The interpretation of powersets in the context of (0,1)-category theory ands its applications to point-set topology
Summary
Given a preordered set \((X, \leq_X)\) we may equip \(X\) with a natural topology called the Alexandrov topology, where the upper subsets of the preordered set are declared to be the open subsets of the induced topology. This construction give a covariant functor from from the category of preordered sets with monotone maps as morphisms to the category of topological spaces.
In an alternative perspective, preordered sets are precisely (0,1)-categories. The points of \(X\) are its objects, and the \(\leq_X\) relationship between points are the morphisms in \(X\). From this point-of-view, the Alexandrov topology takes a category and assigns it to a topological space where the objects are our points and the open sets respect the arrows between them. We discuss this perspective, and ways to understand common, canonical constructions in the category of topological spaces by viewing them through the lens of the Alexandrov topology of a suitable preordered set. We also discuss implications for higher category theory (most specifically, the theory of 1-categories) and also how the various separation axioms for topological spaces may be understood from this perspective.
Background
This note started out as a short piece written for a handful of people on Discord, to try and better-convey an idea that I had tried—and failed—to explain a few days prior. After having time to sort through my thoughts I decided to give them an organized presentation, which bore the previous version of this note. This updated version is expanded to add some other points that I felt were worth addressing, as well as trying to clean up the writing and exposition a bit.