Permutations, power operations, and the center of the category of racks

TL;DR

This paper explores the categorical structures of racks and quandles, revealing their centers and power operations, and providing universal characterizations that deepen understanding of their algebraic and knot classification properties.

Contribution

It develops categorical aspects of racks and quandles, computes their centers, describes power operations, and offers universal property characterizations.

Findings

Computed the centers of the categories of racks and quandles.

Described power operations and their free structures.

Provided universal property characterizations of these theories.

Abstract

Racks and quandles are rich algebraic structures that are strong enough to classify knots. Here we develop several fundamental categorical aspects of the theories of racks and quandles and their relation to the theory of permutations. In particular, we compute the centers of the categories and describe power operations on them, thereby revealing free extra structure that is not apparent from the definitions. This also leads to precise characterizations of these theories in the form of universal properties.