**Title:** The Diamond Lemma through homotopical algebra.

**Abstract:**

The Diamond Lemma is a result indispensable to those studying associative (and other types of) algebras defined by generators and relations. In this talk, I will explain how to approach this celebrated result through the lens of homotopical algebra: we will see how every multigraded resolution of a monomial algebra leads to “its own” Diamond Lemma, which is hard-coded into the Maurer-Cartan equation of its tangent complex. For the reader familiar with homotopical algebra, we hope to provide a conceptual explanation of a very useful but perhaps technical result that guarantees uniqueness of normal forms through the analysis of “overlapping ambiguities”. For a reader familiar with Gröbner bases or term rewriting theory, we hope to offer some intuition behind the Diamond Lemma and at the same time a framework to generalize it to other algebraic structures and optimise it. This is joint work with Vladimir Dotsenko (arXiv:2010.14792).