Fine Moduli Stack of Compact Metric Spaces

• The fine moduli stack of compact metric spaces is a categorical framework classifying families of metric spaces up to isometry via a Grothendieck topology on totally bounded spaces.
• It leverages 1-Lipschitz maps and local submetry conditions to achieve effective descent and the construction of a universal family drawn from the Urysohn universal metric space.
• Its coarse moduli space naturally coincides with the Gromov–Hausdorff space, providing a canonical metric on isometry classes of compact metric spaces.

The fine moduli stack of compact metric spaces is a stack-theoretic structure that encodes families of compact metric spaces up to isometry, reflecting the moduli-theoretic spirit familiar from algebraic geometry in the metric context. Developed through the introduction of a Grothendieck topology on the category of totally bounded metric spaces, this stack—denoted $\GH$—serves as an analogue to classical moduli stacks and exhibits a universal property characterizing it as a fine moduli object. Its coarse moduli space is canonically isometric to the Gromov–Hausdorff space of isometry classes of compact metric spaces (Yuji, 28 Dec 2025).

1. Grothendieck Topology on Totally Bounded Metric Spaces

Let $\operatorname{tbMet}$ denote the category whose objects are totally bounded metric spaces and morphisms are 1-Lipschitz maps. The Grothendieck topology $J_{\mathrm{lsm}}$ on $\operatorname{tbMet}$, called the lsm-site, is defined using "Ism-coverings." An Ism-covering of $X$ is a family $\{f_i : U_i \to X\}_{i \in I}$ of 1-Lipschitz maps, each a local submetry, satisfying a refined lifting condition: For any $x_1, x_2, x_3 \in X$ and $\varepsilon > 0$, there exists an $i \in I$ and lifts $u_j \in f_i^{-1}(x_j)$ $(j=1,2,3)$ such that

$$d_{U_i}(u_1, u_2) < d_X(x_1, x_2) + \varepsilon,\quad d_{U_i}(u_2, u_3) < d_X(x_2, x_3) + \varepsilon.$$

This topology ensures effective descent for 1-Lipschitz maps: any compatible descent data for objects over an Ism-covering can be uniquely glued. Consequently, $\operatorname{tbMet}^{\mathrm{lsm}}$ is a subcanonical site (Yuji, 28 Dec 2025).

2. Definition of the Moduli Stack $\GH$

The category fibered in groupoids $\GH \to \operatorname{tbMet}^{\mathrm{lsm}}$ assigns to each base $B$ the groupoid of proper submetries $p: P \to B$ where $P$ is complete, $p$ is surjective, 1-Lipschitz, and locally onto Hausdorff balls. Morphisms between $(p': P' \to B')$ and $(p: P \to B)$ are given by Cartesian diagrams

$$\begin{CD} P' @>f>> P \ @V p' VV @VV p V \ B' @>>g> B \end{CD}$$

with $P' \cong P \times_B B'$. This structure, combined with the topology on $\operatorname{tbMet}$, yields a stack in groupoids with effective descent for proper submetries (Yuji, 28 Dec 2025).

3. Universal Property and Fine Moduli Structure

The fine moduli property of $\GH$ is established through the existence of a universal family. Let $U_{\mathrm{ry}}$ be the Urysohn universal metric space. The assignment

$$T \mapsto \left\{\, A \subset U_{\mathrm{ry}} \times T\, \bigm|\, A \to T \text{ is a proper submetry} \right\}$$

is representable by a sheaf $\mathrm{Cpt}(U_{\mathrm{ry}})$; the canonical projection $\mathrm{Cpt}(U_{\mathrm{ry}}) \to \GH$ is a formal submetry of stacks, surjective on "points." Thus, $\GH$ is a "naive metric stack" and admits a universal family $\mathcal{U} \to \GH$: pulling back $\mathcal{U}$ along any $T \to \GH$ yields the corresponding family $P \to T$.

The universal property is that, for any stack $X$ with a universal object $\mathcal{X} \to X$, giving a morphism $X \to \GH$ is equivalent to equipping $X$ with a proper submetry $\mathcal{X} \to X$ analogous to pulling back $\mathcal{U}$. The diagonal of $\GH$ is representable by the sheaf of isometries, rendering $\GH$ "rigid" and "metric-space–valued" and exhibiting stack-theoretic properties analogous to those of Deligne–Mumford or Artin stacks (Yuji, 28 Dec 2025).

4. Coarse Moduli Space and the Gromov–Hausdorff Distance

The coarse moduli space of $\GH$, i.e., the sheafification $\pi_0(\GH)$ of isomorphism classes, coincides with the classical Gromov–Hausdorff space. For compact metric spaces $X$ and $Y$, the Gromov–Hausdorff distance is defined as

$$d_{\mathrm{GH}}(X, Y) = \inf \left\{\, d_H^Z(\varphi(X), \psi(Y)) \,\big|\, \varphi: X \hookrightarrow Z,\ \psi: Y \hookrightarrow Z\, \right\}$$

where $d_H^Z$ is the Hausdorff distance in $Z$. The set $\pi_0(\GH)(*)$ is the set of isometry classes $[X]$ of compact metric spaces, with the induced metric structure from the universal family coinciding with $d_{\mathrm{GH}}$. Thus, $\pi_0(\GH)$ is isometric to the Gromov–Hausdorff space (Yuji, 28 Dec 2025).

5. Structural Properties, Examples, and Corollaries

Several structural properties and examples illustrate the behavior of $\GH$ as a fine moduli stack:

• Any trivial family $X \times T \to T$ induces a section $T \to \GH$, exhibiting the "constant moduli" map.
• For a compact group $G$ acting isometrically on compact $X$, the quotient (with the Hausdorff-quotient metric) arises as the pullback of the universal family along $BG \to \GH$.
• Stratification by the number of points in an $n$-pointed family yields stacks $\GH^n$, each a naive metric quotient-type stack over $\GH^{n-1}$.
• Completeness and properness: since $\mathrm{Cpt}(U_{\mathrm{ry}})$ is complete, so too is $\GH$ as a metric stack. Properness of its natural morphisms is inherited in the same manner.

The formal properties match those required of fine moduli stacks in algebraic geometry, with all constructions realized in the context of metric geometry (Yuji, 28 Dec 2025).

6. Significance and Relation to Moduli Theory

The fine moduli stack $\GH$ introduces a rigorous moduli-theoretic framework for compact metric spaces, analogous to the stack-theoretic treatment of algebraic varieties and schemes. Its construction relies essentially on the categorical and topological properties of totally bounded metric spaces and isometries. The identification of the coarse moduli space with the Gromov–Hausdorff space situates $\GH$ as the moduli stack that fully characterizes the moduli problem for compact metric spaces in terms of both families and their isomorphism classes (Yuji, 28 Dec 2025).