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.

中文翻译：

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Tietze对等与弱对等

给定的等分词通常会通过生成器和关系来接受许多演示，而Tietze等价性的概念是当两个演示文稿描述相同的等分词时的特征：一种情况是，一个人可以使用所谓的Tietze变换的两个族将一个演示文稿转换为另一演示文稿。本文的目的是通过在表示类别上构造一个模型结构来提供对这一众所周知的事实的抽象和几何学理解，其中两个表示在呈现相同的类半体动物时几乎是等效的。我们表明，Tietze转换形成了伪生成的平凡纤维化家族，并通过这种情况下的抽象论证证明了这些转换的完整性。