  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
capture 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 $\mathbf{Cat}$.
