The cohomology of a family of algebraic varieties carries a number of interrelated structures, of both Hodge-theoretic and arithmetic flavors. I’ll explain joint work with Josh Lam developing analogues of some of these structures and interrelations for nonabelian cohomology—that is, the space of representations of the fundamental groups of a family of algebraic varieties, in its various incarnations. As an application, I’ll explain a non-abelian version of Katz’s 1972 proof of the p-curvature conjecture for Gauss-Manin connections. Namely, let X→Sbe a smooth proper morphism. We show that if the p-curvature of the isomonodromy foliation on the moduli space of flat bundles on X/Svanishes, then the action of π1(S,s) on the space of conjugacy classes of representations of π1(Xs) into GLn(Z) factors through a finite group.