  Originally conjectured unpublished by Grothendieck, then formulated precisely
by Katz, the $p$-curvature conjecture is a local-global principle for algebraic
differential equations. It is at present open, though various cases are known.
Katz subsequently proved this conjecture in a wide range of cases, for
differential equations corresponding to the Gauss-Manin connection on algebraic
de Rham cohomology.
  This dissertation addresses the non-abelian analogue of Katz' theorem, in the
sense of Simpson's non-abelian Hodge theory, surveyed by Simpson and later
developed in characteristic $p$ by Ogus and Vologodsky. Specifically, there is
a canonical non-abelian Gauss-Manin connection on $M_{dR}$, the stack of vector
bundles with integrable connection, which is the appropriate definition of
non-abelian de Rham cohomology. In this dissertation, I introduce this
connection and its $p$-curvature; this requires the generalization of the
$p$-curvature conjecture due to Bost, Ekedahl and Shepherd-Barron. Then, I
prove that the analogue of the main technical result of Katz' theorem holds for
this connection.
