Graded category

From Exampleproblems

A Graded category is a mathematical concept.

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 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.