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.
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:[1] Let
be an abelian category and
This category theory-related article is a stub.