The Moduli Stack of Compact Metric Spaces

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.

28 pages. Minor corrections to typos, grammar, and exposition