## Definition

Let $S$ be a set. Frequently, $S$ is a group or monoid (usually commutative).

An $S$-graded set is an $S$-indexed family of sets $\{X_s\}_{s\in S}$. This can equivalently be described as a function $S \to Set$, or as a function $X \to S$ (with $X_s$ the fiber over $s\in S$).

The elements of $X_s$ are often said to have degree $s$.

## Examples

The most common choices of $S$ are probably:

• the natural numbers $\mathbb{N}$.
• the integers $\mathbb{Z}$.
• the $2$-element set $\mathbb{Z}/2$. In this case, the elements of degree $0$ are often called even, and those of degree $1$ odd.

## Monoidal structure and enrichment

Suppose $(S,0,+)$ is a monoid, written additively. Then the category $Set^S$ of $S$-graded sets has a closed monoidal structure, where

$(X \otimes Y)_s = \coprod_{u+v = s} (X_u \times Y_v)$

This is a special case of Day convolution.

A $Set^S$-enriched category is a category whose morphisms all have degrees in $S$, and such that identity morphisms have degree $0$ and $deg(g f) = deg(g) + deg(f)$. Note that its underlying ordinary category, in the usual sense of enriched category theory, is the category of degree-$0$ morphisms.

Given any set $S$ and any category $C$, the category of $S$-graded objects of $C$ is simply the functor category $C^S$ (identifying $S$ with its discrete category). This includes graded sets as above, as well as graded vector spaces and graded modules. However, graded rings and graded algebras are not the same (and in particular require $S$ to be a monoid).