Tensor category

In the algebraic convention used here, a tensor category over a field $k$ is a locally finite $k$-linear abelian category with a bilinear monoidal tensor product, left and right duals for every object, and a simple monoidal unit object whose endomorphisms are $k$. A locally finite category has finite-dimensional morphism spaces and objects of finite length. In broader usage, “tensor category” may mean merely a monoidal category; the extra hypotheses should therefore be stated.^[ref]

Finite-dimensional representations of a group over $k$ give a model example: the tensor product has diagonal group action, the unit is the one-dimensional trivial representation, and the dual is the dual vector space with the contragredient action. A tensor category need not have a braiding; swapping two tensor factors compatibly is additional structure.