In mathematics, 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 object. 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 be an abelian category and a monoid. Let be a set of functors from to itself. If
is the identity functor on , for all and is a full and faithful functor for every
we say that is a -graded category.
See also Differential graded category Graded (mathematics) Graded algebra Slice category
References