Chirality and Racemization on Isotopy Classes of Loops: A Groupoid-Based Structural Theory

TL;DR

This paper introduces a groupoid-based theoretical framework for understanding chirality and racemization in isotopy classes of finite loops, linking algebraic symmetries to physical mirror-related phenomena.

Contribution

It develops an intrinsic categorical theory of chirality in loops, establishing a criterion for chirality based on the absence of certain isotopisms, and models racemization as a two-state dynamic.

Findings

Racemization rate vanishes if no isotopism exists between a loop and its opposite.

Chirality is characterized by the non-existence of specific isotopisms.

A variant based on translation symmetries is discussed in the appendix.

Abstract

We develop a theory of chirality and racemization on isotopy classes of finite loops, formulated intrinsically within the loop isotopy groupoid understood in the categorical sense. Motivated by earlier work on quasigroups \cite{InoueQuasiChirality} and by the classical medical paradigm of mirror-related enantiomers, we restrict admissible mirror transitions to those generated by intrinsic, unit-preserving symmetries. Within this framework, racemization is modeled as a two-state dynamics on isotopy classes, with an effective rate determined by the presence of mirror-isotopisms. Our main result shows that this rate vanishes if and only if no loop isotopism exists between a loop and its opposite, providing a structural criterion for chirality. A strengthened variant based on translation-generated symmetries is discussed in the appendix.