The reals and the reals ... and the reals: once more around the track
by
Mark Sioen
Vrije Universiteit Brussel

Categorical Topology, lying on the intersection of two fields, has proved to be a fruitful two-way street over the past decades. On the one hand, categorical methods can be of help tackling some problems in topology and exposing common threads behind at first sight unrelated phenomena (see e.g. the results of Reiterman and Tholen on characterizing effective descent morphisms in Top, the work of Dolecki and Mynard on exploiting convergence-theoretic methods for examining product-stability properties of morphisms and the development of so-called 'metrically generated theories' by Lowen-Colebunders, Lowen et. al. to capture common properties that a lot of familiar topological constructs share), and on the other hand a great deal of insights coming originally form topological spaces and continuous maps have lead to the definition and study of topologically-categorical notions in other categories (such as the use of categorical closure operators by Clementino, Giuli, Tholen et. al. or the 'Topology-via-Lax-algebra's'-program of Clementino, Hofmann and Tholen). Of course the distinction between the two directions in this street is arbitrary and even artificial, most of the examples being quoted in one also having to be quoted in the other. How the spirit of looking at different levels/types of topological-like structures and most importantly, their interrelation, lies at the heart of categorical topology is exhibited in the beautiful paper 'The reals and the reals' (General Topology Appl. 9 (1978), no. 3, 221-232) by Herrlich and Bentley. The category Ap of approach spaces and contractions as introduced by Lowen, provides an answer to the bad productivity properties in Top of metrizablity. It is the aim of this talk, to give an overview of some old and new results concerning interactions between the category of approach spaces with some other categories of interest to (categorical)topologists and functional analysts like the ones of (locally convex) topological vector spaces, convex modules, frames or their quantified approach counterparts. Apart from using some categorical tools such as monads and duality, the common feature will be considering the reals in the different categories, following the spirit of the aforementioned paper.

Date received: June 19, 2006