Definition
Multifusion category
The fusion-category axioms without requiring the tensor unit to be simple.
Over an algebraically closed field , a multifusion category is an essentially small -linear semisimple abelian category with finite-dimensional Hom spaces, finitely many simple isomorphism classes, and a rigid monoidal structure whose tensor product is -bilinear on morphisms. The tensor unit need not be simple. Essentially small means that the isomorphism classes of objects form a set.
Comparison and example
A nonzero multifusion category with simple unit is a fusion category. As with fusion categories, some conventions additionally impose characteristic zero.
The product of two categories of finite-dimensional vector spaces, with componentwise tensor product, is multifusion. Its unit decomposes as , so it is not fusion. This example isolates exactly the role of the simple-unit condition.
References
P. Etingof, S. Gelaki, D. Nikshych, and V. Ostrik, Tensor Categories (2015), §4.1; author-hosted text.