  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, non-linear operations and the generalization of several
classical theorems of calculus. Finally, we extend this category of generalized
functions into 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.
