Let be a semisimplelinear(Vect-enriched) (over a field ) category and a finite group. A -grading on is a direct sum decomposition of into homogeneous components , which are again -linear semisimple categories:
Definition
A graded fusion category is a fusion category with a chosen -grading for some group, such that the monoidal product maps into .
Alternative Definition
Let be the set of equivalence classes of simple objects in . A -grading is a map such that for any two simples and and any simple subobject , we have .
Terminology
The trivial component of the grading is a full fusion subcategory.
A -extension of a fusion category is a -graded fusion category such that .
A grading is faithful if for all . This is a standard assumption.
The adjoint category is the full fusion subcategory of spanned by objects of the form .
Examples
Every fusion category has the trivial grading from the trivial group.
-graded vector spaces for a finite group are naturally a graded fusion category.