Crossed extensions and equivalences of topological 2-groupoids

TL;DR

This paper develops concrete models for morphisms and equivalences of topological 2-groupoids using crossings and crossed extensions, providing a systematic framework and a weak 3-category structure.

Contribution

It introduces the notions of crossings and crossed extensions for topological 2-groupoids and constructs a weak 3-category capturing their morphisms and Morita equivalences.

Findings

Explicit models for generalized morphisms of topological 2-groupoids.

A systematic study of crossings and crossed extensions.

Construction of a weak 3-category of crossed modules.

Abstract

We provide concrete models for generalized morphisms and Morita equivalences of topological 2-groupoids by introducing the notions of crossings and crossed extensions of groupoid crossed modules. A systematic study of these objects is elaborated and an explicit description of how they do yield a groupoid and geometric picture of weak 2-groupoid morphisms is presented. Specifically, we construct a weak 3-category whose objects are crossed modules of topological groupoids and in which weak 1-isomorphisms correspond to Morita equivalences in the "category" of topological 2-groupoids.