In mathematics, if A {\displaystyle {\mathcal {A}}} is a category, then a A {\displaystyle {\mathcal {A}}} -graded category is a category C {\displaystyle {\mathcal {C}}} together with a functor
F : C → A {\displaystyle F\colon {\mathcal {C}}\rightarrow {\mathcal {A}}} . 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 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: Let C {\displaystyle {\mathcal {C}}} be an abelian category and G {\displaystyle G} a monoid. Let S = { S g : g ∈ G } {\displaystyle {\mathcal {S}}=\{S_{g}:g\in G\}} be a set of functors from C {\displaystyle {\mathcal {C}}} to itself. If
S 1 {\displaystyle S_{1}} is the identity functor on C {\displaystyle {\mathcal {C}}} ,
S g S h = S g h {\displaystyle S_{g}S_{h}=S_{gh}} for all g , h ∈ G {\displaystyle g,h\in G} and
S g {\displaystyle S_{g}} is a full and faithful functor for every g ∈ G {\displaystyle g\in G}
we say that ( C , S ) {\displaystyle ({\mathcal {C}},{\mathcal {S}})} is a G {\displaystyle G} -graded category.
See also Differential graded category Graded (mathematics) Graded algebra Slice category
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