Tietze Equivalences as Weak Equivalences

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 structure 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. In order to navigate between the various presentations of a monoid, a very convenient tool is provided by Tietze transformations, originally investigated for groups [12] (see also [9, chapter II]): these are two families of elementary transformations one can perform on a monoid while preserving the presented monoid. Typically, the Knuth-Bendix completion procedure for string rewriting systems uses such transformations in order to turn a presentation of a monoid into another presentation of the same monoid which has the property of being convergent [8, 6], and thus for which the word problem is easily decidable. The Tietze transformations moreover enjoy a completeness property: given any two presentations of a given monoid, there is a way of transforming the first into the second by performing a series of such transformations. In this article, we provide a conceptual and geometrical point of view on Tietze transformations, by showing that they can be abstractly thought of as “continuously deforming” the presentations. In order to make this formal, we consider the category of presentations of monoids with suitably chosen morphisms (it turns out that we need to allow some sort of degeneracies) and construct a model structure on it, where weakly equivalent presentations are presentations of a same monoid. We then show that the Tietze transformations can then be interpreted in this setting as a pseudo-generating family of trivial cofibrations: they generate trivial cofibrations with fibrant codomain. Finally, the classical proof of completeness for Tietze transformations proceeds by constructing some kind of cospan of Tietze transformations between two presentations of the same monoid: we explain here how to reconstruct this proof by purely abstract arguments based on our model structure. The main goal of this article is thus to shed new light on theses well-known concepts and proofs, and advocate the relevance of homotopical methods to people working with presentations of monoids, which is why we have done our best to have a self-contained exposition. We see this work as a first step in order to tackle generalizations of Tietze transformations to higher dimension (e.g. coherent presentations of categories [5, Section 2.1]) or more involved structures (Lawvere theories, operads, etc.). We recall the notion of Tietze transformation between presentations of monoids in section 1, and of model category in section 2. We construct our model structure on the category of presen-