# Graded category

If is a category, then a **-graded category** is a category together with a functor
.

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[edit]

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 be an abelian category and a monoid. Let be a set of functors from to itself. If

- is the identity functor on ,
- for all and
- is a full and faithful functor for every

we say that is a -graded category.

## See also[edit]

This article needs additional citations for verification. (April 2015) |

## References[edit]

**^**Zhang, James J. (1 March 1996). "Twisted graded algebras and equivalences of graded categories" (PDF).*Proceedings of the London Mathematical Society*. s3-72 (2): 281–311. doi:10.1112/plms/s3-72.2.281. hdl:2027.42/135651. MR 1367080.