A classification theorem for normal extensions

TL;DR

This paper establishes a classification theorem for normal extensions within certain Galois structures, linking them to locally split epic and trivial extensions, and interprets the normalization functor as a Kan extension.

Contribution

It provides a new characterization of normal extensions and connects the normalization functor to Kan extensions in the context of Galois theory.

Findings

Normal extensions are exactly those that are locally split epic and trivial.

A Galois theorem for normal extensions is proved.

Normalization functor is interpreted as a Kan extension.

Abstract

For a particular class of Galois structures, we prove that the normal extensions are precisely those extensions that are "locally" split epic and trivial, and we use this to prove a "Galois theorem" for normal extensions. Furthermore, we interpret the normalisation functor as a Kan extension of the trivialisation functor.