# Sheaf theoretic characterization of etale groupoids

## TL;DR

This paper establishes a new sheaf-theoretic framework linking etale groupoids and pseudogroups, generalizing classical concepts and providing a deeper understanding of their correspondence.

## Contribution

It introduces the concept of pseudogroup sheaves and demonstrates their equivalence with etale groupoids, extending the theoretical foundation of sheaf theory in this context.

## Key findings

- Established a correspondence between etale groupoids and pseudogroup sheaves
- Generalized the notion of pseudogroups through sheaf theory
- Provided a new perspective on the duality between groupoids and sheaves

## Abstract

The study of Haeflier suggests that it is natural to regard a pseudogroup as an etale groupoid. We show that any etale groupoid corresponds to a pseudogroup sheaf, a new generalization of a pseudogroup. This correspondence is an analog of the equivalence of the two definitions of a sheaf: as an etale space and as a contravariant functor.