# Asymptotic Filtered Colimits

## TL;DR

This paper investigates the properties of asymptotic filtered colimits of large scale spaces, showing which coarse invariants are preserved and exploring their construction and limitations.

## Contribution

It introduces the concept of asymptotic filtered colimits in large scale spaces and analyzes the preservation of key coarse invariants within this framework.

## Key findings

- Finite asymptotic dimension is preserved.
- Property A is preserved under the colimit.
- Examples illustrate construction and limitations of filtered colimits.

## Abstract

If one has a collection of large scale spaces $\{(X_s,\mathcal{LSS}_s)\}_{s\in S}$ with certain compatibility conditions one may define a large scale space on $X=\bigcup\limits_{s\in S}X_s$ in a way where every function on $X$ is large scale continuous if and only if the function restricted to every $X_s$ is large scale continuous. This large scale structure is called the asymptotic filtered colimit of $\{(X_s,\mathcal{LSS}_s)\}_{s\in S}$. In this paper, we explore a wide variety of coarse invariants that are preserved between $\{(X_s,\mathcal{LSS}_s)\}_{s\in S}$ and the asymptotic filtered colimit $(X,\mathcal{LSS})$. These invariants include finite asymptotic dimension, exactness, property A, and being coarsely embeddable into a separable Hilbert space. We also put forth some questions and show some examples of filtered colimits that give an insight into how to construct filtered colimits and what may not be preserved as well.