# How to centralize and normalize quandle extensions

## TL;DR

This paper explores the categorical structure of quandle coverings and normal extensions, demonstrating their reflective properties and providing concrete constructions using categorical Galois theory.

## Contribution

It establishes that quandle coverings form a reflective subcategory and constructs a centralization congruence, advancing the understanding of quandle extension structures.

## Key findings

- Quandle coverings form a regular epi-reflective subcategory.
- A concrete construction of a centralization congruence is provided.
- Similar results hold for normal quandle extensions.

## Abstract

We show that quandle coverings in the sense of Eisermann form a (regular epi)-reflective subcategory of the category of surjective quandle homomorphisms, both by using arguments coming from categorical Galois theory and by constructing concretely a centralization congruence. Moreover, we show that a similar result holds for normal quandle extensions.