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The group associated to a Tannaka category $T$ over a field is pro-reductive if and only if $T$ is semi-simple.

Pro-reductive groups make sense over any scheme.

Is there an extension of the theory of Tannakian categories over fields to "Tannakian categories over a scheme" such that "semi-simple Tannakian categories" correspond to "pro-reductive group schemes"?

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    $\begingroup$ You seem to be asking whether the representations of pro-reductive groups over schemes are semisimple --- this is not even true over fields of nonzero characteristic. $\endgroup$
    – anon
    Commented Sep 13, 2014 at 21:58

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