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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 using (Rel; IsEquivalence)
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 ⟦_⟧
open IsBisemimoduleHomomorphism isBisemimoduleHomomorphism public
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
_≈_ : {M N : Bisemimodule R S m ℓm} → Rel (BisemimoduleHomomorphism M N) (m ⊔ ℓm)
_≈_ {M} {N} f g = (x : M.Carrierᴹ) → ⟦ f ⟧ x N.≈ᴹ ⟦ g ⟧ x
where
module M = Bisemimodule M
module N = Bisemimodule N
≈-isEquiv : {M N : Bisemimodule R S m ℓm} → IsEquivalence (_≈_ {M} {N})
≈-isEquiv {M} {N} = record
{ refl = λ _ → N.≈ᴹ-refl
; sym = λ f≈g x → N.≈ᴹ-sym (f≈g x)
; trans = λ f≈g g≈h x → N.≈ᴹ-trans (f≈g x) (g≈h x)
}
where
module M = Bisemimodule M
module N = Bisemimodule N
∘-resp-≈
: {M N P : Bisemimodule R S m ℓm}
{f h : BisemimoduleHomomorphism N P}
{g i : BisemimoduleHomomorphism M N}
→ f ≈ h
→ g ≈ i
→ compose M N P f g ≈ compose M N P h i
∘-resp-≈ {M} {N} {P} {f} {g} {h} {i} f≈h g≈i x = P.≈ᴹ-trans (f.⟦⟧-cong (g≈i x)) (f≈h (⟦ i ⟧ x))
where
module P = Bisemimodule P
module f = BisemimoduleHomomorphism f
open Bisemimodule
Bisemimodules : Category (c₁ ⊔ c₂ ⊔ ℓ₁ ⊔ ℓ₂ ⊔ suc (m ⊔ ℓm)) (c₁ ⊔ c₂ ⊔ m ⊔ ℓm) (m ⊔ ℓm)
Bisemimodules = categoryHelper record
{ Obj = Bisemimodule R S m ℓm
; _⇒_ = BisemimoduleHomomorphism
; _≈_ = _≈_
; id = λ {M} → id M
; _∘_ = λ {M N P} f g → compose M N P f g
; assoc = λ {D = D} _ → ≈ᴹ-refl D
; identityˡ = λ {B = B} _ → ≈ᴹ-refl B
; identityʳ = λ {B = B} _ → ≈ᴹ-refl B
; equiv = ≈-isEquiv
; ∘-resp-≈ = λ {f = f} {g h i } → ∘-resp-≈ {f = f} {g} {h} {i}
}
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