A topological fibrewise fundamental groupoid

TL;DR

This paper extends the concept of topological fundamental groupoids to fibrewise contexts, enabling the construction of fibrewise covering spaces and a topological fundamental bigroupoid under certain local connectivity conditions.

Contribution

It introduces a method to lift fibrewise fundamental groupoids to topological groupoids, generalizing classical results to fibrewise topology and categorifying the notion of covering spaces.

Findings

Fibrewise fundamental groupoids can be topologically realized under local connectivity.

Construction of fibrewise simply-connected covering spaces in fibrewise topology.

Definition of a topological fundamental bigroupoid for locally connected spaces.

Abstract

It is well-known that for certain local connectivity assumptions the fundamental groupoid of a topological space can be equipped with a topology making it a topological groupoid. In other words, the fundamental groupoid functor can be lifted through the forgetful functor from topological groupoids to groupoids. This article shows that for a map $Y \to X$ with certain relative local connectivity assumptions, the fibrewise fundamental groupoid can also be lifted to a topological groupoid over the space $X$. This allows the construction of a simply-connected covering space in the setting of fibrewise topology, assuming a local analogue of the definition of an ex-space. When applied to maps which are up-to-homotopy locally trivial fibrations the result is a categorified version of a covering space. The fibrewise fundamental groupoid can also be used to define a topological fundamental bigroupoid of a (suitably locally connected) topological space.