A topological fibrewise fundamental groupoid

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.