Abelian Topological Groups

[Bill Rowan <rowan@transbay.net>]

I am attempting to construct the ideal abelian category within which live
complete, hausdorff abelian topological groups.  The idea is that the
quotients of such a group, in the abelian category, would be completions
of the group with respect to topologies coarser than the given one.  The
subobjects would be those topologies.  Of course, having a topology as an
object in the abelian category means we have to have objects in the category
other than abelian groups.

Of course I want to know if this has been done before.  Also, what other ideas
are there about the ideal abelian category containing these groups?  Mac Lane
felt that compactly-generated spaces formed the ideal base category for
topological algebra.  I seem to be using the category of complete, hausdorff
uniform spaces as a base category.  I wrote a paper on (universal) algebras
with a compatible uniformity, and got some nice results about the congruence
(actually, uniformity) lattices.  But, admittedly, algebras with compatible
uniformities have drawbacks as a foundation for topological algebra because
even something like the complex numbers cannot be formalized as such, the
multiplication not being uniformly continuous.

Bill Rowan