Graded category

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A graded category is a mathematical concept.

If \mathcal{A} is a category, then a \mathcal{A}-graded category is a category \mathcal{C} together with a functor F:\mathcal{C} \rightarrow \mathcal{A}.

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.


There are various different definitions of a graded category, up to the most abstract one given above. A more concrete definition of a semigroup-graded Abelian category is as follows:[1]

Let \mathcal{C} be an Abelian category and \mathbb{G} a semigroup. Let \mathcal{S}=\{ S_{g} : g\in G \} be a set of functors from \mathcal{C} to itself. If

we say that (\mathcal{C},\mathcal{S}) is a \mathbb{G}-graded category.

See also[edit]


  1. ^ Zhang, J. J. (1 March 1996). "Twisted Graded Algebras and Equivalences of Graded Categories". Proceedings of the London Mathematical Society. s3-72 (2): 281–311. doi:10.1112/plms/s3-72.2.281.