Tensor product of (co)monoids

Let $(𝐴, \nabla 𝐴, 𝜂 𝐴)$ and $(𝐵, \nabla 𝐵, 𝜂 𝐵)$ be monoids in a Symmetric monoidal category $𝖢$ with braiding $𝜏$ . Then the tensor product $𝐴 \otimes 𝐵$ is given the structure of a monoid where cat

\nabla 𝐴 \otimes 𝐵 = (\nabla 𝐴 \otimes \nabla 𝐵) (1 \otimes 𝜏 \otimes 1) 𝜂 𝐴 \otimes 𝐵 = 𝜂 𝐴 \otimes 𝜂 𝐵

up to application of the unitor and associator of $𝖢$ . In terms of string diagrams,

By duality (turning the diagrams upside down), one gets the same construction for tensor product of comonoids: If $(𝐴, Δ 𝐴, 𝜖 𝐴)$ and $(𝐵, Δ 𝐵, 𝜖 𝐵)$ are comonoids then $𝐴 \otimes 𝐵$ is given the structure of a comonoid where

Δ 𝐴 \otimes 𝐵 = (1 \otimes 𝜏 \otimes 1) (Δ 𝐴 \otimes Δ 𝐵) 𝜖 𝐴 \otimes 𝐵 = 𝜖 𝐴 \otimes 𝜖 𝐵

up to application of the unitor and associator of $𝖢$ .

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Tensor product of (co)monoids

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