(infinity-)Operads in chains: A tree-like approach

The first part of a talk by Francesca Pratali (University Sorbonne Paris Nord), at the Antwerp Algebra Colloquium on March 8, 2024.

Due to a technical problem, the slides do not update for the first couple of minutes of the video. For the full slides, see the speaker's homepage:
https://sites.google.com/view/frances...

The first part of the talk is meant both as an introduction to operad theory and as a follow-up of previous talks about dg-categories. We will provide definition, motivations and examples of operads enriched in a symmetric monoidal category V, paying special attention to V being the category of spaces or that of chain complexes, and we introduce the notion of 'oo-operads in V', where we relax some of the defining axioms, asking that they hold only 'up to homotopy'. We talk about the study of the homotopy theory of oo-operads in V, and how this drastically depends on the category V. When V=Spaces, several equivalent models have been developed: our interest concerns Cisinki and Moerdijk's model, which generalizes the notion of complete Segal space and is based on a category of trees which extends the simplex category Delta.