Tensor category - Knowlpedia

Fix an algebraically closed field $k$ . Here a tensor category over $k$ is a $k$ -linear abelian category with finite-dimensional Hom spaces and finite-length objects, equipped with a rigid monoidal structure whose tensor product is $k$ -bilinear on morphisms and whose unit is simple. Finite length means admitting a finite filtration with simple successive quotients.

Terminology and nearby conditions

Some authors use “tensor category” simply for a monoidal category. The stronger convention here records the linear, abelian, finiteness, and duality hypotheses explicitly. Neither a braiding nor semisimplicity is part of this definition.

A finite tensor category additionally has a finite underlying abelian category. A fusion category is a semisimple finite tensor category in this convention.

Example

Finite-dimensional representations of a group, with equivariant linear maps and the diagonal action on tensor products, form a tensor category over $k$ . The dual action is $(g\varphi)(v)=\varphi(g^{-1}v)$ . The group need not be finite; finiteness of the underlying abelian category is a separate requirement.

References

P. Etingof, S. Gelaki, D. Nikshych, and V. Ostrik, Tensor Categories (2015), §4.1; author-hosted text.