graded set

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 (with X sX_s the fiber over sSs\in S).

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


The most common choices of SS are probably:

  • the natural numbers \mathbb{N}.
  • the integers \mathbb{Z}.
  • the 22-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 structure 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.

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.

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).

Revised on October 5, 2017 06:08:51 by Urs Schreiber (