nLab
tensor category

Idea

A tensor category is a category equipped with an operation similar to the tensor product in Ab.

The precise definition associated with the term “tensor category” varies somewhat in the literatur.

It may mean:

• any monoidal category,
• specifically a symmetric monoidal category (and then a quasitensor category is a braided monoidal category),
• an additive (symmetric) monoidal category,
• an Ab-enriched or Vect-enriched (symmetric) monoidal category.

Related concepts

• fusion category

References

• Pavel Etingof, Shlomo Gelaki, Dmitri Nikshych, Victor Ostrik, Topics in Lie theory and Tensor categories – 9 Tensor categories, Lecture notes (spring 2009) (pdf web)

Discussion with an eye towards superalgebra is in

• Pierre Deligne, Catégorie Tensorielle (pdf)