  Topos theory is a category-theoretic 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 fibrations, cofibrations and equivalences) satisfying certain
axioms. Functors from the category of Weil algebras to the category 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 containing Weil algebras to the Weil topos
standing for the incarnation of each algebraic entity of the category in the
Weil topos. The Weil functor and the canonical ring object are to be defined in
terms of the Dubuc functor. The principal object in this paper is to present a
category-theoretical axiomatization of the Weil topos with the Dubuc functor
intended to be an adequate framework for axiomatic classical differential
geometry and hopefully comparable with model categories. 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.
