Towards Point-Free Spacetimes

In this work-in-progress talk we highlight some recent developments in our study of spacetimes through the lens of point-free topology. The first half of the talk will gently introduce the notion of an "ordered locale," a type of point-free space equipped with a causal ordering. Throughout, we draw intuition from the causal structure of spacetimes to guide our way. The latter part of the talk explores how two important notions from mathematical relativity might be developed for ordered locales: the so-called "causal ladder," and a "causal boundary" construction. (Based on joint work with Chris Heunen and Prakash Panangaden.)