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 specialized groupoid whose object set is a regular biordered set and the morphisms admit a partial order. A normal category is a specialized small category whose object set is a strict preorder and the morphisms admit a factorization property . A pair of ‘related’ normal categories constitutes a cross-connection. Both inductive groupoids and cross-connections were identified by Nambooripad 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.