Covering groupoids of categorical groups

TL;DR

This paper extends the concept of fundamental groupoids to H-groups, showing they form categorical groups and establishing an equivalence between covering spaces of H-groups and covering groupoids of their fundamental categorical groups.

Contribution

It proves that the fundamental groupoid of an H-group is a categorical group and establishes an equivalence between covering spaces and covering groupoids in this context.

Findings

Fundamental groupoid of an H-group is a categorical group.

Category of covering spaces of an H-group is equivalent to covering groupoids.

Extends classical results to H-groups and categorical groups.

Abstract

If $X$ is a topological group, then its fundamental groupoid $\pi_1X$ is a group-groupoid which is a group object in the category of groupoids. Further if $X$ is a path connected topological group which has a simply connected cover, then the category of covering spaces of $X$ and the category of covering groupoids of $\pi_1X$ are equivalent. In this paper we prove that if $(X,x_0)$ is an $H$-group, then the fundamental groupoid $\pi_1X$ is a categorical group. This enable us to prove that the category of the covering spaces of an $H$-group $(X,x_0)$ is equivalent to the category of covering groupoid of the categorical group $\pi_1X$.