Girish Mahajan (Editor)

Graded category

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A graded category is a mathematical concept.

If A is a category, then a A -graded category is a category C together with a functor F : C → 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.

Definition

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:

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

  • S 1 is the identity functor on A ,
  • S g S h = S g h for all g , h ∈ G and
  • S g is a full and faithful functor for every g ∈ G

    • we say that ( C , S ) is a G -graded category.

    References

    Graded category Wikipedia
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