Graded category

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:

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


 * $$S_{1}$$ is the identity functor on $$\mathcal{C}$$,
 * $$S_{g}S_{h}=S_{gh}$$ for all $$g,h \in \mathbb{G}$$ and
 * $$S_{g}$$ is a full and faithful functor for every $$g\in \mathbb{G}$$

we say that $$(\mathcal{C},\mathcal{S})$$ is a $$\mathbb{G}$$-graded category.