# The homology of principally directed ordered groupoids

## TL;DR

This paper explores homological properties of a relation on ordered groupoids, generalizing known results for inverse semigroups and establishing a connection between the homology of a groupoid and its quotient.

## Contribution

It introduces a generalized relation on ordered groupoids and proves that their homology is determined by the homology of their quotients, extending Loganathan's work.

## Key findings

- Homology of ordered groupoids is determined by their quotients.
- Constructs adjoint functors between module categories.
- Generalizes results from inverse semigroups to ordered groupoids.

## Abstract

We present some homological properties of a relation $\beta$ on ordered groupoids that generalises the minimum group congruence for inverse semigroups. When $\beta$ is a transitive relation on an ordered groupoid $G$, the quotient $G / \beta$ is again an ordered groupoid, and construct a pair of adjoint functors between the module categories of $G$ and of $G / \beta$. As a consequence, we show that the homology of $G$ is completely determined by that of $G / \beta$, generalising a result of Loganathan for inverse semigroups.