Characterization of the circle within metric spaces

Question

There are various characterizations of the circle. To be precise, there is not the circle. There are several categories which contain an object we refer to as "the circle". In $\mathsf{Top}$ the circle is the unique $1$-dimensional closed connected manifold. In $\mathsf{Grp}$ the circle group is the unique divisible abelian group whose cardinality is the continuum and whose torsion subgroup is $\cong \mathbb{Q}/\mathbb{Z}$; in particular it is non-canonically isomorphic to $\mathbb{C}^{\times}$, which of course does not happen in $\mathsf{TopGrp}$. Now what about $\mathsf{Met}$, the category of metric spaces with a suitable class of morphisms different from continuous or equicontinuous maps? There are various choices, for example isometries, short maps, Lipschitz maps. Does the unit circle, endowed with the "obvious" metric (for example, the distance between $(1,0)$ and $(-1,0)$ is $\pi$), have a concise characterization (up to isomorphism, i.e. isometry) in one of these categories of metric spaces? For example, the curvature is $1$, but I know almost nothing about metric geometry to say more.

Answer 1

As seems to be my habit, I'm going to say a lot of things here whose details I have not checked. Use at your own risk :).

Given a metric space $(X,d)$, there is a geodesic remetrization $(X,d_G)$, equipped with a bijective short map $(X,d_G) \to (X,d)$. The geodesic metric is given by

$d_G(p,q) = \sup_{\epsilon>0} \inf_{\{p = p_0,p_1,\dots,p_n=q \,\mid\, d(p_i,p_{i+1})<\epsilon\}} \sum_i d(p_i,p_{i+1})$

Geodesic remetrization is an idempotent procedure, so we say that $(X,d)$ is a geodesic metric space if $d = d_G$. The idea is that $d_G(p,q)$ can be approximated by adding up distances along a "path" from $p$ to $q$, where the "steps" of the path are arbitrarily small. For example, a manifold with a Riemannian metric is geodesic.

The unit circle is the unique geodesic metric on (the topological space) $S^1$ with diameter $\pi$.

(Here the diameter of a metric space is the maximum distance between two points.) The reason is that if a metric on a 1-dimensional manifold is geodesic, then you can parameterize each connected component of the manifold by arc length, which sets up an isometry to an interval, the real line, or a dilation of the unit circle.

There should be some more conceptual description of where the geodesic remetrization "comes from". Lawvere alludes to some such description in the second-to-last paragraph of the introduction here. I was thinking that it could be the comonad induced by the obvious adjunction between $\mathcal V$-categories and $\mathcal V$-graphs when $\mathcal V = \mathbb R_{\geq 0}$. But on further thought, I think this ends up being the identity comonad. I'm still fond of the idea that it's some kind of density comonad for the inclusion of some category of free metric spaces.

EDIT

Actually, I think Lawvere is saying that geodesic re-metrization is the density comonad associated to the full subcategory of metric spaces given by the intervals. If so, then I think his notion of geodesic re-metrization is more restrictive than mine: his geodesic metric will be given by

$d^{\mathrm{Lawvere}}_G(p,q) = \inf_{\{ \gamma: [0,a] \to X \,\mid\, \gamma(0)=p,\gamma(a) = q\}} a$

(where the paths $\gamma$ are required to be short maps) but they should agree when we restrict attention to metric spaces which are locally path-connected, or something like that. I think my definition of the geodesic metric agrees on compact metric spaces with the density comonad for dense subsets of real intervals.