Quasi-categories vs. Segal spaces: Cartesian edition (Q2063063)

The title of the article under review alludes to the paper [\textit{A. Joyal} and \textit{M. Tierney}, Contemp. Math. 431, 277--326 (2007; Zbl 1138.55016)] in which -- among other things -- the authors prove that quasi-categories and complete Segal spaces are equivalent models of \((\infty,1)\)-categories by constructing two Quillen equivalences \[ p_1^* \colon (sSet)^{\mathrm{Joy}} \rightleftarrows (s\mathcal{S})^{\mathrm{CSS}} \,:\!i_1^* \qquad\text{and}\qquad t_! \colon (s\mathcal{S})^{\mathrm{CSS}} \rightleftarrows (sSet)^{\mathrm{Joy}} \,:\!t^! \] between the Joyal model structure \((sSet)^{\mathrm{Joy}}\) on simplicial sets and the model structure \((s\mathcal{S})^{\mathrm{CSS}}\) for complete Segal spaces on simplicial spaces. The article under review establishes analogous Quillen equivalences \[ (p_1^+)^* \colon (sSet^+)^{\mathrm{Joy}^+} \rightleftarrows (s\mathcal{S}^+)^{\mathrm{CSS}^+} \,:\!(i_1^+)^* \qquad\text{and}\qquad (t^+)_! \colon (s\mathcal{S}^+)^{\mathrm{CSS}^+} \rightleftarrows (sSet^+)^{\mathrm{Joy}^+} \,:\!(t^+)^! \] between the marked model structure \((sSet^+)^{\mathrm{Joy}^+}\) on marked simplicial sets and a new model structure \((s\mathcal{S}^+)^{\mathrm{CSS}^+}\) for marked complete Segal spaces. Moreover, building on these equivalences and Lurie's straightening and unstraightening functors, a similar comparison result is proven for four different models of cartesian fibrations, i.\,e.\ \((\infty,1)\)-categorical analogues of Grothendieck fibrations. Just as Grothendieck fibrations enable us to speak of pseudofunctors into the \(2\)-category \(\mathrm{Cat}\) of categories in terms of ordinary functors, cartesian fibrations enable us to speak of \((\infty,1)\)-functors into the category of \((\infty,1)\)-categories in terms of maps between \((\infty,1)\)-categories. Cartesian fibrations for quasi-categories were first defined in [\textit{J. Lurie}, Higher topos theory. Princeton, NJ: Princeton University Press (2009; Zbl 1175.18001)]. In order to obtain a model structure whose fibrant objects are (roughly) the cartesian fibrations, Lurie considers marked simplicial sets and eventually obtains a Quillen equivalence \[ \mathrm{St}_S^+ \colon (sSet_{/S}^+)^{\mathrm{Cart}} \rightleftarrows \mathrm{Fun}\bigl(\mathfrak{C}[S]^{\mathrm{op}}, ( sSet^+)^{\mathrm{Joy}^+} \bigr)^\mathrm{Proj } \,:\!\mathrm{Un}_S^+ \] between the cartesian model structure on marked simplicial sets over \(S\) and the projective model structure on functors \(\mathfrak{C}[S] \to (sSet^+)^{\mathrm{Joy}^+}\), where \(\mathfrak{C}\) is left adjoint to the simplicial nerve. In the precursory paper [\textit{N. Rasekh}, ``Cartesian fibrations of complete Segal spaces'', Preprint, \url{arXiv:2102.05190}], the author investigates a novel approach to cartesian fibrations: As internal categories in set-valued presheaves on some ordinary category \(S\) are nothing but presheaves of categories on \(S\), the author proposes to view internal \((\infty,1)\)-categories in a category of space-valued presheaves on \(S\) as presheaves on \(S\) taking values in \((\infty,1)\)-categories, i.\,e.\ as cartesian fibrations over~\(S\). This program is carried out in [loc. cit.] for the contravariant model structure \((s\mathcal{S}_{/X})^{\mathrm{Contra}}\) from [\textit{N. Rasekh}, ''Yoneda lemma for simplicial spaces'', Preprint, \url{arXiv:1711.03160}] with a cartesian model category \((ss\mathcal{S}_{/X})^{\mathrm{Cart}}\) on bisimplicial spaces over \(X\) as result. In the paper under review, the author works through this program for the contravariant model structure \((sSet_{/S})^{\mathrm{Contra}}\) on simplicial sets over \(S\) and thus obtains a cartesian model structure \((ssSet_{/S})^{\mathrm{Cart}}\) on bisimplicial sets over \(S\). Moreover, the equivalences between complete Segal spaces and quasi-categories due to Joyal and Tierney induce Quillen equivalences \[ (ss\mathcal{S}_{/X})^{\mathrm{Cart}} \rightleftarrows (ssSet_{st_!X})^{\mathrm{Cart}} \qquad\text{and}\qquad (ssSet_{/S})^{\mathrm{Cart}} \rightleftarrows (ss\mathcal{S}_{sp_1^*S})^{\mathrm{Cart}}. \] The paper continues with the construction of the model structure \((s\mathcal{S}^+)^{\mathrm{CSS}^+}\) for marked complete Segal spaces and the Quillen equivalences mentioned at the beginning of this review. These Quillen equivalences and Lurie's cartesian model structure eventually yield a cartesian model structure \((s\mathcal{S}^+_{/X})^{\mathrm{Cart}}\) on marked simplicial spaces over some simplicial space \(X\) together with Quillen equivalences \[ (p_1^+)^* \colon (sSet^+_{/S})^{\mathrm{Cart}} \rightleftarrows (s\mathcal{S}^+_{/(p_1^+)^* S})^{\mathrm{Cart}} \,:\!(i_1^+)^* \qquad\text{and}\qquad (t^+)_! \colon (s\mathcal{S}^+_{/X})^{\mathrm{Cart}} \rightleftarrows (sSet^+_{/(t^+)_!X})^{\mathrm{Cart}} \,:\!(t^+)^!. \] The article concludes with the observation that all these Quillen equivalences together with Lurie's straightening and unstraightening construction and the invariance of the projective model structure under Quillen equivalences eventually establish the equivalence of all four models \((sSet^+_{/S})^{\mathrm{Cart}}\), \((s\mathcal{S}^+_{/X})^{\mathrm{Cart}}\), \((ssSet_{/S})^{\mathrm{Cart}}\) and \((ss\mathcal{S}_{/X})^{\mathrm{Cart}}\) of cartesian fibrations.