ident)
(((h ∘f g) ∘f f) .isFunctor .ident)
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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 ])
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((h ∘f (g ∘f f)) .isFunctor .distrib) (((h ∘f g) ∘f f) .isFunctor .distrib)
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assc : h ∘f (g ∘f f) (h ∘f g) ∘f f
assc = Functor≡ eq* eq→ eqI eqD
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module _ { 𝔻 : Category '} {F : Functor 𝔻} where
module _ where
private
eq* : (func* F) (func* (identity {C = })) func* F
eq* = refl
-- lemmm : func→ {C = A} {D = B} (f ∘f identity) ≡ func→ f
eq→ : PathP
(λ i
{x y : Object } Arrow x y Arrow 𝔻 (func* F x) (func* F y))
(func→ (F ∘f identity)) (func→ F)
eq→ = refl
postulate
eqI-r : PathP (λ i {c : .Object}
PathP (λ _ Arrow 𝔻 (func* F c) (func* F c)) (func→ F ( .𝟙)) (𝔻 .𝟙))
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((F ∘f identity) .isFunctor .ident) (F .isFunctor .ident)
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eqD-r : PathP
(λ i
{A B C : .Object} {f : .Arrow A B} {g : .Arrow B C}
eq→ i ( [ g f ]) 𝔻 [ eq→ i g eq→ i f ])
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((F ∘f identity) .isFunctor .distrib) (F .isFunctor .distrib)
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ident-r : F ∘f identity F
ident-r = Functor≡ eq* eq→ eqI-r eqD-r
module _ where
private
postulate
eq* : (identity ∘f F) .func* F .func*
eq→ : PathP
(λ i {x y : Object } .Arrow x y 𝔻 .Arrow (eq* i x) (eq* i y))
((identity ∘f F) .func→) (F .func→)
eqI : PathP (λ i {A : .Object} eq→ i ( .𝟙 {A}) 𝔻 .𝟙 {eq* i A})
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((identity ∘f F) .isFunctor .ident) (F .isFunctor .ident)
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eqD : PathP (λ i {A B C : .Object} {f : .Arrow A B} {g : .Arrow B C}
eq→ i ( [ g f ]) 𝔻 [ eq→ i g eq→ i f ])
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((identity ∘f F) .isFunctor .distrib) (F .isFunctor .distrib)
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ident-l : identity ∘f F F
ident-l = Functor≡ eq* eq→ eqI eqD
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Cat : Category (lsuc ( ')) ( ')
Cat =
record
{ Object = Category '
; Arrow = Functor
; 𝟙 = identity
; _∘_ = _∘f_
-- What gives here? Why can I not name the variables directly?
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; isCategory = record
{ assoc = λ {_ _ _ _ f g h} assc {f = f} {g = g} {h = h}
; ident = ident-r , ident-l
}
}
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module _ { ' : Level} where
Catt = Cat '
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module _ ( 𝔻 : Category ') where
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private
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:Object: = .Object × 𝔻 .Object
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:Arrow: : :Object: :Object: Set '
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:Arrow: (c , d) (c' , d') = Arrow c c' × Arrow 𝔻 d d'
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:𝟙: : {o : :Object:} :Arrow: o o
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:𝟙: = .𝟙 , 𝔻 .𝟙
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_:⊕:_ :
{a b c : :Object:}
:Arrow: b c
:Arrow: a b
:Arrow: a c
_:⊕:_ = λ { (bc∈C , bc∈D) (ab∈C , ab∈D) [ bc∈C ab∈C ] , 𝔻 [ bc∈D ab∈D ]}
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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
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open module C = IsCategory ( .isCategory)
open module D = IsCategory (𝔻 .isCategory)
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:product: : Category '
:product: = record
{ Object = :Object:
; Arrow = :Arrow:
; 𝟙 = :𝟙:
; _∘_ = _:⊕:_
}
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proj₁ : Arrow Catt :product:
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proj₁ = record { func* = fst ; func→ = fst ; isFunctor = record { ident = refl ; distrib = refl } }
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proj₂ : Arrow Catt :product: 𝔻
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proj₂ = record { func* = snd ; func→ = snd ; isFunctor = record { ident = refl ; distrib = refl } }
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module _ {X : Object Catt} (x₁ : Arrow Catt X ) (x₂ : Arrow Catt X 𝔻) where
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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
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; isFunctor = record
{ ident = lift-eq x₁.ident x₂.ident
; distrib = lift-eq x₁.distrib x₂.distrib
}
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}
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where
open module x = IsFunctor (x₁ .isFunctor)
open module x = IsFunctor (x₂ .isFunctor)
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-- 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 : Catt [ proj₁ x ] x₁
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-- isUniqL = Functor≡ refl refl {!!} {!!}
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postulate isUniqR : Catt [ proj₂ x ] x₂
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-- isUniqR = Functor≡ refl refl {!!} {!!}
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isUniq : Catt [ proj₁ x ] x₁ × Catt [ proj₂ x ] x₂
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isUniq = isUniqL , isUniqR
uniq : ∃![ x ] (Catt [ proj₁ x ] x₁ × Catt [ proj₂ x ] x₂)
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uniq = x , isUniq
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instance
isProduct : IsProduct (Cat ') proj₁ proj₂
isProduct = uniq
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product : Product { = (Cat ')} 𝔻
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product = record
{ obj = :product:
; proj₁ = proj₁
; proj₂ = proj₂
}
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module _ { ' : Level} where
instance
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hasProducts : HasProducts (Cat ')
hasProducts = record { product = product }
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-- Basically proves that `Cat ` is cartesian closed.
