Graded category

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

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

• $S_1$ is the identity functor on ${𝒞}$,
• $S_g{S}_h=S_{gh}$ 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 $({𝒞},{𝒮})$ is a $\mathbb{𝔾}$-graded category.

See also

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