Graded category
In mathematics, if <math>\mathcal{A}</math> is a category, then a <math>\mathcal{A}</math>-graded category is a category <math>\mathcal{C}</math> together with a functor <math>F\colon\mathcal{C} \rightarrow \mathcal{A}</math>.
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:<ref>Template:Cite journal</ref>
Let <math>\mathcal{C}</math> be an abelian category and <math>G</math> a monoid. Let <math>\mathcal{S} = \{ S_g : g \in G \}</math> be a set of functors from <math>\mathcal{C}</math> to itself. If
- <math>S_1</math> is the identity functor on <math>\mathcal{C}</math>,
- <math>S_g S_h = S_{gh}</math> for all <math>g,h \in G</math> and
- <math>S_g</math> is a full and faithful functor for every <math>g\in G</math>
we say that <math>(\mathcal{C},\mathcal{S})</math> is a <math>G</math>-graded category.