Homology, Homotopy and Applications

Volume 22 (2020)

Number 2

The homology of principally directed ordered groupoids

Pages: 163 – 172

DOI: https://dx.doi.org/10.4310/HHA.2020.v22.n2.a10


B.O. Bainson (Department of Mathematics, Kwame Nkrumah University of Science and Technology, Kumasi, Ghana)

N.D. Gilbert (School of Mathematical and Computer Sciences and the Maxwell Institute for the Mathematical Sciences, Heriot-Watt University, Edinburgh, United Kingdom)


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 we 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.


groupoid, homology, colimit

2010 Mathematics Subject Classification

18G60, 20J05, 20L05

Some of these results are presented in a different form as part of the first author’s PhD thesis [2]. The support of a MACS Global Platform Studentship from Heriot-Watt University is gratefully acknowledged.

Received 12 April 2017

Published 15 April 2020