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moved the statement that a profinite group is a group internal to profinite sets from the “Examples”-section to “Definition”-section
I vaguely recall that it’s not entirely trivial that pro-(finite algebras) are the same as internal algebras in profinite sets – there’s some restriction on the signature and axiomatisation, I think.
Stone Spaces chapter VI, section 2.9 has a general result in this direction.
I added the remark that finite index subgroups of a profinite group are not generally open, and gave the usual example involving non-principal ultrafilters. I also briefly discussed when a group is isomorphic to its own profinite completion.
Dumb question:
Is it right that the (integral, say) group cohomology of the profinite integers is the colimit of the group cohomologies of the cyclic groups
$H^k\big( \widehat \mathbb{Z} ;\, \mathbb{Z}\big) \;\simeq\; H^k\big( \underset{\longleftarrow}{\lim} C_\bullet ;\, \mathbb{Z}\big) \;\simeq\; \underset{\longrightarrow}{\lim} H^k\big( C_\bullet ;\, \mathbb{Z}\big)$?
Hence that, in particular,
$H^2\big( \widehat \mathbb{Z} ;\, \mathbb{Z}\big) \;\simeq\; \underset{\longrightarrow}{\lim} H^2\big( C_\bullet ;\, \mathbb{Z}\big) \;\simeq\; \underset{\longrightarrow}{\lim} C_\bullet \;\simeq\; \mathbb{Q}/\mathbb{Z}$?
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