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

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:^[1]

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

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

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

See also

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

References