A Double Categorical View on Representations of Etendues

TL;DR

This paper presents a novel double categorical framework for ordered groupoids, establishing a biequivalence with certain categories and analyzing functors that induce geometric morphisms between sheaf categories.

Contribution

It introduces a double categorical perspective on ordered groupoids and extends Lawson's correspondence to a biequivalence, with applications to sheaf theory and site maps.

Findings

Established a biequivalence between ordered groupoids and double categories.

Identified conditions under which ordered functors induce geometric morphisms.

Proved a Comparison Lemma for maps between Ehresmann sites.

Abstract

In this paper we introduce a description of ordered groupoids as a particular type of double categories. This enables us to turn Lawson's correspondence between ordered groupoids and left-cancellative categories into a biequivalence. We use this to identify which ordered functors are maps of sites in the sense that they give rise to geometric morphisms between the induced sheaf categories, and establish a Comparison Lemma for maps between Ehresmann sites.