Macro- and micro-aspects of categorical topology
by
Walter Tholen
York University, Toronto, Canada

The macro-aspect of categorical topology is the study of (large) categories like Top, Unif or Loc

of topological or uniform spaces and of locales by predominantly categorical means and to give unified

presentations of such studies. They have been pursued in the realm of concrete categories, topological functors

and closure operators by many authors, including H. Herrlich. During the first half of this talk we concentrate on

the very general approach that starts off with an abstract category whose objects are regarded as 'spaces'

and whose morphisms are treated as 'continuous maps', together with an axiomatically given class of special maps, normally to be

to be thought of as the 'closed' or even 'perfect' maps in the category,

but which may also be taken to be the 'open' maps in the category.

We discuss old and new results in this setting on separation, (local) compactness and

exponentiability (e.g. the ability to form function spaces), with applications to fibred topology.

The micro-aspect goes back to F. W. Lawvere who, in the early 1970s, presented individual metric spaces as (small)

categories, enriched over the set of non-negative reals (ordered by the reversed natural order), the idea of which is

to capitalize on the formal similarity between

hom(x,y) x hom(y,z) --> hom(x,z),

d(x,y) + d(y,z) >= d(x,z).

Taken in conjunction with work by E. G. Manes and M. Barr on presenting topological spaces via ultra-filter convergence,

this work has recently been greatly extended and allows us to regard individual 'topologial' objects as so-called

(T,V)-categories, where T is a suitable Set-monad and V a 'good' lattice, and to mimic ideas from enriched category theory in this setting. For example, Lawvere's elegant

and effective notion of Cauchy-type completeness may be pursued in this context. There are natural notions of 'open'

and of 'proper' maps between such (T,V)-categories, which then take us back to the macro-aspect

described earlier. We will report on recent and on-going work, pursued in part in collaboration with

M. M. Clementino, D. Hofmann, C. Schubert, and G. Seal.

Date received: May 25, 2006