A bigroupoid's topology (or, Topologising the homotopy bigroupoid of a space)

TL;DR

This paper explores the topology of the fundamental bigroupoid of a space, extending classical concepts of fundamental groupoids to higher homotopy types with a focus on topological and local triviality properties.

Contribution

It generalizes the topological structure of the fundamental bigroupoid for semilocally 2-connected spaces, building on Brown and Danesh-Naruie's work for fundamental groupoids.

Findings

Constructs a topological bigroupoid for semilocally 2-connected spaces.

Shows local triviality of the topological bigroupoid for locally relatively contractible spaces.

Extends classical topological groupoid concepts to higher homotopy structures.

Abstract

The fundamental bigroupoid of a topological space is one way of capturing its homotopy 2-type. When the space is semilocally 2-connected, one can lift the construction to a bigroupoid internal to the category of topological spaces, as Brown and Danesh-Naruie lifted the fundamental groupoid to a topological groupoid. For locally relatively contractible spaces the resulting topological bigroupoid is locally trivial in a way analogous to the case of the topologised fundamental groupoid.