Covariant model structures and simplicial localization

Abstract

In this paper we prove that for any simplicial set B, there is a Quillen equiv- alence between the covariant model structure on S/B and a certain localization of the projective model structure on the category of simplicial presheaves on the simplex category ∆/B of B. We extend this result to give a new Quillen equivalence between this covariant model structure and the projective model structure on the category of simplicial presheaves on the simplicial category C[B]. We study the relationship with Lurie’s straightening theorem. Along the way we also prove some results on localizations of simplicial categories and quasi-categories.