Asymptotic 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.