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.