  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.
