Partially multiplicative quandles and simplicial Hurwitz spaces

TL;DR

This paper introduces partially multiplicative quandles (PMQ), explores their properties, and constructs associated Hurwitz spaces that generalize classical spaces, with detailed analysis of symmetric group-based PMQs.

Contribution

It develops the theory of PMQs, defines Hurwitz spaces parametrizing branched coverings, and analyzes symmetric group examples, extending classical concepts.

Findings

Defined and studied properties of free and complete PMQs

Constructed Hurwitz spaces parametrizing branched coverings

Analyzed symmetric group-based PMQs and computed their enveloping groups

Abstract

We introduce partially multiplicative quandles (PMQ), a generalisation of both partial monoids and quandles. We set up the basic theory of PMQs, focusing on the properties of free PMQs and complete PMQs. For a PMQ $\mathcal{Q}$ with completion $\hat{\mathcal{Q}}$, we introduce the category of $\hat{\mathcal{Q}}$-crossed topological spaces, and define the Hurwitz space $\mathrm{Hur}^{\Delta}(\mathcal{Q})$: it is a $\hat{\mathcal{Q}}$-crossed space, and it parametrises $\mathcal{Q}$-branched coverings of the plane. The definition recovers classical Hurwitz spaces when $\mathcal{Q}$ is a discrete group $G$. Finally, we analyse the class of PMQs $\mathfrak{S}_d^{\mathrm{geo}}$ arising from the symmetric groups $\mathfrak{S}_d$, and we compute their enveloping groups and their PMQ completions.