= 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.<Tensor Categories by Pavel Etingof and others>[section 4.1, tensor and multitensor category definitions]{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.
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