Limits in categories of \'etale groupoids and pseudogroups

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.