Platonic and alternatinc 2-groups

TL;DR

This paper develops a theory of categorical extensions of groups, establishing the universality of String 2-groups and analyzing their restrictions to finite subgroups, including those related to Spin groups and the stable 3-stem.

Contribution

It introduces an analogous theory for categorical group extensions, proving the universality of String 2-groups and examining their restrictions to specific finite subgroups.

Findings

String 2-groups are universal in categorical extensions.

Restrictions to finite subgroups relate to the stable 3-stem.

Categorical extensions of spin double covers are characterized by the stable 3-stem.

Abstract

We recall Schur's work on universal central extensions and develop the analogous theory for categorical extensions of groups. We prove that the String 2-groups are universal in this sense and study in detail their restrictions to the finite subgroups of the Spin groups. Of particular interest are subgroups of the 3-sphere Spin(3), as well as the spin double covers of the alternating groups, whose categorical extensions turn out to be governed by the stable 3-stem.