A new characterisation of groups amongst monoids

TL;DR

This paper provides a new characterization of groups within monoids by showing that a monoid is a group if and only if all points over it are strong in the category of monoids, simplifying previous characterizations.

Contribution

It introduces a novel categorical criterion for identifying groups among monoids based on the strength of points, refining earlier characterizations.

Findings

Monoids are groups iff all points over them are strong in the monoid category.

Simplifies previous characterizations of groups among monoids.

Provides a sharper categorical condition for group identification.

Abstract

We prove that a monoid $M$ is a group if and only if, in the category of monoids, all points over $M$ are strong. This sharpens and greatly simplifies a result of Montoli, Rodelo and Van der Linden which characterises groups amongst monoids as the protomodular objects.