Homotopical models for metric spaces and completeness

TL;DR

This paper develops model structures for categories of Lawvere metric spaces, capturing features like completeness and symmetry, and proves the uniqueness of some of these structures.

Contribution

It introduces new model structures on Lawvere metric spaces categories that characterize completeness and symmetry, and establishes their uniqueness.

Findings

Fibrant-cofibrant objects correspond to extended and Cauchy complete metric spaces.

Constructed three distinct model structures for Lawvere metric spaces.

Proved the uniqueness of two model structures similar to the canonical model on Cat.

Abstract

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 $\mathbf{Cat}$.