FF-isocrystals and homotopy types

We study a positive characteristic analogue of the nonabelian Hodge structure constructed by Katzarkov, Pantev, and Toen on the homotopy type of a complex algebraic variety. Given a proper smooth scheme XX over a perfect field of characteristic pp and a Tannakian category CC of isocrystals on XX, we construct an object XCXC in a suitable homotopy category of simplicial presheaves whose category of local systems is equivalent to CC in a manner compatible with cohomology. We then study FF-isocrystal structure on these simplicial presheaves. As applications of the theory, we prove a pp-adic analogue of a result of Hain on relative Malcev completions, a generalization to the level of homotopy types of a theorem of Katz relating pp-adic étale local systems and FF-isocrystals, as well as a pp-adic version of the formality theorem in homotopy theory. We have also included a new proof based on reduction modulo pp of the formality theorem for complex algebraic varieties.