A categorical approach to loops, neardomains and nearfields

TL;DR

This paper explores the categorical relationships between loops, neardomains, and nearfields, establishing equivalences with categories of permutation sets and sharply 2-transitive groups, thus providing a new perspective on their structural connections.

Contribution

It introduces a categorical framework that generalizes known equivalences, revealing new insights into the structure of neardomains and related algebraic objects.

Findings

Category of neardomains is equivalent to sharply 2-transitive groups

Categories of loops, neardomains, and nearfields are equivalent to permutation set categories

New categorical equivalences extend previous known results

Abstract

In this paper we study loops, neardomains and nearfields from a categorical point of view. By choosing the right kind of morphisms, we can show that the category of neardomains is equivalent to the category of sharply 2-transitive groups. The other categories are also shown to be equivalent with categories whose objects are sets of permutations with suitable extra properties. Up to now the equivalence between neardomains and sharply 2-transitive groups was only known when both categories were equipped with the obvious isomorphisms as morphisms. We thank Hubert Kiechle for this observation.