Normalizers in the non-pointed context: a weak case of extremal decomposition

TL;DR

This paper explores the structural conditions under which normalizers exist in non-pointed categories, revealing a deep connection with properties of fibrations and internal groupoids, especially in quasi-pointed and protomodular contexts.

Contribution

It characterizes the existence of normalizers via a property of the fibration of points and links this to internal groupoids, extending understanding in non-pointed categorical settings.

Findings

Normalizers exist under specific fibrational properties.

The property of the fibration of points characterizes normalizer existence in certain categories.

A new equivalence between fibrational properties and internal groupoid structures is established.

Abstract

The aim of this work is to point out a strong structural phenomenon hidden behind the existence of normalizers through the investigation of this property in the non-pointed context: given any category E, a certain property of the fibration of points: Pt(E) --> E guarentees the existence of normalizers. This property becomes a characterization of this existence when E is quasi-pointed and protomodular. This property is also showed to be equivalent to a property of the category GrdE of internal groupoids in E which is a kind of opposite, for the monomorphic internal functors, of the comprehensive factorization.