Noninvertibility, semisupermanifolds and categories regularization

TL;DR

This paper explores categories with noninvertible morphisms, introducing regularization techniques and generalizing key algebraic structures to handle noninvertibility, resulting in a 2-category framework.

Contribution

It introduces concepts of regular n-cycles, obstruction, and regularization for categories with noninvertible morphisms, extending algebraic structures to the regular case.

Findings

Regularization of categories forms a 2-category.

Generalization of functors and Yang-Baxter equation to regular categories.

Framework for noninvertible morphisms analogous to semisupermanifolds.

Abstract

The categories with noninvertible morphisms are studied analogously to the semisupermanifolds with noninvertible transition functions. The concepts of regular n-cycles, obstruction and the regularization procedure are introduced and investigated. It is shown that the regularization of a category with nonivertible morphisms and obstruction form a 2-category. The generalization of functors, Yang-Baxter equation, (co-) algebras, (co-) modules and some related structures to the regular case is given.