43 lines
1.3 KiB
Agda
43 lines
1.3 KiB
Agda
{-# OPTIONS --allow-unsolved-metas #-}
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module Cat.Category.Free where
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open import Agda.Primitive
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open import Cubical hiding (Path)
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open import Data.Product
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open import Cat.Category as C
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module _ {ℓ ℓ' : Level} (ℂ : Category ℓ ℓ') where
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private
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open module ℂ = Category ℂ
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postulate
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Path : (a b : ℂ.Object) → Set ℓ'
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emptyPath : (o : ℂ.Object) → Path o o
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concatenate : {a b c : ℂ.Object} → Path b c → Path a b → Path a c
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private
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module _ {A B C D : ℂ.Object} {r : Path A B} {q : Path B C} {p : Path C D} where
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postulate
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p-assoc : concatenate {A} {C} {D} p (concatenate {A} {B} {C} q r)
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≡ concatenate {A} {B} {D} (concatenate {B} {C} {D} p q) r
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module _ {A B : ℂ.Object} {p : Path A B} where
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postulate
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ident-r : concatenate {A} {A} {B} p (emptyPath A) ≡ p
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ident-l : concatenate {A} {B} {B} (emptyPath B) p ≡ p
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RawFree : RawCategory ℓ ℓ'
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RawFree = record
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{ Object = ℂ.Object
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; Arrow = Path
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; 𝟙 = λ {o} → emptyPath o
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; _∘_ = λ {a b c} → concatenate {a} {b} {c}
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}
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RawIsCategoryFree : IsCategory RawFree
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RawIsCategoryFree = record
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{ assoc = p-assoc
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; ident = ident-r , ident-l
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; arrow-is-set = {!!}
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; univalent = {!!}
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}
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