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47 lines
1.4 KiB
Agda
47 lines
1.4 KiB
Agda
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{-# OPTIONS --without-K --safe #-}
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open import Categories.Category hiding (_[_,_])
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open import Level
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open import Function using (_$_)
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module Coproduct {o ℓ e} (𝒞 : Category o ℓ e) where
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open Category 𝒞
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open import Categories.Morphism.Reasoning 𝒞
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open import Categories.Morphism 𝒞
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open HomReasoning
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open import Categories.Object.Coproduct
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private
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variable
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A B C D : Obj
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f g h : A ⇒ B
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module _ {A B : Obj} where
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open Coproduct {𝒞 = 𝒞} {A = A} {B = B} renaming ([_,_] to _[_,_])
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repack : (p₁ p₂ : Coproduct 𝒞 A B) → A+B p₁ ⇒ A+B p₂
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repack p₁ p₂ = p₁ [ i₁ p₂ , i₂ p₂ ]
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repack∘ : (p₁ p₂ p₃ : Coproduct 𝒞 A B) → repack p₂ p₃ ∘ repack p₁ p₂ ≈ repack p₁ p₃
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repack∘ p₁ p₂ p₃ = ⟺ $ unique p₁
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(glueTrianglesˡ (inject₁ p₂) (inject₁ p₁))
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(glueTrianglesˡ (inject₂ p₂) (inject₂ p₁))
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repack≡id : (p : Coproduct 𝒞 A B) → repack p p ≈ id
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repack≡id = η
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repack-cancel : (p₁ p₂ : Coproduct 𝒞 A B) → repack p₁ p₂ ∘ repack p₂ p₁ ≈ id
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repack-cancel p₁ p₂ = repack∘ p₂ p₁ p₂ ○ repack≡id p₂
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up-to-iso : ∀ (p₁ p₂ : Coproduct 𝒞 A B) → Coproduct.A+B p₁ ≅ Coproduct.A+B p₂
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up-to-iso p₁ p₂ = record
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{ from = repack p₁ p₂
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; to = repack p₂ p₁
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; iso = record
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{ isoˡ = repack-cancel p₂ p₁
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; isoʳ = repack-cancel p₁ p₂
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}
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}
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