Weak Pullbacks of Topological Groupoids

TL;DR

This paper introduces the category HG of topological groupoids with measure-theoretic data, defines weak pullbacks within this category, and discusses their role in topological groupoidification.

Contribution

It formalizes the concept of weak pullbacks in the category of topological groupoids with measures, advancing the framework for topological groupoidification.

Findings

Defined the category HG with Haar systems and measures

Introduced and studied weak pullbacks in HG

Provided foundational work for topological groupoidification

Abstract

We introduce the category HG, whose objects are topological groupoids endowed with compatible measure theoretic data: a Haar system and a measure on the unit space. We then define and study the notion of weak-pullback in the category of topological groupoids, and subsequently in HG. The category HG is the setting for topological groupoidification, which we present in separate papers, and in which the weak pullback is a key ingredient.