Graded category

A graded category is a mathematical concept.

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

Monoids and groups can be thought of as categories with a single element. 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 semigroup-graded Abelian category is as follows:^[1]

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

• $S_{{1}}$ is the identity functor on ${\mathcal {A}}$,
• $S_{{g}}S_{{h}}=S_{{gh}}$ 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

References

1. Zhang, J. J. (1 March 1996). "Twisted Graded Algebras and Equivalences of Graded Categories". Proceedings of the London Mathematical Society. s3-72 (2): 281–311. doi:10.1112/plms/s3-72.2.281.