Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The Moduli Stack of Compact Metric Spaces

Tomoki Yuji

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.

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.
Paper Structure (5 sections, 43 equations)

This paper contains 5 sections, 43 equations.

Table of Contents

  1. The Category of Metric Spaces
  2. The Hausdorff Distance
  3. The lsm-Topology on tbMet
  4. Naive Metric Stack
  5. The Gromov--Hausdorff Stack

Theorems & Definitions (40)

  • Remark
  • proof
  • proof
  • proof
  • proof
  • proof
  • proof
  • proof
  • proof
  • proof
  • ...and 30 more