A Tale of Two Categories: Inductive groupoids and Cross-connections
Abstract
A groupoid is a small category in which all morphisms are isomorphisms. An inductive groupoid is a specialised groupoid whose object set is a regular biordered set and the morphisms admit a partial order. A normal category is a specialised small category whose object set is a strict preorder and the morphisms admit a factorisation property. A pair of `related' normal categories constitutes a cross-connection. Both inductive groupoids and cross-connections were identified by Nambooripad \cite{mem,cross} as categorical models of regular semigroups. We explore the inter-relationship between these seemingly different categorical structures and prove a direct category equivalence between the category of inductive groupoids and the category of cross-connections.
Cite
@article{arxiv.1901.05731,
title = {A Tale of Two Categories: Inductive groupoids and Cross-connections},
author = {P. A. Azeef Muhammed and Mikhail V. Volkov},
journal= {arXiv preprint arXiv:1901.05731},
year = {2021}
}