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{-# OPTIONS --without-K --safe #-}
open import Categories.Category using (Category)
open import Categories.Category.Monoidal using (Monoidal)
open import Categories.Category.Monoidal.Symmetric using (Symmetric)
open import Level using (Level; _⊔_)
module Object.Bimonoid {o ℓ e : Level} {C : Category o ℓ e} {M : Monoidal C} (S : Symmetric M) where
import Categories.Category.Monoidal.Interchange.Braided as Interchange
open import Categories.Category.Monoidal.Properties M using () renaming (monoidal-Op to Mᵒᵖ)
open import Categories.Category.Monoidal.Symmetric.Properties S using () renaming (symmetric-Op to Sᵒᵖ)
open import Categories.Category.Monoidal.Utilities M using (module Shorthands)
open import Categories.Object.Monoid using (IsMonoid)
open import Object.Monoid.Commutative using (IsCommutativeMonoid)
open Category C
open Symmetric S
open Interchange braided using (module swapInner)
open Shorthands using (λ⇒; λ⇐)
open swapInner renaming (from to i⇒)
record IsBimonoid (A : Obj) : Set (ℓ ⊔ e) where
field
monoid : IsMonoid M A
comonoid : IsMonoid Mᵒᵖ A
open IsMonoid monoid
using (μ; η)
renaming (assoc to μ-assoc; identityˡ to μ-η-identityˡ; identityʳ to μ-η-identityʳ)
public
open IsMonoid comonoid
using ()
renaming (μ to δ; η to ϵ; assoc to δ-assoc; identityˡ to δ-ϵ-identityˡ; identityʳ to δ-ϵ-identityʳ)
public
field
μ-δ-compat : δ ∘ μ ≈ μ ⊗₁ μ ∘ i⇒ ∘ δ ⊗₁ δ
μ-ϵ-compat : ϵ ∘ μ ≈ λ⇒ ∘ ϵ ⊗₁ ϵ
δ-η-compat : δ ∘ η ≈ η ⊗₁ η ∘ λ⇐
extra : ϵ ∘ η ≈ id
record Bimonoid : Set (o ⊔ ℓ ⊔ e) where
field
Carrier : Obj
isBimonoid : IsBimonoid Carrier
open IsBimonoid isBimonoid public
record IsBicommutativeBimonoid (A : Obj) : Set (ℓ ⊔ e) where
field
commutativeMonoid : IsCommutativeMonoid S A
cocommutativeComonoid : IsCommutativeMonoid Sᵒᵖ A
open IsCommutativeMonoid commutativeMonoid
using (μ; η; commutative)
renaming (assoc to μ-assoc; identityˡ to μ-η-identityˡ; identityʳ to μ-η-identityʳ)
public
open IsCommutativeMonoid cocommutativeComonoid
using ()
renaming (μ to δ; η to ϵ; assoc to δ-assoc; identityˡ to δ-ϵ-identityˡ; identityʳ to δ-ϵ-identityʳ; commutative to cocommutative)
public
field
μ-η-compat : δ ∘ μ ≈ μ ⊗₁ μ ∘ i⇒ ∘ δ ⊗₁ δ
μ-ϵ-compat : ϵ ∘ μ ≈ λ⇒ ∘ ϵ ⊗₁ ϵ
δ-η-compat : δ ∘ η ≈ η ⊗₁ η ∘ λ⇐
extra : ϵ ∘ η ≈ id
record BicommutativeBimonoid : Set (o ⊔ ℓ ⊔ e) where
field
Carrier : Obj
isBicommutativeBimonoid : IsBicommutativeBimonoid Carrier
open IsBicommutativeBimonoid isBicommutativeBimonoid public
record IsSpecialBicommutativeBimonoid (A : Obj) : Set (ℓ ⊔ e) where
field
bimonoid : IsBicommutativeBimonoid A
open IsBicommutativeBimonoid bimonoid public
field
special : μ ∘ δ ≈ id
record SpecialBicommutativeBimonoid : Set (o ⊔ ℓ ⊔ e) where
field
Carrier : Obj
isSpecialBicommutativeBimonoid : IsSpecialBicommutativeBimonoid Carrier
open IsSpecialBicommutativeBimonoid isSpecialBicommutativeBimonoid public
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