PD Higgs crystals and Higgs cohomology in characteristic $p$

For a smooth scheme $X$ over a perfect field of characteristic $p$, we define two sites and show that if a lifting of $X$ over the ring of Witt vectors is given, the categories of crystals on these sites are equivalent to the categories of certain $\mathcal {D}$-modules and certain Higgs modules respectively. We construct a natural functor between these sites and show that the morphism of topoi associated to this functor induces an equivalence between the categories of quasi-coherent crystals on these sites. We also study a filtered version of these results and give interpretations of the de Rham cohomology and the Higgs cohomology in terms of filtered crystals. We use it to give an alternative proof of the cohomology comparison theorem due to Ogus and Vologodsky.