Abstract differential calculus

Abstract

We consider differentiable maps in the framework of Abstract Differential Geometry and we prove a number of calculus-type results pertaining to the chain rule, restrictions of differentiable maps to subspaces, and the differentiability of maps to and from cartesian products of spaces. We elaborate the latter in the case of functional structure sheaves to obtain a situation similar to that of smooth manifolds. Since the $\varepsilon,\delta$-approach does not make sense here, our machinery is the existence of a number of limits in the category of differential triads