Limits in categories of \'etale groupoids and pseudogroups

TL;DR

This paper demonstrates that the category of sober étale groupoids and actors has all small limits by analyzing the equivalent pseudogroup category, providing an alternative proof of a key adjunction, and clarifying the limit structures involved.

Contribution

It establishes the existence of all small limits in the category of sober étale groupoids and actors, using an analysis of pseudogroups and an alternative proof of a known adjunction.

Findings

The category of sober étale groupoids and actors admits all small limits.

Limits in this category can be computed via the equivalent pseudogroup category.

An alternative proof of the adjunction between étale groupoids and pseudogroups is provided.

Abstract

We show that the category of sober \'etale 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 \'etale groupoids and pseudogroups which is a central tool for computing limits of sober \'etale groupoids.