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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.

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    $\begingroup$ You wouldn't count discrete spaces or compact spaces, I guess... And the category of locally compact abelian Hausdorff groups is still quasi-abelian, which is amply sufficient for most purposes (it is a meta-principle that outside of essentially algebraic contexts, the theory of abelian categories is hopelessly inadequate). $\endgroup$
    – t.b.
    Mar 31 '12 at 11:05
  • $\begingroup$ @t.b.: "amply sufficient for most purposes" please expand, I'm interested. Thanks for your neat comment! $\endgroup$
    – fosco
    Mar 31 '12 at 11:29
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    $\begingroup$ The category of internal abelian groups in a Barr-exact category is automatically an abelian category, so it is enough to find a Barr-exact subcategory of $\textbf{Top}$. $\endgroup$
    – Zhen Lin
    Mar 31 '12 at 11:42
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    $\begingroup$ You could try this article, for example and to follow up on @ZhenLin 's comment: see here. $\endgroup$
    – t.b.
    Mar 31 '12 at 11:45
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    $\begingroup$ Dear @t.b.: would you mind expanding on your meta-principle? Thanks! $\endgroup$ Jan 16 '14 at 14:23
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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.

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