Homotopical models for metric spaces and completeness

Categories enriched in the opposite poset of non-negative reals can be viewed as generalizations of metric spaces, known as Lawvere metric spaces. In this article, we develop model structures on the categories $\mathbb{R}_+\text-\mathbf{Cat}$ and $\mathbb{R}_+\text-\mathbf{Cat}^{\mathrm{sym}}$ of Lawvere metric spaces and symmetric Lawvere metric spaces, each of which captures different features pertinent to the study of metric spaces. More precisely, in the three model structures we construct, the fibrant-cofibrant objects are the extended metric spaces (in the usual sense), the Cauchy complete Lawvere metric spaces, and the Cauchy complete extended metric spaces, respectively. Finally, we show that two of these model structures are unique in a similar way to the canonical model structure on $\mathbf{Cat}$.