Higher regularity properties of mappings and morphisms

TL;DR

This paper generalizes invertibility to regularity in categories, introduces higher regularity and semicommutative diagrams, and extends structures like functors and braidings to noninvertible contexts, including a noninvertible Yang-Baxter equation.

Contribution

It introduces a new framework for regularity in categories, extending invertibility concepts and generalizing structures like functors and braidings to noninvertible cases.

Findings

Higher regularity conditions and semicommutative diagrams are defined.

A noninvertible Yang-Baxter equation is proposed.

Generalization of functors and braidings to regular, noninvertible contexts.

Abstract

We propose to extend ``invertibility'' to ``regularity'' for categories in general abstract algebraic manner. Higher regularity conditions and ``semicommutative'' diagrams are introduced. Distinction between commutative and ``semicommutative'' cases is measured by non-zero obstruction proportional to the difference of some self-mappings (obstructors) $e^{(n)}$ from the identity. This allows us to generalize the notion of functor and to ``regularize'' braidings and related structures in monoidal categories. A ``noninvertible'' analog of the Yang-Baxter equation is proposed.