Progress in operad-like theories with a focus on Feynman categories

Abstract: In the past years, several genereralization of operads have been considered. We will briefly give a quick overview and then turn to a particularly effective formulation — Feynman categories.

The basic setup is categorical, as the name suggests, and this allows to consider many natural constructions. One important aspect are representations, which are functors in this setting. These include algebras, operads and other more intricate or less sophisticated gadgets. In this respect the theory is analogous to representations of groups with restriction, induction and Frobenius reciprocity.

We will give a gentle introduction and as time allows, we may highlight various other constructions and applications achieved ranging from categorical considerations like comprehension schemes and Hopf-algebraic aspects to moduli spaces of curves.

These results are partly in collaboration with B. Ward, J. Lucas, I. Gálvez-Carrillo, A. Tonks, C. Berger, M. Monaco, M. Markl and M. Batanin (in chronological order).

Mathematics

Audience: researchers in the topic

Opening Workshop (IRP Higher Homotopy Structures 2021, CRM-Bellaterra)