cat/src/Cat/Categories/Fam.agda

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{-# OPTIONS --allow-unsolved-metas #-}
module Cat.Categories.Fam where
open import Agda.Primitive
open import Data.Product
open import Cubical
import Function
open import Cat.Category
open import Cat.Equality
open Equality.Data.Product
module _ (a b : Level) where
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private
Obj' = Σ[ A Set a ] (A Set b)
Arr : Obj' Obj' Set (a b)
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Arr (A , B) (A' , B') = Σ[ f (A A') ] ({x : A} B x B' (f x))
one : {o : Obj'} Arr o o
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proj₁ one = λ x x
proj₂ one = λ b b
_∘_ : {a b c : Obj'} Arr b c Arr a b Arr a c
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(g , g') (f , f') = g Function.∘ f , g' Function.∘ f'
_⟨_∘_⟩ : {a b : Obj'} (c : Obj') Arr b c Arr a b Arr a c
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c g f = _∘_ {c = c} g f
module _ {A B C D : Obj'} {f : Arr A B} {g : Arr B C} {h : Arr C D} where
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isAssociative : (D h C g f ) D D h g f
isAssociative = Σ≡ refl refl
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module _ {A B : Obj'} {f : Arr A B} where
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ident : B f one f × B one {B} f f
ident = (Σ≡ refl refl) , Σ≡ refl refl
RawFam : RawCategory (lsuc (a b)) (a b)
RawFam = record
{ Object = Obj'
; Arrow = Arr
; 𝟙 = one
; _∘_ = λ {a b c} _∘_ {a} {b} {c}
}
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instance
isCategory : IsCategory RawFam
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isCategory = record
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{ isAssociative = λ {A} {B} {C} {D} {f} {g} {h} isAssociative {D = D} {f} {g} {h}
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; ident = λ {A} {B} {f} ident {A} {B} {f = f}
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; arrowIsSet = {!!}
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; univalent = {!!}
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}
Fam : Category (lsuc (a b)) (a b)
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Category.raw Fam = RawFam