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

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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

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References

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  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.