Sheaf of connection algebras

This article defines a sheaf that can be associated to a differential manifold. The global analog of this sheaf, which is also the same as the object of the sheaf associated to the whole manifold, is: connection algebra

Definition

Let $M$ be a differential manifold and $E$ a vector bundle over $M$. The sheaf of connection algebras of $E$ is defined as follows:

• For every open subset $U$ of $M$, the object associated to $E$ is the connection algebra associated to the restriction of $E$ to a vector bundle over $U$
• The restriction map is defined as follows: Fill this in later

Sometimes the sheaf of connection algebras is termed the connection algebra, though the latter term is sometimes used for the global object.