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{-# OPTIONS --without-K --safe #-}
open import Algebra using (Semiring)
open import Level using (Level; suc; _⊔_)
module Category.Instance.Bisemimodules {c₁ c₂ ℓ₁ ℓ₂ m ℓm : Level} (R : Semiring c₁ ℓ₁) (S : Semiring c₂ ℓ₂) where
import Algebra.Module.Morphism.Construct.Composition as Compose
import Algebra.Module.Morphism.Construct.Identity as Identity
open import Algebra.Module using (Bisemimodule)
open import Algebra.Module.Morphism.Structures using (IsBisemimoduleHomomorphism)
open import Categories.Category using (Category)
open import Categories.Category.Helper using (categoryHelper)
open import Relation.Binary.PropositionalEquality as ≡ using (_≗_)
record BisemimoduleHomomorphism (M N : Bisemimodule R S m ℓm) : Set (c₁ ⊔ c₂ ⊔ m ⊔ ℓm) where
private
module M = Bisemimodule M
module N = Bisemimodule N
field
⟦_⟧ : M.Carrierᴹ → N.Carrierᴹ
isBisemimoduleHomomorphism : IsBisemimoduleHomomorphism M.rawBisemimodule N.rawBisemimodule ⟦_⟧
id : (M : Bisemimodule R S m ℓm) → BisemimoduleHomomorphism M M
id M = record
{ isBisemimoduleHomomorphism = Identity.isBisemimoduleHomomorphism M.rawBisemimodule M.≈ᴹ-refl
}
where
module M = Bisemimodule M
compose
: (M N P : Bisemimodule R S m ℓm)
→ BisemimoduleHomomorphism N P
→ BisemimoduleHomomorphism M N
→ BisemimoduleHomomorphism M P
compose M N P f g = record
{ isBisemimoduleHomomorphism =
Compose.isBisemimoduleHomomorphism
P.≈ᴹ-trans
g.isBisemimoduleHomomorphism
f.isBisemimoduleHomomorphism
}
where
module f = BisemimoduleHomomorphism f
module g = BisemimoduleHomomorphism g
module P = Bisemimodule P
open BisemimoduleHomomorphism
Bisemimodules : Category (c₁ ⊔ c₂ ⊔ ℓ₁ ⊔ ℓ₂ ⊔ suc (m ⊔ ℓm)) (c₁ ⊔ c₂ ⊔ m ⊔ ℓm) m
Bisemimodules = categoryHelper record
{ Obj = Bisemimodule R S m ℓm
; _⇒_ = BisemimoduleHomomorphism
; _≈_ = λ f g → ⟦ f ⟧ ≗ ⟦ g ⟧
; id = λ {M} → id M
; _∘_ = λ {M N P} f g → compose M N P f g
; assoc = λ _ → ≡.refl
; identityˡ = λ _ → ≡.refl
; identityʳ = λ _ → ≡.refl
; equiv = record
{ refl = λ _ → ≡.refl
; sym = λ f≈g x → ≡.sym (f≈g x)
; trans = λ f≈g g≈h x → ≡.trans (f≈g x) (g≈h x)
}
; ∘-resp-≈ = λ {f = f} {h g i} eq₁ eq₂ x → ≡.trans (≡.cong ⟦ f ⟧ (eq₂ x)) (eq₁ (⟦ i ⟧ x))
}
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