Pseudogroups and their etale groupoids

TL;DR

This paper develops the theory of pseudogroups and their etale groupoids, establishing dualities and applications to C*-algebras, including non-commutative Stone duality and constructions of important algebraic structures.

Contribution

It introduces a duality framework between pseudogroups and etale groupoids, extending non-commutative Stone duality and applying it to C*-algebra theory.

Findings

Established a duality between spatial pseudogroups and sober etale groupoids.

Developed a non-commutative Stone duality involving boolean inverse semigroups.

Connected the theory to C*-algebras, Cuntz algebras, and Thompson groups.

Abstract

A pseudogroup is a complete infinitely distributive inverse monoid. Such inverse monoids bear the same relationship to classical pseudogroups of transformations as frames do to topological spaces. The goal of this paper is to develop the theory of pseudogroups motivated by applications to group theory, C*-algebras and aperiodic tilings. Our starting point is an adjunction between a category of pseudogroups and a category of etale groupoids from which we are able to set up a duality between spatial pseudogroups and sober etale groupoids. As a corollary to this duality, we deduce a non-commutative version of Stone duality involving what we call boolean inverse semigroups and boolean etale groupoids, as well as a generalization of this duality to distributive inverse semigroups. Non-commutative Stone duality has important applications in the theory of C*-algebras: it is the basis for the construction of Cuntz and Cuntz-Krieger algbras and in the case of the Cuntz algebras it can also be used to construct the Thompson groups. We then define coverages on inverse semigroups and the resulting presentations of pseudogroups. As applications, we show that Paterson's universal groupoid is an example of a booleanization, and reconcile Exel's recent work on the theory of tight maps with the work of the second author.