Generalizing quasi-categories via model structures on simplicial sets

Abstract

We use Cisinski's machinery to construct and study model structures on the category of simplicial sets whose classes of fibrant objects generalize quasi-categories. We identify a lifting condition which captures the homotopical behavior of quasi-categories without the algebraic aspects and show that there is a model structure whose fibrant objects are precisely those which satisfy this condition. We also identify a localization of this model structure whose fibrant objects satisfy a "special horn lifting" condition similar to the one satisfied by quasi-categories. This special horn model structure leads to a conjecture characterization of the bijective-on-0-simplices trivial cofibrations of the Joyal model structure. We also discuss how these model structures all relate to one another and to the minimal model structure.