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Categorical approach to rigidity of Roe-like algebras of coarse spaces

Kostyantyn Krutoy

TL;DR

The paper develops a categorical framework linking Roe-like operator algebras for locally finite coarse spaces to coarse geometry via approximable categories. It proves that every fully faithful $*$-functor between approximable categories induces a coarse embedding between underlying spaces, and fully characterizes such functors up to central unitaries. A domain invariant $\operatorname{dom}_{\kappa}(C)$ is introduced to capture faithfulness and ampleness of coarse modules, and domain invariance under invertible maps is established, enabling rigidity results without strict faithfulness. A functor $\mathcal{A}$ from locally finite coarse spaces to approximable categories is shown to be full and faithful, with corollaries linking coarse embeddings to category-level equivalences and several open questions highlighted for quasi-local and related frameworks.

Abstract

We demonstrate that any full and faithful $*$-functor between approximable categories of locally finite coarse spaces induces a coarse embedding between the underlying spaces. Furthermore, we establish a general characterisation of such $*$-functors between approximable categories and prove that the functor associating each locally finite coarse space with its approximable category is full and faithful.

Categorical approach to rigidity of Roe-like algebras of coarse spaces

TL;DR

The paper develops a categorical framework linking Roe-like operator algebras for locally finite coarse spaces to coarse geometry via approximable categories. It proves that every fully faithful -functor between approximable categories induces a coarse embedding between underlying spaces, and fully characterizes such functors up to central unitaries. A domain invariant is introduced to capture faithfulness and ampleness of coarse modules, and domain invariance under invertible maps is established, enabling rigidity results without strict faithfulness. A functor from locally finite coarse spaces to approximable categories is shown to be full and faithful, with corollaries linking coarse embeddings to category-level equivalences and several open questions highlighted for quasi-local and related frameworks.

Abstract

We demonstrate that any full and faithful -functor between approximable categories of locally finite coarse spaces induces a coarse embedding between the underlying spaces. Furthermore, we establish a general characterisation of such -functors between approximable categories and prove that the functor associating each locally finite coarse space with its approximable category is full and faithful.

Paper Structure

This paper contains 9 sections, 37 theorems, 88 equations.

Table of Contents

  1. Preliminaries
  2. Coarse geometry
  3. Coarse geometric modules
  4. Approximable category
  5. Rigidity of $*$-isomorphisms for countably generated spaces
  6. Domains of coarse modules
  7. Rigidity of fully faithful $*$-functors
  8. Structure of fully faithful $*$-functors
  9. Corollaries and open questions

Key Result

Theorem 1

Let $\mathcal{X}$ and $\mathcal{Y}$ be locally finite countably generated coarse spaces with at most countably many connected components and $F \colon \mathcal{A}(\mathcal{X}) \to \mathcal{A}(\mathcal{Y})$ be a full and faithful $*$-functor. Then $F$ induces a coarse embedding $f^F \colon \mathcal{X

Theorems & Definitions (86)

  • Theorem 1: Theorem \ref{['*-functor => coarse emb']} and Corollary \ref{['ff *-funct => coarse emb (2)']}
  • Theorem 2
  • Corollary 3: Corollary \ref{['corollary: initial goal']}
  • Corollary 4: Corollary \ref{['corollary: cat equiv']}
  • Definition 1.1
  • Definition 1.2
  • Definition 1.3: MVmodules, Definition 3.11
  • Definition 1.4: MVmodules, section 3.2
  • Remark 1.5
  • Lemma 1.6: MVmodules, Corollary 3.42
  • ...and 76 more