The Moduli Stack of Compact Metric Spaces

TL;DR

This paper develops a stack-theoretic framework for compact metric spaces, establishing a moduli stack whose coarse space aligns with the Gromov--Hausdorff space, bridging topology and metric geometry.

Contribution

It introduces a Grothendieck topology on totally bounded metric spaces and constructs a moduli stack whose coarse space matches the Gromov--Hausdorff space.

Findings

The coarse moduli space of the stack is isometric to the Gromov--Hausdorff space.

A Grothendieck topology on totally bounded metric spaces is established.

A fine moduli stack of compact metric spaces is defined.

Abstract

In this paper, we introduce a Grothendieck topology on the category of totally bounded metric spaces and develop a theory of stacks with respect to this topology. We further define the fine moduli stack of compact metric spaces and prove that its coarse moduli space is isometric to the Gromov--Hausdorff space.