A model structure for Grothendieck fibrations

Abstract

We construct two model structures, whose fibrant objects capture the notions of discrete fibrations and of Grothendieck fibrations over a category C. For the discrete case, we build a model structure on the slice Cat/C, Quillen equivalent to the projective model structure on [Cop,Set] via the classical category of elements construction. The cartesian case requires the use of markings, and we define a model structure on the slice Cat/C+, Quillen equivalent to the projective model structure on [Cop,Cat] via a marked version of the Grothendieck construction.We further show that both of these model structures have the expected interactions with their ∞-counterparts; namely, with the contravariant model structure on sSet/NC and with Lurie's cartesian model structure on sSet/NC+.