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$s$.

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.