Graded category

If formula_1 is a category, then a
formula_1-graded category
is a category formula_3 together with a functor
formula_4.
Monoids and groups can be thought of as categories with a single element. A monoid-graded or group-graded category is therefore one in which to each morphism is attached an element of a given monoid (resp. group), its grade. This must be compatible with composition, in the sense that compositions have the product grade.
Definition.
There are various different definitions of a graded category, up to the most abstract one given above. A more concrete definition of a graded Abelian category is as follows:
Let formula_3 be an Abelian category and formula_6 a monoid. Let formula_7 be a set of functors from formula_3 to itself. If
we say that formula_15 is a formula_6-graded category.