Pursuing the analogy between variations of Hodge structures in characteristic zero and F-crystals in characteristic $p$, we introduce and study the category of “T-crystals”, which are the crystalline manifestation of modules with integrable connection and filtration satisfying Griffiths transversality. We construct a functor from the category of F-crystals (or more generally F-spans) to the category of T-crystals, on any smooth logarithmic scheme in characteristic $p$. This functor is shown to commute with the formation of higher direct images–a generalization of Mazur's fundamental theorem on Frobenius and Hodge filtration to the case of crystalline cohomology with coefficients. Applications include results about Newton and Hodge polygons (“Katz's Conjecture”) and the degeneration of the Hodge spectral sequence (“Hodge Decomposition”), in both cases for the cohomology of a variety with coefficients in an F-crystal.