Isotropy Groups of Free Racks and Quandles

TL;DR

This paper characterizes the isotropy groups of free racks and quandles, revealing that inner automorphisms are exactly those extendible coherently, and computes the automorphism groups of their categories.

Contribution

It provides a detailed characterization of isotropy groups for free racks and quandles and computes their global automorphism groups, advancing understanding of their symmetries.

Findings

Inner automorphisms are exactly the coherently extendible automorphisms.

Computed the global isotropy groups of racks and quandles.

Characterized covariant isotropy groups for free, finitely generated racks and quandles.

Abstract

In this article, we characterize the (covariant) isotropy groups of free, finitely generated racks and quandles. As a consequence, we show that the usual inner automorphisms of such racks and quandles are precisely those automorphisms that are "coherently extendible". We then use this result to compute the global isotropy groups of the categories of racks and quandles, i.e. the automorphism groups of the identity functors of these categories.