# A Large Scale Approach to Decomposition Spaces

## TL;DR

This paper explores the geometric structure of decomposition spaces using large scale geometry, introducing concepts like geometric embedding to compare and analyze these spaces based on their coverings.

## Contribution

It introduces a large scale geometric framework for analyzing decomposition spaces, emphasizing quasi-isometry and geometric embeddings to compare different spaces.

## Key findings

- Decomposition spaces with quasi-isometric coverings share many geometric features.
- The notion of geometric embedding formalizes how one decomposition space can be embedded into another respecting coverings.
- Large scale geometric tools like asymptotic dimension and hyperbolicity are useful in studying decomposition spaces.

## Abstract

Decomposition spaces are a class of function spaces constructed out of well-behaved coverings and partitions of unity of a set. The structure of the covering of the set determines the properties of the decomposition space. Besov spaces, shearlet spaces and modulation spaces are well-known decomposition spaces. In this paper we focus on the geometric aspects of decomposition spaces and utilize that these are naturally captured by the large scale properties of a metric space, the covered space, associated to a covering of a set. We demonstrate that decomposition spaces constructed out of quasi-isometric covered spaces have many geometric features in common. The notion of geometric embedding is introduced to formalize the way one decomposition space can be embedded into another decomposition space while respecting the geometric features of the coverings. Some consequences of the large scale approach to decomposition spaces are (i) comparison of coverings of different sets, (ii) study of embeddings of decomposition spaces based on the geometric features and the symmetries of the coverings and (iii) the use of notions from large scale geometry, such as asymptotic dimension or hyperbolicity, to study the properties of decomposition spaces.