Covariant model structures and simplicial localization

Abstract

In this paper we prove that for any simplicial set \(B\), there is a Quillen equivalence between the covariant model structure on \(\mathbf{S}/B\) and a certain localization of the projective model structure on the category of simplicial presheaves on the simplex category \(\Delta/B\) 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 \(\mathfrak{C}[B]\). 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.