# nLab comparison

## In higher category theory

A comparison on a set $A$ is a (binary) relation $\sim$ on $A$ such that in every pair of related elements, any other element is related to one of the original elements in the same order as the original pair:

$\forall \left(x,y,z:A\right),\phantom{\rule{thickmathspace}{0ex}}x\sim z\phantom{\rule{thickmathspace}{0ex}}⇒\phantom{\rule{thickmathspace}{0ex}}x\sim y\phantom{\rule{thickmathspace}{0ex}}\vee \phantom{\rule{thickmathspace}{0ex}}y\sim z$\forall (x, y, z: A),\; x \sim z \;\Rightarrow\; x \sim y \;\vee\; y \sim z

which generalises from $3$ to any (finite, positive) number of elements. To include the case where $n=0$, we must explicitly state that the relation is irreflexive.

Comparisons are most often studied in constructive mathematics. In particular, the relation $<$ on the (located Dedekind) real numbers is a comparison, even though its negation $\ge$ is not constructively total. (Indeed, $<$ is a linear order, even though $\ge$ is not constructively a total order.)

Revised on August 24, 2012 20:04:53 by Urs Schreiber (89.204.138.8)