Graded category

If [math]\displaystyle{ \mathcal{A} }[/math] is a category, then a [math]\displaystyle{ \mathcal{A} }[/math]-graded category is a category [math]\displaystyle{ \mathcal{C} }[/math] together with a functor [math]\displaystyle{ 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:

Let [math]\displaystyle{ \mathcal{C} }[/math] be an abelian category and [math]\displaystyle{ \mathbb{G} }[/math] a monoid. Let [math]\displaystyle{ \mathcal{S}=\{ S_{g} : g\in \mathbb{G} \} }[/math] be a set of functors from [math]\displaystyle{ \mathcal{C} }[/math] to itself. If

• [math]\displaystyle{ S_{1} }[/math] is the identity functor on [math]\displaystyle{ \mathcal{C} }[/math],
• [math]\displaystyle{ S_{g}S_{h}=S_{gh} }[/math] for all [math]\displaystyle{ g,h \in \mathbb{G} }[/math] and
• [math]\displaystyle{ S_{g} }[/math] is a full and faithful functor for every [math]\displaystyle{ g\in \mathbb{G} }[/math]

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

See also

• Differential graded category
• Graded (mathematics)
• Graded algebra
• Slice category