46 lines
1.2 KiB
Agda
46 lines
1.2 KiB
Agda
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module Cat.Category.Monoid where
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open import Agda.Primitive
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open import Cat.Category
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open import Cat.Category.Product
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open import Cat.Category.Functor
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import Cat.Categories.Cat as Cat
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-- TODO: Incorrect!
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module _ (ℓa ℓb : Level) where
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private
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ℓ = lsuc (ℓa ⊔ ℓb)
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-- Might not need this to be able to form products of categories!
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postulate unprovable : IsCategory (Cat.RawCat ℓa ℓb)
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open HasProducts (Cat.hasProducts unprovable)
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record RawMonoidalCategory : Set ℓ where
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field
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category : Category ℓa ℓb
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open Category category public
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field
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{{hasProducts}} : HasProducts category
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mempty : Object
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-- aka. tensor product, monoidal product.
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mappend : Functor (category × category) category
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record MonoidalCategory : Set ℓ where
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field
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raw : RawMonoidalCategory
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open RawMonoidalCategory raw public
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module _ {ℓa ℓb : Level} (ℂ : MonoidalCategory ℓa ℓb) where
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private
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ℓ = ℓa ⊔ ℓb
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module MC = MonoidalCategory ℂ
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open HasProducts MC.hasProducts
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record Monoid : Set ℓ where
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field
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carrier : MC.Object
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mempty : MC.Arrow (carrier × carrier) carrier
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mappend : MC.Arrow MC.mempty carrier
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