nLab graded set

Graded sets

Graded sets


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

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

The elements of X sX_s are often said to have degree ss.

Given a pair of SS-graded sets {X s} sS\{X_s\}_{s\in S} and {Y s} sS\{Y_s\}_{s\in S}, a homomorphism between them is an SS-indexed family of functions {f s:X sY s} sS\{f_s \colon X_s \to Y_s \}_{s\in S}. This can equivalently be described as a natural transformation between the two associated functions SSetS \to Set, or as a function from XX to YY that make the diagram with the associated functions XSX \to S and YSY \to S commute.


The most common choices of SS are probably:

  • the natural numbers \mathbb{N}.

  • the integers \mathbb{Z}.

  • the 2-element set /2\mathbb{Z}/2. In this case, the elements of degree 00 are often called even, and those of degree 11 odd.

Monoidal structures and enrichment

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

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

This is a special case of Day convolution.

Furthermore, this monoidal structure laxly interchanges with the pointwise product of graded sets:

(X×Y) s:=X s×Y s (X \times Y)_s := X_s \times Y_s

where the lax interchange maps (X×Y)(Z×W)(XZ)×(YW)(X \times Y) \otimes (Z \times W) \to (X \otimes Z) \times (Y \otimes W) are given by inclusions. These interacting monoidal structures make Set SSet^S into a duoidal category.

A Set SSet^S-enriched category is a category whose morphisms all have degrees in SS, and such that identity morphisms have degree 00 and deg(gf)=deg(g)+deg(f)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-00 morphisms.

More generally, we may grade by a monoidal category. This leads to the notion of locally graded category.

Graded objects

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

Last revised on June 29, 2023 at 10:10:57. See the history of this page for a list of all contributions to it.