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Berthelot's comparison theorem connects the algebraic de Rham cohomology of a $\mathbb{Z}_p$-scheme and the crystalline cohomology of its special fiber. Is there a statement on the level of homotopy types which implies Berthelot comparison?

One would have to define crystalline and de Rham homotopy types first, and I am not completely sure how to do that.

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Yes. A Google search will immediately give you lots of articles in varying generality (see e.g. work of Shiho), but one article I'm particularly fond of is Kim and Hain's "A de Rham-Witt approach to crystalline rational homotopy theory" in Compositio 2001. See Theorem 2 for the comparison of crystalline and de Rham homotopy types.

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