The group of strong Galois objects associated to a cocommutative Hopf quasigroup

TL;DR

This paper introduces the concept of strong Galois H-objects for cocommutative Hopf quasigroups in a symmetric monoidal category, establishing a group structure analogous to classical Galois groups.

Contribution

It defines strong Galois H-objects in a categorical setting and proves their isomorphism classes form a group monoid, extending classical Galois theory to Hopf quasigroups.

Findings

The set of isomorphism classes forms a group monoid.

This structure generalizes the Galois group concept from Hopf algebras.

Establishes a categorical framework for Galois objects.

Abstract

Let H be a cocommutative faithfully flat Hopf quasigroup in a strict symmetric monoidal category with equalizers. In this paper we introduce the notion of (strong) Galois H-object and we prove that the set of isomorphism classes of (strong) Galois H-objects is a (group) monoid which coincides, in the Hopf algebra setting, with the Galois group of H-Galois objects introduced by Chase and Sweedler.