Duality of Scales

We establish an interaction between the large scale and small scale using two types of maps from large scale spaces to small scale spaces. First we use slowly oscillating maps, which can be described as those having arbitrarily small variation at infinity. These lead to a Galois connection between certain collections of large scale structures and small scale structures on a given set. Slowly oscillating functions can also be used to define to the notion of a dual pair of scale structures on a space. A dual pair consists of a large and a small scale structure on a space which is maximal with respect to the identity map being slowly oscillating. Finally, slowly oscillating functions and dual pairs are used to explain several well-known classes of large scale structures. The second type of maps studied are pinch-spacing maps. These are maps which respect the large scale structure of the domain, but only at a fixed scale. We use pinch-spacing to characterize and explain connections between the large scale properties of finite asymptotic dimension, property A, coarse embeddability into Hilbert space, exactness, and large scale paracompactness.