Tensor product of representations over a symmetric bimonoidal category

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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