# Tensor product of representations over a symmetric bimonoidal category

## Definition

Suppose $G$ is a group and $\mathcal{C}$ is a symmetric bimonoidal category with operation $\oplus$ playing the role analogous to addition and operation $\otimes$ playing a role analogous to multiplication. Suppose $A_1,A_2$ are objects of $\mathcal{C}$ and $\rho_1:G \to \operatorname{Aut}(A_1)$ and $\rho_2:G \to \operatorname{Aut}(A_2)$ are group homomorphisms. We can thus think of $(A_1,\rho_1)$ and $(A_2,\rho_2)$ as representations of $G$ over $\mathcal{C}$. We define the tensor product of these representations, as a group homomorphism $\rho_1 \otimes \rho_2:G \to A_1 \otimes A_2$ where $(\rho_1 \otimes \rho_2)(g) = \rho_1(g) \otimes \rho_2(g)$ where the $\otimes$ on the right is the natural mapping $\operatorname{Aut}(A_1) \times \operatorname{Aut}(A_2) \to \operatorname{Aut}(A_1 \times A_2)$.

## Particular cases

Symmetric bimonoidal category that we're starting with Meaning of tensor product in that category
category of sets, where $\oplus$ is the coproduct (disjoint union) and $\otimes$ is the product (Cartesian product) tensor product of permutation representations
category of vector spaces over a field $K$ , where $\oplus$ is the direct sum of vector spaces (both the product and the coproduct) and $\otimes$ is the tensor product of vector spaces tensor product of linear representations