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The tilting equivalence for perfectoid algebras gives an equivalence of categories $$K\text{-perf} \cong K^\flat\text{-perf}$$ where the left-hand-side are algebras in characteristic zero and the right-hand-side are in characteristic $p$.

Are there any other non-trivial examples of an equivalence of categories relating characteristic zero things (rings, or schemes, or whatever) on one side to characteristic $p$ things on the other?

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  • $\begingroup$ A trivial example: vector spaces. Slightly less trivial: representations. $\endgroup$
    – WhatsUp
    Sep 28, 2019 at 13:14
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    $\begingroup$ How are vector spaces an example? $\endgroup$
    – Kim
    Sep 28, 2019 at 13:19
  • $\begingroup$ I think the category of split reductive algebraic groups, with morphisms isomorphisms, is an example, I think, because the objects and morphisms may be described combinatorially. $\endgroup$
    – Will Sawin
    Sep 28, 2019 at 23:30
  • $\begingroup$ I'd like to consider such examples (where one artificially restricts the set of morphisms) to be trivial. $\endgroup$
    – Kim
    Sep 29, 2019 at 0:30

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