My question is simple:

Is the category of profinite groups an accessible category?

Thank you

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    $\begingroup$ @PeteL.Clark That only shows that profinite groups are not closed under colimits in topological groups, however. For instance, the category of compact Hausdorff spaces is cocomplete but not closed under colimits in the category of all topological spaces. $\endgroup$ – Zhen Lin Aug 17 '13 at 12:16
  • $\begingroup$ @user48900 I'm inclined to believe the answer is no, but for a different reason: the category of profinite groups is the pro-completion of the category of finite groups, so its opposite is an accessible category. There is a result that says that the opposite of a locally presentable category is locally presentable if and only if it is a preorder, but unfortunately that is not applicable in this case. $\endgroup$ – Zhen Lin Aug 17 '13 at 12:21
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    $\begingroup$ Yes, in the category of topological spaces. But in the category of compact Hausdorff spaces, the coproduct of countably many copies of $1$ is the ultrafilter space $\beta \mathbb{N}$. $\endgroup$ – Zhen Lin Aug 17 '13 at 12:24
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    $\begingroup$ Maybe this question should be asked on MathOverflow, by the way... $\endgroup$ – Pete L. Clark Aug 17 '13 at 12:28
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    $\begingroup$ @user48900 Yes, as I said. My point is that there is no reason to expect it to be accessible – if it is then it is some kind of accident. (For example, the category of profinite abelian groups is definitely not accessible: because it is equivalent to the opposite of the category of torsion abelian groups, which is locally presentable.) $\endgroup$ – Zhen Lin Aug 17 '13 at 12:44

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