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35 lines
1 KiB
Agda
35 lines
1 KiB
Agda
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{-# OPTIONS --without-K --safe #-}
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-- Commutative Monad on a symmetric monoidal category
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-- https://ncatlab.org/nlab/show/commutative+monad
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module Monad.Commutative where
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open import Level
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open import Data.Product using (_,_)
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open import Categories.Category.Core
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open import Categories.Category.Monoidal
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open import Categories.Category.Monoidal.Symmetric
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open import Categories.Monad
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open import Categories.Monad.Strong
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private
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variable
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o ℓ e : Level
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record CommutativeMonad {C : Category o ℓ e} {V : Monoidal C} (S : Symmetric V) (T : StrongMonad V) : Set (o ⊔ ℓ ⊔ e) where
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open Category C
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open Symmetric S
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open StrongMonad T
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private
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σ : ∀ {X Y} → X ⊗₀ M.F.₀ Y ⇒ M.F.₀ (X ⊗₀ Y)
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σ {X} {Y} = strengthen.η (X , Y)
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τ : ∀ {X Y} → M.F.₀ X ⊗₀ Y ⇒ M.F.₀ (X ⊗₀ Y)
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τ {X} {Y} = M.F.₁ (braiding.⇐.η (X , Y)) ∘ σ ∘ braiding.⇒.η (M.F.₀ X , Y)
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field
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commutes : ∀ {X Y} → M.μ.η (X ⊗₀ Y) ∘ M.F.₁ τ ∘ σ ≈ M.μ.η (X ⊗₀ Y) ∘ M.F.₁ σ ∘ τ
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