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 $R^n$ 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.
Keywords: Differential restriction categories, Rational functions and the Rational monad, Join completion, Classical completion
2000 MSC: 14A20,18D10, 18C20, 12H05, 32W99,58A99
Theory and Applications of Categories, Vol. 25, 2011, No. 21, pp 537-613.