Homotopical models for metric spaces and completeness (Q6631560)

Categories enriched in the opposite poset of non-negative reals can be viewed as generalization of metric spaces, known as \textit{Lawvere metric spaces} [\textit{F. W. Lawvere}, Rend. Semin. Mat. Fis. Milano 43, 135--166 (1974; Zbl 0335.18006)]. This paper aims to develop model categories that give insight into the theory of various kinds of Lawvere metric spaces specifically extended and Cauchy complete metric spaces.\N\NThe synopsis of the paper goes as follows.\N\N\begin{itemize}\N\item[\S 2] provides background and preliminaries.\N\N\item[\S 3] establishes the existence of a model structure on \(\boldsymbol{Cat}\), which characterizes the idempotent completion of categories and which is called the \textit{Karoubian model structure}. The model structure was constructed in [\textit{G. Caviglia} and \textit{J. J. Gutiérrez}, Forum Math. 31, No. 3, 661--684 (2019; Zbl 1422.55039), \S 1] as a Bousfield localization of the canonical model structure on \(\boldsymbol{Cat}\), where it is called the \textit{Morita model structure}. The authors provide an alternative proof for the existence of this model structure, which serves as a warm-up for their later work. The Karoubian model structure parallels their construction of the Cauchy model structure on \(\mathbb{R}_{+}\)-\(\boldsymbol{Cat}^{\mathrm{sym}}\).\N\N\item[\S 4] is devoted to the construction of the metric model structure on \(\mathbb{R}_{+}\)-\(\boldsymbol{Cat}\)\ (resp. \(\mathbb{R}_{+}\)-\(\boldsymbol{Cat}^{\mathrm{sym}}\)), establishing the following theorem.\N\NTheorem. There is a unique model structure on \(\mathbb{R}_{+}\)-\(\boldsymbol{Cat}\)\ (resp. \(\mathbb{R}_{+}\)-\(\boldsymbol{Cat}^{\mathrm{sym}}\))\ such that\N\begin{itemize}\N\item[(1)] the weak equivalences are the fully faithful and essentially surjective \(\mathbb{R}_{+}\)-functors, and\N\item[(2)] not every object is both fibrant and cofibrant.\N\end{itemize}\N\NMoreover, the fibrant-cofibrant objects in this model structure are precisely the gaunt \(\mathbb{R}_{+}\)-categories (resp. the symmetric gaunt \(\mathbb{R}_{+}\)-categories).\N\N\item[\S 5] is devoted to the construction of the Cauchy model structure on \(\mathbb{R}_{+}\)-\(\boldsymbol{Cat}^{\mathrm{sym}}\), establishing the following theorem.\N\NTheorem. There is a model structure on \(\mathbb{R}_{+}\)-\(\boldsymbol{Cat}^{\mathrm{sym}}\)\ such that\N\begin{itemize}\N\item[(1)] the weak equivalences are the fully faithful and dense \(\mathbb{R}_{+}\)-functors, and\N\item[(2)] the fibrant-cofibrant objects are the Cauchy-complete and symmetric \(\mathbb{R}_{+}\)-categories.\N\end{itemize}\N\N\item[\S 6] is devoted to the construction of the Cauchy-metric model structure on \(\mathbb{R}_{+}\)-\(\boldsymbol{Cat}^{\mathrm{sym}}\), establishing the following theorem.\N\NTheorem. There is a unique model structure on \(\mathbb{R}_{+}\)-\(\boldsymbol{Cat}^{\mathrm{sym}}\)\ such that\N\begin{itemize}\N\item[(1)] the weak equivalences are the fully faithful and dense \(\mathbb{R}_{+}\)-functors, and\N\item[(2)]every fibrant-cofibrant object is gaunt.\N\end{itemize}\N\N\end{itemize}