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module _ ( : Level) where
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private
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open Data.Product
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open import Cat.Categories.Fun
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Cat : Category (lsuc ( )) ( )
Cat = Cat
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module _ ( 𝔻 : Category ) where
private
:obj: : Cat .Object
:obj: = Fun { = } {𝔻 = 𝔻}
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:func*: : Functor 𝔻 × .Object 𝔻 .Object
:func*: (F , A) = F .func* A
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module _ {dom cod : Functor 𝔻 × .Object} where
private
F : Functor 𝔻
F = proj₁ dom
A : .Object
A = proj₂ dom
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G : Functor 𝔻
G = proj₁ cod
B : .Object
B = proj₂ cod
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:func→: : (pobj : NaturalTransformation F G × .Arrow A B)
𝔻 .Arrow (F .func* A) (G .func* B)
:func→: ((θ , θNat) , f) = result
where
θA : 𝔻 .Arrow (F .func* A) (G .func* A)
θA = θ A
θB : 𝔻 .Arrow (F .func* B) (G .func* B)
θB = θ B
F→f : 𝔻 .Arrow (F .func* A) (F .func* B)
F→f = F .func→ f
G→f : 𝔻 .Arrow (G .func* A) (G .func* B)
G→f = G .func→ f
l : 𝔻 .Arrow (F .func* A) (G .func* B)
l = 𝔻 [ θB F→f ]
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r : 𝔻 .Arrow (F .func* A) (G .func* B)
r = 𝔻 [ G→f θA ]
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-- There are two choices at this point,
-- but I suppose the whole point is that
-- by `θNat f` we have `l ≡ r`
-- lem : 𝔻 [ θ B ∘ F .func→ f ] ≡ 𝔻 [ G .func→ f ∘ θ A ]
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-- lem = θNat f
result : 𝔻 .Arrow (F .func* A) (G .func* B)
result = l
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_×p_ = product
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module _ {c : Functor 𝔻 × .Object} where
private
F : Functor 𝔻
F = proj₁ c
C : .Object
C = proj₂ c
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-- NaturalTransformation F G × .Arrow A B
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-- :ident: : :func→: {c} {c} (identityNat F , .𝟙) 𝔻 .𝟙
-- :ident: = trans (proj₂ 𝔻.ident) (F .ident)
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-- where
-- open module 𝔻 = IsCategory (𝔻 .isCategory)
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-- Unfortunately the equational version has some ambigous arguments.
