Graded category

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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.


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]


  1. ^ 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.