Crossed modules as maps between connected components of topological groups

TL;DR

This paper characterizes when a group homomorphism can be realized as a map between connected components of topological groups, linking it to the concept of crossed modules and their topological realizations.

Contribution

It establishes a precise criterion for when a discrete group homomorphism arises from a topological group inclusion via crossed modules.

Findings

A homomorphism arises from a topological group inclusion iff it admits a crossed module structure.

Such realizations are essentially unique and can be constructed using homotopically discrete topological groups.

Abstract

The purpose of this note is to observe that a homomorphism of discrete groups $f:\Gamma\to G$ arises as the induced map $\pi_0(\mathfrak{M})\to \pi_0(\mathfrak{X})$ on path components of some closed normal inclusion of topological groups $\mathfrak{M}\subseteq \mathfrak{X},$ if and only if the map $f$ can be equipped with a crossed module structure. In that case an essentially unique realization $\mathfrak{M}\subseteq \mathfrak{X}$ exists by homotopically discrete topological groups.