Quasi-categories are particular simplicial sets which behave like categories up to homotopy. Their theory has been massively developed in the past two decades, thanks largely due to Joyal and Lurie, and they have become vital tools in many areas of algebraic topology, algebraic geometry, and beyond. Due to the success of quasi-categories, it would be nice to extend the theory to up-to-homotopy versions of objects more general than categories, such as the 2-Segal sets of Dyckerhoff-Kapranov and Gàlvez-Kock-Tonks. Such a generalization would ideally come with an associated model structure on the category of simplicial sets, but finding a model structure with a more general class of fibrant objects than a given model structure is a nontrivial and open-ended task. In this talk, I will explain how to use Cisinski’s machinery to construct model structures on the category of simplicial sets whose fibrant objects generalize quasi-categories. In particular, one of these model structures has fibrant objects precisely the simplicial sets that satisfy a lifting condition which captures the homotopical behavior of quasi-categories without the algebraic aspects.

This talk is in-person, but a zoom link is available as well. The meeting ID is 954 8701 7543, with password 123708.