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Article Dans Une Revue Applied Categorical Structures Année : 2022

Tietze Equivalences as Weak Equivalences

Résumé

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.
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hal-04244388 , version 1 (16-10-2023)

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Simon Henry, Samuel Mimram. Tietze Equivalences as Weak Equivalences. Applied Categorical Structures, 2022, 30 (3), pp.453-483. ⟨10.1007/s10485-021-09662-w⟩. ⟨hal-04244388⟩
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