Algebras and their bimodules form a 2-category in which 2-morphisms are certain zero-th Hochschild cohomology groups. When we derive this structure (i.e., use Hochschild cochains instead of HH^0 for 2-morphisms), we find that algebras form a category in dg cocategories. The Hochschild-Kostant-Rosenberg theorem and non-commutative calculus give a rich algebraic...