F-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 X over a perfect field of characteristic p and a Tannakian category image of isocrystals on X, we construct an object image in a suitable homotopy category of simplicial presheaves whose category of local systems is equivalent to image in a manner compatible with cohomology. We then study F-isocrystal structure on these simplicial presheaves. As applications of the theory, we prove a p-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 p-adic étale local systems and F-isocrystals, as well as a p-adic version of the formality theorem in homotopy theory. We have also included a new proof based on reduction modulo p of the formality theorem for complex algebraic varieties.