An explicit comparison between 2-complicial sets and \(\Theta_2\)-spaces (Q6657456)

There have been various approaches to model \((\infty, n)\)-categories, and this paper aims to give an explicit comparison between two of these models, namely, the complete Segal \(\Theta_{n}\)-spaces [\textit{C. Rezk}, Geom. Topol. 14, No. 1, 521--571 (2010; Zbl 1203.18015); Geom. Topol. 14, No. 4, 2301--2304 (2010; Zbl 1203.18016)] and \(n\)-complicial sets [\textit{D. R. B. Verity}, Adv. Math. 219, No. 4, 1081--1149 (2008; Zbl 1158.18007); \textit{D. Verity}, Contemp. Math. 431, 441--467 (2007; Zbl 1137.18005); \textit{E. Riehl}, MATRIX Book Ser. 1, 49--76 (2018; Zbl 1409.18018); \textit{V. Ozornova} and \textit{M. Rovelli}, Algebr. Geom. Topol. 20, No. 3, 1543--1600 (2020; Zbl 1441.18032)]. The author gives such a comparison when \(n=2\), establishing the following theorem.\N\NTheorem. There is a Quillen equivalence between complete Segal \(\Theta_{2}\)-spaces, presented by the model category \(s\mathcal{S}et_{p, (\infty, 2)}^{\Theta_{2}^{\mathrm{op}}}\) [\textit{C. Rezk}, Trans. Am. Math. Soc. 353, No. 3, 973--1007 (2001; Zbl 0961.18008)] and the \(2\)-complicial sets, presented by the model category \(ms\mathcal{S}et_{(\infty, 2)}^{{}}\) [\textit{A. Gagna} et al., J. Lond. Math. Soc., II. Ser. 106, No. 3, 1920--1982 (2022; Zbl 1518.18020)].\N\NWhile the existence of such a direct Quillen equivalence follows formally, for example using the methods of [\textit{D. Dugger}, Adv. Math. 164, No. 1, 144--176 (2001; Zbl 1009.55011)], the author gives an explicit description. The main ingredients of the proof go as follows.\N\N\begin{itemize}\N\item[(1)] The author uses the compatibility of the \(2\)-categorical nerve valued in marked simplicial sets [\textit{V. Ozornova} and \textit{M. Rovelli}, High. Struct. 6, No. 1, 403--438 (2022; Zbl 1502.18050)] to construct a left Quillen functor\N\[\NL:s\mathcal{S}et_{p, (\infty, 2)}^{\Theta_{2}^{\mathrm{op}}}\rightarrow ms\mathcal{S}et_{(\infty, 2)}^{{}}\N\]\N\N\item[(2)] To show that this left Quillen functor is really a Quillen equivalence, the author uses a result in [\textit{C. Barwick} and \textit{C. Schommer-Pries}, J. Am. Math. Soc. 34, No. 4, 1011--1058 (2021; Zbl 1507.18025)] to reduce the problem to showing that it preserves cells in dimensions 0, 1, and 2.\N\end{itemize}