If $mathcal\{A\}$ is a category, then a $mathcal\{A\}$-graded category is a category $mathcal\{C\}$ together with a functor $F:mathcal\{C\}\; rightarrow\; mathcal\{A\}$.

Monoids and groups can be thought of 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.

See also

• graded algebra.