Foundations of topological racks and quandles

TL;DR

This paper develops foundational concepts for topological racks and quandles, introducing structures like ideals, kernels, and automorphisms, and exploring modules, bundles, and extensions to pave the way for a cohomology theory.

Contribution

It provides the first comprehensive framework for topological racks and quandles, including definitions and initial structures for cohomology development.

Findings

Defined ideals, kernels, units, and automorphisms in topological racks.

Explored topological rack modules and principal bundles.

Introduced central extensions as a step towards cohomology theory.

Abstract

We give a foundational account on topological racks and quandles. Specifically, we define the notions of ideals, kernels, units, and inner automorphism group in the context of topological racks. Further, we investigate topological rack modules and principal rack bundles. Central extensions of topological racks are then introduced providing a first step towards a general continuous cohomology theory for topological racks and quandles.