Abstract. Bicategories of spans are characterized as cartesian bicategories in which every comonad has an Eilenberg-Moore object and every left adjoint arrow is comonadic. 1.

Abstract. We combine two recent ideas: cartesian differential categories, and restriction categories. The result is a new structure which axiomatizes the category of smooth maps defined on open subsets of Rn in a way that is completely algebraic. We also give other models for the resulting structure, discuss what it means for a partial map to be additive or linear, and show that differential restriction structure can be lifted through various completion operations.