This article gives a global construction for a differential manifold. There exists a sheaf analog of it, that associates a similar construct to every open subset. This sheaf analog is termed: sheaf of connection algebras
Let be a differential manifold. The connection algebra of , denoted , is defined as follows. Consider the Lie algebra of first-order differential operators on , and treat it as a -bimodule. Take the tensor algebra generated by this as a -bimodule, and quotient it by the two-sided ideal generated by . Here is the differential operator obtained as multiplication by the constant function .
The quotient algebra we get is termed the connection algebra on .
The term connection algebra is also sometimes used for the sheaf of connection algebras, which is a sheaf that associates to every open subset, the connection algebra over that open subset.