A symmetric approach to higher coverings in categorical Galois theory

TL;DR

This paper introduces a symmetric, absolute characterization of higher coverings in categorical Galois theory, simplifying their understanding and applying to algebraic structures like quandles.

Contribution

It provides a non-inductive, symmetric description of higher coverings in Galois structures, broadening the theoretical framework and applications.

Findings

Higher coverings are characterized absolutely within Galois towers.

The approach simplifies understanding of coverings in categorical Galois theory.

Application to algebraic structures like quandles enhances algebraic comprehension.

Abstract

In the context of a tower of (strongly Birkhoff) Galois structures in the sense of categorical Galois theory, we show that the concept of a higher covering admits a characterisation which is at the same time absolute (with respect to the base level in the tower), rather than inductively defined relative to extensions of a lower order; and symmetric, rather than depending on a perspective in terms of arrows pointing in a certain chosen direction. This result applies to the Galois theory of quandles, for instance, where it helps us characterising the higher coverings in purely algebraic terms.