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{-# OPTIONS --without-K --safe #-}
open import Categories.Category.Core using (Category)
open import Categories.Category.Monoidal.Core using (Monoidal)
open import Categories.Category.Monoidal.Symmetric using (Symmetric)
open import Level using (Level; _⊔_)
module Object.Monoid.Commutative {o ℓ e : Level} {𝒞 : Category o ℓ e} {M : Monoidal 𝒞} (sym : Symmetric M) where
open import Categories.Object.Monoid M using (IsMonoid; Monoid; Monoid⇒)
-- a commutative monoid object in a symmetric monoidal category
open Category 𝒞
open Symmetric sym using (module braiding; _⊗₁_)
record IsCommutativeMonoid (M : Obj) : Set (ℓ ⊔ e) where
field
isMonoid : IsMonoid M
open IsMonoid isMonoid public
field
commutative : μ ≈ μ ∘ braiding.⇒.η _
record CommutativeMonoid : Set (o ⊔ ℓ ⊔ e) where
field
Carrier : Obj
isCommutativeMonoid : IsCommutativeMonoid Carrier
open IsCommutativeMonoid isCommutativeMonoid public
monoid : Monoid
monoid = record { isMonoid = isMonoid }
open CommutativeMonoid
record CommutativeMonoid⇒ (M M′ : CommutativeMonoid) : Set (ℓ ⊔ e) where
module M = CommutativeMonoid M
module M′ = CommutativeMonoid M′
field
monoid⇒ : Monoid⇒ M.monoid M′.monoid
open Monoid⇒ monoid⇒ public
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