A Grothendieck topos of generalized functions I: basic theory

The main aim of the present work is to arrive at a mathematical theory close to the historically original conception of generalized functions, i.e. set-theoretical functions defined on, and with values in, a suitable ring of scalars and sharing a number of fundamental properties with smooth functions, in particular with respect to composition and nonlinear operations. This is how they are still used in informal calculations in physics. We introduce a category of generalized functions as smooth set-theoretical maps on (multidimensional) points of a ring of scalars containing infinitesimals and infinities. This category extends Schwartz distributions. The calculus of these generalized functions is closely related to classical analysis, with point values, composition, nonlinear operations and the generalization of several classical theorems of calculus. Finally, we extend this category of generalized functions to a Grothendieck topos of sheaves over a concrete site. This topos hence provides a suitable framework for the study of spaces and functions with singularities. In this first paper, we present the basic theory; subsequent ones will be devoted to the resulting theory of ODE and PDE.