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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.Monoid.Frobenius {o ℓ e : Level} {C : Category o ℓ e} {M : Monoidal C} (S : Symmetric M) where
open import Categories.Category.Monoidal.Symmetric.Properties S using () renaming (symmetric-Op to Sᵒᵖ)
open import Object.Monoid.Commutative using (IsCommutativeMonoid)
open Category C
record IsSpecialCommutativeFrobeniusMonoid (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
special : μ ∘ δ ≈ id
record SpecialCommutativeFrobeniusMonoid : Set (o ⊔ ℓ ⊔ e) where
field
Carrier : Obj
isSpecialCommutativeFrobeniusMonoid : IsSpecialCommutativeFrobeniusMonoid Carrier
open IsSpecialCommutativeFrobeniusMonoid isSpecialCommutativeFrobeniusMonoid public
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