# Tensor product of representations over a symmetric bimonoidal category

From Groupprops

## Definition

Suppose is a group and is a symmetric bimonoidal category with operation playing the role analogous to addition and operation playing a role analogous to multiplication. Suppose are objects of and and are group homomorphisms. We can thus think of and as representations of over . We define the **tensor product** of these representations, as a group homomorphism where where the on the right is the natural mapping .

## Particular cases

Symmetric bimonoidal category that we're starting with | Meaning of tensor product in that category |
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category of sets, where is the coproduct (disjoint union) and is the product (Cartesian product) | tensor product of permutation representations |

category of vector spaces over a field , where is the direct sum of vector spaces (both the product and the coproduct) and is the tensor product of vector spaces | tensor product of linear representations |