TITLE: Is there an abelian category of topological groups?
QUESTION [12 upvotes]: There are lots of reasons why the category of topological abelian groups (i.e. internal abelian groups in $\bf Top$) is not an abelian category. So I'm wondering:

Is there a "suitably well behaved" subcategory of $\bf Top$, say $\bf T$, such that $\bf Ab(T)$ is an abelian category? 

My first guess was to look for well behaved topological spaces (locally compact Hausdorff, compactly generated Hausdorff, and so on...) Googling a little shows me that compactly generated topological groups are well known animals, but the web seems to lack of a more categorical point of view.
Any clue? Thanks in advance.

REPLY [6 votes]: This was alluded to in the comments and may not be what you're looking for, but it surely deserves mention that you can take $\mathbf{T}$ to be the category of compact Hausdorff spaces.  The category $\mathbf{Ab}(\mathbf{T})$ is the the category of compact abelian groups, which is equivalent to $\mathbf{Ab}^{op}$ and hence abelian by Pontryagin duality.