Cohomology and extensions of ordered groupoids

TL;DR

This paper extends cohomology theory to ordered groupoids, deriving a five-term exact sequence that classifies their extensions via second cohomology, using structural methods rather than cocycle computations.

Contribution

It generalizes Loganathan's results to ordered groupoids and introduces a new classification approach for extensions through cohomology.

Findings

Derived a five-term exact sequence in cohomology

Classified extensions of ordered groupoids via second cohomology

Used structural cohomology methods instead of cocycle calculations

Abstract

We adapt and generalise results of Loganathan on the cohomology of inverse semigroups to the cohomology of ordered groupoids. We then derive a five-term exact sequence in cohomology from an extension of ordered groupoids, and show that this sequence leads to a classification of extensions by a second cohomology group. Our methods use structural ideas in cohomology as far as possible, rather than computation with cocycles.