50 lines
2.0 KiB
Plaintext
50 lines
2.0 KiB
Plaintext
{-# OPTIONS --cubical --safe --exact-split #-}
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module Cubical.Structures.Str where
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open import Cubical.Foundations.Everything
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open import Cubical.Foundations.Equiv
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open import Cubical.Functions.Image
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open import Cubical.HITs.PropositionalTruncation as P
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open import Cubical.Data.Nat
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open import Cubical.Data.List as L
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open import Cubical.Data.Sigma
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open import Cubical.Reflection.RecordEquiv
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open import Cubical.HITs.SetQuotients as Q
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open import Agda.Primitive
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open import Cubical.Structures.Sig
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record struct {f a : Level} (n : Level) (σ : Sig f a) : Type (ℓ-max f (ℓ-max a (ℓ-suc n))) where
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constructor <_,_>
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field
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car : Type n
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alg : sig σ car -> car
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open struct public
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module _ {f a x y : Level} {σ : Sig f a} (𝔛 : struct x σ) (𝔜 : struct y σ) where
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structIsHom : (h : 𝔛 .car -> 𝔜 .car) -> Type (ℓ-max f (ℓ-max a (ℓ-max x y)))
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structIsHom h =
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((f : σ .symbol) -> (i : σ .arity f -> 𝔛 .car) -> 𝔜 .alg (f , h ∘ i) ≡ h (𝔛 .alg (f , i)))
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structHom : Type (ℓ-max f (ℓ-max a (ℓ-max x y)))
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structHom = Σ[ h ∈ (𝔛 .car -> 𝔜 .car) ] structIsHom h
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structHom≡ : (g h : structHom) -> isSet (𝔜 .car) -> g .fst ≡ h .fst -> g ≡ h
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structHom≡ (g-f , g-hom) (h-f , h-hom) isSetY =
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Σ≡Prop (\fun -> isPropΠ \f -> isPropΠ \o -> isSetY (𝔜 .alg (f , fun ∘ o)) (fun (𝔛 .alg (f , o))))
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module _ {f a x : Level} {σ : Sig f a} (𝔛 : struct x σ) where
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idHom : structHom 𝔛 𝔛
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idHom = idfun _ , \f i -> refl
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module _ {f a x y z : Level} {σ : Sig f a} (𝔛 : struct x σ) (𝔜 : struct y σ) (ℨ : struct z σ) where
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structHom∘ : (g : structHom 𝔜 ℨ) -> (h : structHom 𝔛 𝔜) -> structHom 𝔛 ℨ
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structHom∘ (g-f , g-hom) (h-f , h-hom) = g-f ∘ h-f , lemma-α
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where
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lemma-α : structIsHom 𝔛 ℨ (g-f ∘ h-f)
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lemma-α eqn i =
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ℨ .alg (eqn , g-f ∘ h-f ∘ i) ≡⟨ g-hom eqn (h-f ∘ i) ⟩
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g-f (𝔜 .alg (eqn , h-f ∘ i)) ≡⟨ cong g-f (h-hom eqn i) ⟩
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_ ∎
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