A relative version of the etale homotopy type, spectral arithmetic duality, and rational points

In the first part, we will introduce a relative version of the etale homotopy type due to Schlank and Barnea. In particular, to each variety X over a number field K, the machine assigns a pro-finite space or spectrum Et(X) with an action by the absolute Galois group of K. In the second part, we will discuss a spectrum-level version of Tate-Poitou duality, whose application to Et(X) gives information about the existence of rational points on X.