Grothendieck connection

In algebraic geometry and synthetic differential geometry, a Grothendieck connection is a way of viewing connections in terms of descent data from infinitesimal neighbourhoods of the diagonal.
Introduction and motivation.
The Grothendieck connection is a generalization of the Gauss–Manin connection constructed in a manner analogous to that in which the Ehresmann connection generalizes the Koszul connection. The construction itself must satisfy a requirement of "geometric invariance", which may be regarded as the analog of covariance for a wider class of structures including the schemes of algebraic geometry. Thus the connection in a certain sense must live in a natural sheaf on a Grothendieck topology. In this section, we discuss how to describe an Ehresmann connection in sheaf-theoretic terms as a Grothendieck connection.
Let "M" be a manifold and π : "E" → "M" a surjective submersion, so that "E" is a manifold fibred over "M". Let J1("M","E") be the first-order jet bundle of sections of "E". This may be regarded as a bundle over "M" or a bundle over the total space of "E". With the latter interpretation, an Ehresmann connection is a section of the bundle (over "E") J1("M","E") → "E". The problem is thus to obtain an intrinsic description of the sheaf of sections of this vector bundle.
Grothendieck's solution is to consider the diagonal embedding Δ : "M" → "M" × "M". The sheaf "I" of ideals of Δ in "M" × "M" consists of functions on "M" × "M" which vanish along the diagonal. Much of the infinitesimal geometry of "M" can be realized in terms of "I". For instance, Δ* ("I"/"I"2) is the sheaf of sections of the cotangent bundle. One may define a "first-order infinitesimal neighborhood" "M"(2) of Δ in "M" × "M" to be the subscheme corresponding to the sheaf of ideals "I"2. (See below for a coordinate description.) 
There are a pair of projections "p"1, "p"2 : "M" × "M" → "M" given by projection the respective factors of the Cartesian product, which restrict to give projections "p"1, "p"2 : "M"(2) → "M". One may now form the pullback of the fibre space "E" along one or the other of "p"1 or "p"2. In general, there is no canonical way to identify "p"1*"E" and "p"2*"E" with each other. A Grothendieck connection is a specified isomorphism between these two spaces. One may proceed to define curvature and p-curvature of a connection in the same language.