The theory of $F$-zips is a positive characteristic analog of the theory of integral Hodge-structures. As shown by Moonen and Wedhorn, one can associate to any proper smooth scheme with degenerating Hodge-de Rham spectral sequence and finite locally free Hodge cohomologies an $F$-zips, via its $n$-th de Rham cohomology.
Using the theory of derived algebraic geometry, we can work with the de Rham hypercohomology and show that it has a derived analog of an $F$-zip structure. We call these structures derived $F$-zips. We can attach to any proper smooth morphism a derived $F$-zip and analyze families of proper smooth morphisms via their underlying derived $F$-zip.