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 $\mathcal{C}$. For the discrete case, we build a model structure on the slice ${\mathrm{C}\mathrm{a}\mathrm{t}}_{\mathrm{/}\mathcal{C}}$, Quillen equivalent to the projective model structure on $[{\mathcal{C}}^{\mathrm{o}\mathrm{p}},\mathrm{S}\mathrm{e}\mathrm{t}]$ via the classical category of elements construction. The cartesian case requires the use of markings, and we define a model structure on the slice ${\mathrm{C}\mathrm{a}\mathrm{t}}_{\mathrm{/}\mathcal{C}}^{+}$, Quillen equivalent to the projective model structure on $[{\mathcal{C}}^{\mathrm{o}\mathrm{p}},\mathrm{C}\mathrm{a}\mathrm{t}]$ via a marked version of the Grothendieck construction. We further show that both of these model structures have the expected interactions with their $\mathrm{\infty }$-counterparts; namely, with the contravariant model structure on ${\mathrm{s}\mathrm{S}\mathrm{e}\mathrm{t}}_{\mathrm{/}N\mathcal{C}}$ and with Lurie’s cartesian model structure on ${\mathrm{s}\mathrm{S}\mathrm{e}\mathrm{t}}_{\mathrm{/}N\mathcal{C}}^{+}$.