# A Tale of Two Categories: Inductive groupoids and Cross-connections

## TL;DR

This paper establishes a categorical equivalence between inductive groupoids and cross-connections, two models used to represent regular semigroups, thereby unifying different categorical approaches in semigroup theory.

## Contribution

It proves a direct category equivalence between the category of inductive groupoids and the category of cross-connections, clarifying their relationship in the theory of regular semigroups.

## Key findings

- Proves a categorical equivalence between inductive groupoids and cross-connections.
- Unifies two categorical models of regular semigroups.
- Enhances understanding of the structure of regular semigroups.

## 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.