  In "On the calculation of some differential Galois groups" (Invent. Math. 87
(1987), no. 1), Katz defines the notion of a special flat connection on the
complex affine line minus the origin, and he shows that the functor which
restricts a flat connection to the punctured disc around the point at infinity
gives rise to an equivalence between the category of special flat connections
and the category of differential modules on the field of Laurent series over
the complex numbers.
  In this article, we prove the corresponding statement over an algebraically
closed field k of positive characteristic. The role of flat connections is
played by vector bundles carrying an action of the (full) ring of differential
operators. We call such objects stratified bundles. The formal local variant on
the field of Laurent series k((t)) is called iterated differential module. We
define the notion of a special stratified bundle on the affine line over k
minus the origin, and show that restriction to the punctured disc around the
point at infinity induces an equivalence between the category of special
stratified bundles and iterated differential modules. This extends one of the
main results of "Local-to-global extensions of representations of fundamental
groups" (N. Katz, Ann. Inst. Fourier (Grenoble) 36 (1986), no. 4), and has
several interesting consequences which extend well-known statements about
\'etale fundamental groups to higher dimensional fundamental groups.
