Re: Uniform spaces

[Michael Barr <barr@barrs.org>]

Any study of the category must begin with Isbell's wonderful book on the
subject.  Although John's exposition could be difficult, it was not so in
that book.  I don't recall about limits and colimits (but they ought to be
easy), but there is a lot of discussion of internal homs (which do not
always exist and are not symmetric when they do).  The category is not
cartesian closed.  I am pretty sure the forgetful functor to Top has a
left adjoint and therefore preserves limits.  It preserves sums for sure,
but not coequalizers since a quotient space of a hausdorff uniform space
can be hausdorff without being completely regular.  At least, that is what
I think I remember.

Michael

On Wed, 27 Aug 2003, Tom Leinster wrote:

> 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
>
>
>
>