Definitions

# Graded category

A graded category is a mathematical concept.

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

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.
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