A forager's guide to infinity operads (with recipes)

Sophie Raynor – 4 March 2020

As category theorists, our ideas of homotopy theory and higher structures are framed in the language of infinity-categories and model category theory. These immensely powerful tools have provided us profound insights into our subject and the systems it can model. However, there are many mathematicians working with higher structures who are untrained in categorical methods (and even some who actively go out of their way to avoid them). And, even for us, it can be valuable to come down to earth and consider what's happening at a more nuts-and-bolts level.

This talk is about some characteristics to help identify infinity operads when we encounter them in the wild. More precisely, I'll give a number of necessary and sufficient conditions on a graded space (simplicial set) for it to underly a (monochrome, planar) dendroidal Segal space. Depending on time, recipes, in the form of direct (independent of model theoretic techniques) proofs of these equivalences will be discussed, and we may look at a nice example that illustrates some of these features.