Graded Category

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

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 $\mathcal$ be an abelian category and $G$ a monoid. Let $\mathcal = \$ be a set of functors from $\mathcal$ to itself. If

* $S_1$ is the identity functor on $\mathcal$,
* $S_g S_h = S_$ for all $g,h \in G$ and
* $S_g$ is a full and faithful functor for every $g\in G$

we say that $(\mathcal,\mathcal)$ is a $G$-graded category.

See also

* Differential graded category
* Graded (mathematics)
* Graded algebra
* Slice category

References

Category theory

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