:ident: : :func→: {c} {c} (identityNat F , .𝟙 {o = proj₂ c}) 𝔻 .𝟙
:ident: = begin
:func→: {c} {c} ((:obj: ×p ) .Product.obj .𝟙 {c}) ≡⟨⟩
:func→: {c} {c} (identityNat F , .𝟙) ≡⟨⟩
𝔻 [ identityTrans F C F .func→ ( .𝟙)] ≡⟨⟩
𝔻 [ 𝔻 .𝟙 F .func→ ( .𝟙)] ≡⟨ proj₂ 𝔻.ident
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F .func→ ( .𝟙) ≡⟨ F.ident
𝔻 .𝟙
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where
open module 𝔻 = IsCategory (𝔻 .isCategory)
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open module F = IsFunctor (F .isFunctor)
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module _ {F×A G×B H×C : Functor 𝔻 × .Object} where
F = F×A .proj₁
A = F×A .proj₂
G = G×B .proj₁
B = G×B .proj₂
H = H×C .proj₁
C = H×C .proj₂
-- Not entirely clear what this is at this point:
_P⊕_ = (:obj: ×p ) .Product.obj .Category._∘_ {F×A} {G×B} {H×C}
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module _
-- NaturalTransformation F G × .Arrow A B
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{θ×f : NaturalTransformation F G × .Arrow A B}
{η×g : NaturalTransformation G H × .Arrow B C} where
private
θ : Transformation F G
θ = proj₁ (proj₁ θ×f)
θNat : Natural F G θ
θNat = proj₂ (proj₁ θ×f)
f : .Arrow A B
f = proj₂ θ×f
η : Transformation G H
η = proj₁ (proj₁ η×g)
ηNat : Natural G H η
ηNat = proj₂ (proj₁ η×g)
g : .Arrow B C
g = proj₂ η×g
ηθNT : NaturalTransformation F H
ηθNT = Fun .Category._∘_ {F} {G} {H} (η , ηNat) (θ , θNat)
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ηθ = proj₁ ηθNT
ηθNat = proj₂ ηθNT
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:distrib: :
𝔻 [ 𝔻 [ η C θ C ] F .func→ ( [ g f ] ) ]
𝔻 [ 𝔻 [ η C G .func→ g ] 𝔻 [ θ B F .func→ f ] ]
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:distrib: = begin
𝔻 [ (ηθ C) F .func→ ( [ g f ]) ]
≡⟨ ηθNat ( [ g f ])
𝔻 [ H .func→ ( [ g f ]) (ηθ A) ]
≡⟨ cong (λ φ 𝔻 [ φ ηθ A ]) (H.distrib)
𝔻 [ 𝔻 [ H .func→ g H .func→ f ] (ηθ A) ]
≡⟨ sym assoc
𝔻 [ H .func→ g 𝔻 [ H .func→ f ηθ A ] ]
≡⟨ cong (λ φ 𝔻 [ H .func→ g φ ]) assoc
𝔻 [ H .func→ g 𝔻 [ 𝔻 [ H .func→ f η A ] θ A ] ]
≡⟨ cong (λ φ 𝔻 [ H .func→ g φ ]) (cong (λ φ 𝔻 [ φ θ A ]) (sym (ηNat f)))
𝔻 [ H .func→ g 𝔻 [ 𝔻 [ η B G .func→ f ] θ A ] ]
≡⟨ cong (λ φ 𝔻 [ H .func→ g φ ]) (sym assoc)
𝔻 [ H .func→ g 𝔻 [ η B 𝔻 [ G .func→ f θ A ] ] ] ≡⟨ assoc
𝔻 [ 𝔻 [ H .func→ g η B ] 𝔻 [ G .func→ f θ A ] ]
≡⟨ cong (λ φ 𝔻 [ φ 𝔻 [ G .func→ f θ A ] ]) (sym (ηNat g))
𝔻 [ 𝔻 [ η C G .func→ g ] 𝔻 [ G .func→ f θ A ] ]
≡⟨ cong (λ φ 𝔻 [ 𝔻 [ η C G .func→ g ] φ ]) (sym (θNat f))
𝔻 [ 𝔻 [ η C G .func→ g ] 𝔻 [ θ B F .func→ f ] ]
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where
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open IsCategory (𝔻 .isCategory)
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open module H = IsFunctor (H .isFunctor)
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:eval: : Functor ((:obj: ×p ) .Product.obj) 𝔻
:eval: = record
{ func* = :func*:
; func→ = λ {dom} {cod} :func→: {dom} {cod}
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; isFunctor = record
{ ident = λ {o} :ident: {o}
; distrib = λ {f u n k y} :distrib: {f} {u} {n} {k} {y}
}
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}
module _ (𝔸 : Category ) (F : Functor ((𝔸 ×p ) .Product.obj) 𝔻) where
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open HasProducts (hasProducts {} {}) using (parallelProduct)
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postulate
transpose : Functor 𝔸 :obj:
eq : Cat [ :eval: (parallelProduct transpose (Cat .𝟙 {o = })) ] F
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catTranspose : ∃![ F~ ] (Cat [ :eval: (parallelProduct F~ (Cat .𝟙 {o = }))] F )
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catTranspose = transpose , eq
:isExponential: : IsExponential Cat 𝔻 :obj: :eval:
:isExponential: = catTranspose
-- :exponent: : Exponential (Cat ) A B
:exponent: : Exponential Cat 𝔻
:exponent: = record
{ obj = :obj:
; eval = :eval:
; isExponential = :isExponential:
}
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hasExponentials : HasExponentials (Cat )
hasExponentials = record { exponent = :exponent: }