Weil diffeology I: Classical differential geometry

Topos theory is a category-theoretical axiomatization of set theory. Model categories are a category-theoretical framework for abstract homotopy theory. They are complete and cocomplete categories endowed with three classes of morphisms (called brations, co brations and weak equivalences) satisfying certain axioms. We would like to present an abstract framework for classical di erential geometry as an extension of topos theory, hopefully comparable with model categories for homotopy theory. Functors from the category W of Weil algebras to the category Sets of sets are called Weil spaces by Wolfgang Bertram and form the Weil topos after Eduardo J. Dubuc. The Weil topos is endowed intrinsically with the Dubuc functor, a functor from a larger category eW of cahiers algebras to the Weil topos standing for the incarnation of each algebraic entity of eW in the Weil topos. The Weil functor and the canonical ring object are to be de ned in terms of the Dubuc functor. The principal objective of this paper is to present a category-theoretical axiomatization of theWeil topos with the Dubuc functor intended to be an adequate framework for axiomatic classical di erential geometry. We will give an appropriate formulation and a rather complete proof of a generalization of the familiar and desired fact that the tangent space of a microlinear Weil space is a module over the canonical ring object