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Classification of metric fibrations

Yasuhiko Asao

TL;DR

The paper develops a comprehensive theory of metric fibrations by recasting them as enriched Grothendieck fibrations over metric spaces, producing a precise classification in parallel with classical topological fibrations.Core ideas include the Grothendieck construction to move between metric actions and fibrations, the introduction of metric groups and the metric fundamental group $\pi_1^m(X)$, and the equivalences ${\sf PMet}_X^\mathcal{G} \simeq {\sf Tors}_X^\mathcal{G} \simeq {\sf Hom}(\pi_1^m(X, x_0), \mathcal{G})$ that classify principal fibrations by holonomy representations.The work treats both bounded and unbounded fibers via finite and extended metric groups, respectively, and culminates in a cohomological interpretation through a 1-Cech-like cohomology category ${\sf H}^1(X; \mathcal{G})$ that is equivalent to the torsor category, offering a robust toolkit for analyzing metric fibrations.Overall, the results bridge metric geometry, enriched category theory, and cohomology to provide a structural, computational framework for metric fibrations with potential implications for magnitude theory and enriched categorical perspectives.

Abstract

In this paper, we study `a fibration of metric spaces' that was originally introduced by Leinster in the study of the magnitude and called metric fibrations. He showed that the magnitude of a metric fibration splits into the product of those of the fiber and the base, which is analogous to the Euler characteristic and topological fiber bundles. His idea and our approach is based on Lawvere's suggestion of viewing a metric space as an enriched category. Actually, the metric fibration turns out to be the restriction of the enriched Grothendieck fibrations to metric spaces. We give a complete classification of metric fibrations by several means, which is parallel to that of topological fiber bundles. That is, the classification of metric fibrations is reduced to that of `principal fibrations', which is done by the `1-Cech cohomology' in an appropriate sense. Here we introduce the notion of torsors in the category of metric spaces, and the discussions are analogous to the sheaf theory. Further, we can define the `fundamental group $π^m_1(X)$' of a metric space $X$, which is a group object in metric spaces, such that the conjugation classes of homomorphisms $Hom(π^m_1(X), G)$ corresponds to the isomorphism classes of `principal $G$-fibrations' over $X$. Namely, it is classified like topological covering spaces.

Classification of metric fibrations

TL;DR

The paper develops a comprehensive theory of metric fibrations by recasting them as enriched Grothendieck fibrations over metric spaces, producing a precise classification in parallel with classical topological fibrations.Core ideas include the Grothendieck construction to move between metric actions and fibrations, the introduction of metric groups and the metric fundamental group $\pi_1^m(X)$, and the equivalences ${\sf PMet}_X^\mathcal{G} \simeq {\sf Tors}_X^\mathcal{G} \simeq {\sf Hom}(\pi_1^m(X, x_0), \mathcal{G})$ that classify principal fibrations by holonomy representations.The work treats both bounded and unbounded fibers via finite and extended metric groups, respectively, and culminates in a cohomological interpretation through a 1-Cech-like cohomology category ${\sf H}^1(X; \mathcal{G})$ that is equivalent to the torsor category, offering a robust toolkit for analyzing metric fibrations.Overall, the results bridge metric geometry, enriched category theory, and cohomology to provide a structural, computational framework for metric fibrations with potential implications for magnitude theory and enriched categorical perspectives.

Abstract

In this paper, we study `a fibration of metric spaces' that was originally introduced by Leinster in the study of the magnitude and called metric fibrations. He showed that the magnitude of a metric fibration splits into the product of those of the fiber and the base, which is analogous to the Euler characteristic and topological fiber bundles. His idea and our approach is based on Lawvere's suggestion of viewing a metric space as an enriched category. Actually, the metric fibration turns out to be the restriction of the enriched Grothendieck fibrations to metric spaces. We give a complete classification of metric fibrations by several means, which is parallel to that of topological fiber bundles. That is, the classification of metric fibrations is reduced to that of `principal fibrations', which is done by the `1-Cech cohomology' in an appropriate sense. Here we introduce the notion of torsors in the category of metric spaces, and the discussions are analogous to the sheaf theory. Further, we can define the `fundamental group ' of a metric space , which is a group object in metric spaces, such that the conjugation classes of homomorphisms corresponds to the isomorphism classes of `principal -fibrations' over . Namely, it is classified like topological covering spaces.
Paper Structure (17 sections, 46 theorems, 57 equations, 1 figure)

This paper contains 17 sections, 46 theorems, 57 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Conventions
  3. Categories
  4. Metric spaces
  5. ${\sf Met}_X \simeq {\sf Fib}_X$
  6. The metric fundamental group of a metric space
  7. Metric groups
  8. The metric fundamental group
  9. ${\sf PMet}_X^\mathcal{G} \simeq {\sf Tors}_X^\mathcal{G} \simeq {\sf Hom} (\pi^m_1(X, x_0), \mathcal{G})$
  10. ${\sf PMet}_X^\mathcal{G} \simeq {\sf Tors}_X^\mathcal{G}$
  11. ${\sf PMet}_X^\mathcal{G} \simeq {\sf Hom} (\pi^m_1(X, x_0), \mathcal{G})$
  12. Example
  13. Classification of metric fibrations
  14. The functor $\widehat{(-)}^{x_0}$
  15. Classification for the case of bounded fibers
  16. ...and 2 more sections

Key Result

Theorem 1.1

For a metric space $X$, the Grothendieck construction gives an equivalence of categories where we denote the category of metric actions $X \longrightarrow {\sf Met}$ by ${\sf Met}_X$ and the category of metric fibrations over $X$ by ${\sf Fib}_X$ (Definitions metacdef, metfibdef).

Figures (1)

  • Figure 1: The left is $K_3\times K_2$, and the right is isomorphic to $K_{3, 3}$. They both have magnitude equal to $\frac{6}{1 + 3q + 2q^2}$.

Theorems & Definitions (117)

  • Theorem 1.1: Proposition \ref{['metfib']}
  • Theorem 1.2: Proposition \ref{['pmettor']}
  • Theorem 1.3: Proposition \ref{['hompri']}
  • Proposition 1.4: Proposition \ref{['cyclegraph']}
  • Theorem 1.5: Corollary \ref{['pmetcore']}
  • Theorem 1.6: Corollary \ref{['betaeq']}
  • Definition 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Remark 2.4
  • ...and 107 more