Tensor product of representations over a symmetric bimonoidal category
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 .
|Symmetric bimonoidal category that we're starting with||Meaning of tensor product in that category|
|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|