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 $\mathcal{C}$. 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 $\mathrm{sSet}^+_{/ N\mathcal{C}}$.