Re: Uniform spaces

[Oswald Wyler <owyler@suscom-maine.net>]

On Wed, 27 Aug 2003, Tom Leinster wrote:

> Date: Wed, 27 Aug 2003 16:51:55 +0200 (CEST)
> From: Tom Leinster <leinster@ihes.fr>
> To: categories@mta.ca
> Subject: categories: Uniform spaces
> 
> Hello,
> 
> Does anyone know of any account of the basic properties of the category of
> uniform spaces?  I'm after things like (co)limits, cartesian closure, and
> (co)limit-preservation by the forgetful functor to Top.  Bourbaki gets me
> some of the way, but his decision not to use categorical language and
> the resulting circumlocutions make it a struggle.
> 
> Thanks,
> Tom

Hi Tom,

The category UNIF of uniform spaces, without a separation axiom, is
topological over sets, and hence complete and cocomplete, with concrete
limits and colimits.  UNIF is not cartesian closed.

Cook and Fischer, Math. Ann. 173 (1967), 290-306, defined uniform convergence
structures of a set X as sets \scrF of filters on XxX satisfying five
axioms.  With the obvious definition of uniform continuity, sets with a
uniform convergence structure in this sense form a topological category
over sets, but Gazik, Kent and Richardson in Bull.Austral.Math.soc 11 (1974),
413-424, showed that this category is not cartesian closed.

In LNM 378, 591-637, I replaced the Cook-Fischer axiom that the principal
filter generated by the diagonal of XxX is in \scrF by the less demanding
axion that the principal filter generated by (x,x), for every x \in X,
is in \scrF.  This is now part of the accepted definition of uniform
convergence spaces.  In Bull.Austral.Math.Soc. 15 (1976), 461-465 my
student R.S. Lee showed that the category of uniform convergence spaces
with this definition is cartesian closed; this is not the cartesian closed
hull of UNIF.

For quasitoposes, we must go to semiuniform spaces which have partial
morphisms -- relations (m,g) with m an embedding -- represented by
one-point extensions.  Semiuniform convergence spaces and their uniformly
continuous maps form a quasitopos, but not the quasitopos hull of UNIF.
This has been determined by Adámek and Reiterman, The quasitopos hull of
the category of uniform spaes -- a correction, in the journal Topology
and its Applications.

For more information and literature, see my book Lecture Notes on Topoi
and Quasitopoi.

Oswald Wyler