Quandle cohomology is a Quillen cohomology

TL;DR

This paper demonstrates that quandle cohomology aligns with Quillen cohomology, enabling the use of advanced homological tools for racks and quandles.

Contribution

It establishes the equivalence between quandle cohomology and Quillen cohomology, expanding the toolkit for algebraic and topological analysis.

Findings

Quandle cohomology groups match Quillen cohomology groups.

Enables use of long exact sequences and excision isomorphisms.

Provides a unified framework for algebraic theories of racks and quandles.

Abstract

We show that the cohomology groups usually associated with racks and quandles agree with the Quillen cohomology groups for the algebraic theories of racks and quandles, respectively. We also explain how this makes available the entire range of tools that comes with a Quillen homology theory, such as long exact sequences (transitivity) and excision isomorphisms (flat base change).