1 May 06:37 2001

### Abelian Topological Groups

Bill Rowan <rowan <at> transbay.net>

2001-05-01 04:37:18 GMT

2001-05-01 04:37:18 GMT

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