We construct two model structures, whose fibrant objects capture the notions of discrete fibrations and of Grothendieck fibrations over a category $\mathcal{ C}$. For the discrete case, we build a model structure on the slice $\mathrm{ Cat}_{ /\mathcal{ C}}$, Quillen equivalent to the projective model structure on $[\mathcal{ C}^{ \mathrm{ op}},\mathrm{ Set}]$ via the classical category of elements construction. The cartesian case requires the use of markings, and we define a model structure on the slice $\mathrm{ Cat}^+_{ /\mathcal{ C}}$, Quillen equivalent to the projective model structure on $[\mathcal{ C}^{ \mathrm{ op}},\mathrm{ Cat}]$ via a marked version of the Grothendieck construction. We further show that both of these model structures have the expected interactions with their $\infty$-counterparts; namely, with the contravariant model structure on $\mathrm{ sSet}_{ / N\mathcal{ C}}$ and with Lurie's cartesian model structure on $\mathrm{ sSet}^+_{ / N\mathcal{ C}}$.
Categories: math.CT math.AT