Graded category

In mathematics, if ${\mathcal {A}}$ is a category, then a ${\mathcal {A}}$ -graded category is a category ${\mathcal {C}}$ together with a functor $F\colon {\mathcal {C}}\rightarrow {\mathcal {A}}$ .

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:^[1]

Let ${\mathcal {C}}$ be an abelian category and $G$ a monoid. Let ${\mathcal {S}}=\{S_{g}:g\in G\}$ be a set of functors from ${\mathcal {C}}$ to itself. If