Crossed modules as maps between connected components of topological groups

摘要

The purpose of this paper is to observe that a homomorphism of discrete groups f : Γ → G {f:Gammao G} arises asthe induced map π 0 ? ( ?? ) → π 0 ? ( ?? ) {pi_{0}(mathfrak{M})opi_{0}(mathfrak{X})} on path components of some closed normalinclusion of topological groups ?? ? ?? {mathfrak{M}subseteqmathfrak{X}} if and only ifthe map f can be equipped with a crossed module structure.In that case an essentially u