On Friday, February 20th, there will be two lectures on infinity-operads at Utrecht University, given by Kensuke Arakawa (University of Kyoto) and Victor Carmona (Max Planck Institute, Leipzig).

The seminars will take place from 10:00–12:00 am and 2:00–4:00 pm at the Freudenthal Institute of UU, room 6.11. 

The first seminar will include an introductory part on operads and their algebras. You can find titles and abstracts below.

V. Carmona: The classical viewpoint on the homotopy theory of operadic algebras. 

Abstract: The homotopy theory of operadic algebras provides a powerful framework for studying algebraic structures up to coherent homotopy, allowing for a flexible and robust treatment of deformation, classification, and invariants in contexts ranging from algebraic topology to mathematical physics. The classical way to access such homotopy theory is by endowing categories of algebras over an operad with suitable (semi)model structures. This will be the point of view that we adopt in this talk, which is structured in two parts: (1) we will start with a short introduction to operads and their algebras to refresh some basics for this and the following talk by K. Arakawa, and (2) a deeper exploration of foundational questions in the homotopy theory of operadic algebras with a special emphasis on the role played by enveloping operads. To give a taste of the kind of questions we are referring to, we can ask: is it possible to always construct such (semi)model structures? Are they Quillen equivalent when the underlying operads are equivalent, or when we change the ambient model category?

K. Arakawa: How to compare models of enriched infinity operads 

Abstract: Symmetric monoidal categories used to enrich operads often have a natural class of "weak equivalences." In such situations, there is an induced notion of "weak equivalence" of operads. We want to regard two weakly equivalent operads as the same, and the theory of enriched operads provides a flexible setting for this kind of identification. There are, however, many different models of enriched infinity operads, reflecting the complexity of algebraic structures involved. These models have complementary strengths and weaknesses, and moving between them is essential. This raises a basic question: do they actually describe the same homotopy theory? While several comparisons are known, a complete picture is still missing. This talk focuses on the comparison of these models. In the first part, we will review the existing models and the comparison results currently available. In the second part, we will offer a new perspective that combines the classical model-categorical viewpoint and the modern infinity-categorical viewpoint. This viewpoint often allows us to conclude that models of enriched infinity operads are equivalent as long as they reproduce ordinary operads for ordinary symmetric monoidal model categories. As an illustration, we will provide a new equivalence of two models.