Graded category

A graded category is a mathematical concept. A graded category is a mathematical concept. If A {displaystyle {mathcal {A}}} is a category, then a A {displaystyle {mathcal {A}}} -graded categoryis a category C {displaystyle {mathcal {C}}} together with a functor F : C → A {displaystyle F:{mathcal {C}} ightarrow {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: Let C {displaystyle {mathcal {C}}} be an Abelian category and G {displaystyle mathbb {G} } a semigroup. Let S = { S g : g ∈ G } {displaystyle {mathcal {S}}={S_{g}:gin G}} be a set of functors from C {displaystyle {mathcal {C}}} to itself. If we say that ( C , S ) {displaystyle ({mathcal {C}},{mathcal {S}})} is a G {displaystyle mathbb {G} } -graded category.