Topological vistas from considering spaces as small categories

Easy examples of small categories include monoids (as one-object categories) and preordered sets (as categories whose hom-sets are at most singleton sets). It came as a surprise in the early 1970s when Lawvere presented every metric space as a small enriched category (a category enriched in the extended real half-line ). The discovery certainly raises the question whether other types of “spaces”, including topological spaces, may be individually considered as (some kind of) small categories, and whether there are any benefits arising from such a presentation. 
A first lead to answering the question comes from Manes’ 1967 algebraic axiomatization of a compact Hausdorff space in terms of two monoid-like laws for the operation that assigns to every ultrafilter of the underlying set its limit point. Replacing this operation by a mere relation, Barr showed in 1970 that every topological space is described by its ultrafilter convergence relation, satisfying a reflexivity and transitivity condition which entail the two axioms for preordered sets when considered as Alexandroff spaces. (It may be argued that there are visionary hints at such a presentation already in Hausdorff’s 1914 book!)
Monoidal Topology (as largely presented in the 2014 book of this title) provides a common environment for Lawvere’s metric spaces and Barr’s topological spaces. It considers a quantale (i.e., a suitably structured complete lattice) instead of , and a -monad (i.e., a suitably structured endofunctor of ) instead of the ultrafilter monad and considers small so-called -categories which, for and , reproduce Lawvere’s metric spaces, and for and (the two-element chain) Barr’s topological spaces.
The (large) category of all (small) -categories is a topological category over and, by variation of its parameters, leads to an interesting array of categories of old and new types of “spaces”. It allows us to study fundamental topological concepts in this categorical environment, such as Hausdorff separation, compactness, and perfectness, and to establish fundamental facts about them, often fairly easily so, such as Tychonoff’s Theorem and the Frol\'{i}k-Bourbaki Theorem for perfect maps. The interaction of topological and categorical ideas proceeds in both ways. For example, the categorically fundamental Yoneda embedding of a small category has important ramifications for -categories.
In Part 1 of this largely expository talk we plan to carefully introduce the syntax needed for forming the category – and present some important examples and properties of this category. Part 2 will be concerned with the pursuit of topological concepts inside –. We will also reconcile, in fair generality, the inherent convergence point of view of the theory with the more common geometric view of a topology as given by open sets or a closure operation. We finally point to some current work and open problems in the area.