Tietze Equivalences as Weak Equivalences

TL;DR

This paper provides an abstract and geometric framework for understanding Tietze equivalences of monoid presentations by constructing a model structure where such equivalences are characterized as weak equivalences.

Contribution

It introduces a model structure on the category of monoid presentations, interpreting Tietze equivalences as weak equivalences and establishing their completeness through an abstract approach.

Findings

Tietze transformations form a pseudo-generating family of trivial cofibrations.

The model structure captures when two presentations define the same monoid.

The approach offers a geometric and abstract understanding of Tietze equivalences.

Abstract

A given monoid usually admits many presentations by generators and relations and the notion of Tietze equivalence characterizes when two presentations describe the same monoid: it is the case when one can transform one presentation into the other using the two families of so-called Tietze transformations. The goal of this article is to provide an abstract and geometrical understanding of this well-known fact, by constructing a model structuree on the category of presentations, in which two presentations are weakly equivalent when they present the same monoid. We show that Tietze transformations form a pseudo-generating family of trivial cofibrations and give a proof of the completeness of these transformations by an abstract argument in this setting.