Graded category

This article is about a concept in category theory. For the concept in cognitive psychology and philosophy, see prototype theory.

If ${\mathcal {A}}$ is a category, then a ${\mathcal {A}}$-graded category is a category ${\mathcal {C}}$ together with a functor $F\colon {\mathcal {C}}\rightarrow {\mathcal {A}}$.

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

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

we say that $({\mathcal {C}},{\mathcal {S}})$ is a $\mathbb {G}$-graded category.

See also

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