Tietze Equivalences as Weak Equivalences

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