Graded category
If is a category, then a
-graded category
is a category together with a functor
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, 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 be an Abelian category and a monoid. Let be a set of functors from to itself. If
we say that is a -graded category.
