With the introduction of toposes as a categorical surrogate of set theory whose underlying logic is Heyting and where no appeal to an axiom of choice is permitted, it became possible to formally introduce infinitesimals in the study of differential geometry in the same spirit as that of the work carried out by Charles Ehresmann and AndréWeil in the 50s. Whereas the known models that are well adapted for the applications to classical mathematics are necessarily Grothendieck toposes, the theory of such is not enough to express the richness of the synthetic theory. For the latter one needs the internal language that is part of the theory of toposes and that is based on the axiom of the existence of a subobjects classifier. This first part is an introduction to topos theory and to synthetic differential geometry, both of which originated in the work of F.W. Lawvere. These introductory presentations will gradually be enriched as we proceed, but only as needed for the purposes of this book.

In this second part we illustrate the principles of synthetic differential geometry and topology in two distinct areas. The first example is a theory of connections and sprays, where we show that—unlike the classical situation—the passage from connections to geodesic sprays need not involve integration, except in infinitesimal form. Moreover, the validity of the Ambrose-Palais-Singer theorem within SDG extends well beyond the classical one. In our second example we show how in SDG one can develop a calculus of variations ‘without variations’, except for those of an infinitesimal nature. Once again, the range of applications of the calculus of variations within SDG extends beyond the classical one. Indeed, in both examples, we work with infinitesimally linear objects—a class closed under finite limits, exponentiation, and ´etale descent. The existence of well adapted models of SDG guarantees that those theories developed in its context are indeed relevant to the corresponding classical theories.

Our goals in this book are twofold. The first is to achieve conceptual simplicity by a judicious choice of axioms in the setting of topos theory. The second is to make sure that our results apply to classical mathematics. To this end we revisit the notion of a well adapted model of SDG, extend it to SDT, and then assume the existence of one such. An application of the existence of such a model to the theory of unfoldings is then given.

In this third part we introduce and then make essential use of the intrinsic local and infinitesimal concepts available in any topos. By analogy with the synthetic theory of differential geometry, where jets of smooth maps are assumed representable by suitable infinitesimal objects, the axioms of synthetic differential topology express the representability of germs of smooth mappings, also by infinitesimal objects. However, whereas the nature of the infinitesimals of synthetic differential geometry are algebraic, those of synthetic differential topology are defined using the full force of the logic of a topos. In both cases, the use of Heyting (instead of Boolean) logic introduces an unexpected conceptual richness. To the basic axioms of synthetic differential topology we add a postulate of infinitesimal inversion and a postulate of the (logical) infinitesimal integration of vector fields. To these axioms and postulates we add others that shall be needed in the theory of stable mappings and their singularities to be dealt with in the fourth part of this book. The validity of all of these axioms and postulates in a single model will be shown in the fifth part of the book.

In this fourth part we present, within the context of synthetic differential topology (SDT), a theory of stable germs of smooth maps including Mather's theorem on the equivalence between stability and infinitesimal stability, followed by Morse theory. Germ representability by logico-infinitesimal objects brings about a considerable simplification of the subject.