Limits in categories of étale groupoids and pseudogroups
Jonathan Taylor
TL;DR
The paper establishes that sober étale groupoids with actors form a complete category by leveraging an adjunction between étale groupoids and pseudogroups and the fact that the forgetful functor to sets creates limits. Limits are computed by passing to the set-theoretic realm, lifting to pseudogroups, and applying spatialisation to recover a sober étale groupoid, enabling concrete constructions such as products and pullbacks (e.g., graph groupoid pullbacks). The authors provide an accessible account of the adjunction, its restriction to sober/spatial subcategories, and explicit limit computations, including a detailed pullback of graph groupoids. They also discuss how limits interact with subcategories of groups and address colimits, situating their results within the broader Cockett–Garner framework.
Abstract
We show that the category of sober étale groupoids and actors admits all small limits. This is achieved by computing the limits in the equivalent category of pseudogroups with pseudogroup morphisms, which we show admits a forgetful functor to the category of sets which creates limits. We give an alternative proof of the adjunction of Cockett and Garner in the specific setting of étale groupoids and pseudogroups which is a central tool for computing limits of sober étale groupoids.