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 prove some results on localizations of simplicial categories and quasi-categories.

v3: corrected some errors, final version, 49 pages

References in corpus (2)

• Higher Topos Theory
• Left fibrations and homotopy colimits II

Cited by in corpus (5)

• Model Structures for Correspondences and Bifibrations
• Quasi-Categories vs. Segal Spaces: Cartesian Edition
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• Joyal's cylinder conjecture
• Goodwillie's cosimplicial model for the space of long knots and its applications