A model structure for Grothendieck fibrations

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 [C^op,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 [C^op,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⁺.