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 element. 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:[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.
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See also
- Differential graded category
- Graded (mathematics)
- Graded algebra
- Slice category
References
- 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. MR 1367080.
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