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 $$\pi : E \to M$$ a surjective submersion, so that $$E$$ is a manifold fibred over $$M.$$ Let $$J^1(M, E)$$ 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 $$E$$) $$J^1(M, E) \to E.$$  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 $$\Delta : M \to M \times M.$$ The sheaf $$I$$ of ideals of $$\Delta$$ in $$M \times M$$ consists of functions on $$M \times M$$ which vanish along the diagonal. Much of the infinitesimal geometry of $$M$$ can be realized in terms of $$I.$$ For instance, $$\Delta^*\left(I, I^2\right)$$ is the sheaf of sections of the cotangent bundle. One may define a first-order infinitesimal neighborhood $$M^{(2)}$$ of $$\Delta$$ in $$M \times M$$ to be the subscheme corresponding to the sheaf of ideals $$I^2.$$ (See below for a coordinate description.)

There are a pair of projections $$p_1, p_2 : M \times M \to M$$ given by projection the respective factors of the Cartesian product, which restrict to give projections $$p_1, p_2 : M^{(2)} \to M.$$ 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.