Weakly $\omega$-Categorified Models of Algebraic Theories

TL;DR

This paper constructs weakly $oldsymbol{ extomega}$-categorified models of algebraic theories like groups and quandles, replacing traditional homotopies with higher-order homotopies as categorical arrows, advancing the understanding of higher-dimensional algebraic structures.

Contribution

It introduces explicit constructions of weakly $oldsymbol{ extomega}$-categorified models for algebraic theories, extending the framework of fundamental groups and quandles with higher-order homotopies.

Findings

Constructed weakly $oldsymbol{ extomega}$-categorified models for groups and quandles.

Replaced homotopies with higher-order homotopies as categorical arrows.

Discussed related constructions of weakly $oldsymbol{ extomega}$-categorified algebras.

Abstract

We provide the expected constructions of weakly $\omega$-categorified models (in the sense of Bressie) of the theory of groups and quandles which arise by replacing the homotopies used to give equivalence relations in the theory of fundamental groups, fundamental quandles, and knot quandles with homotopies of all orders used as arrows of categorical dimensions one and greater, and discuss other related constructions of weakly $\omega$-categorifed algebras.