A fusion category over an algebraically closed field 
is a monoidal category
that is also a 
-linear
abelian category, with finite-dimensional morphism spaces and a tensor product bilinear on morphisms.
Every object is a finite direct sum of simple objects,
there are finitely many isomorphism classes of simple objects, the unit object is
simple, and every object has left and right duals. Here a simple object is a nonzero
object with no proper nonzero subobjects (Etingof et al. 2015, p. 65).
For example, the category of finite-dimensional complex representations of a finite group is a fusion category.
