nLab
tensor category

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

We tend not to use this term, since it means different things to different people. 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.