Abstract

Synthetic Differential Geometry (SDG) is an approach to differential geometry that builds on an algebro-geometric theory of infinitesimals. The notion of infinitesimal is made rigorous by the Kock-Lawvere axiom scheme. One strength of this approach is that the formalization agrees with the intuitive use of infinitesimals in differential geometry as employed by geometers like E. Cartan and S. Lie.

The simplicity and intuition of this approach comes with a price: It turns out that SDG has no non-trivial models in the realm of classical logic, i.e., in any boolean topos. However, one can construct toposes that yield models of SDG . Of particular interest are the so called well-adapted models. These are topos models together with a ‘nice’ embedding of the category of smooth manifolds.

This talk will consist of three parts. In the first part I shall introduce the basic concepts and notions of SDG , in particular the Kock-Lawvere axiom scheme. In the second part I will introduce well-adapted models axiomatically and present some consequences of the axioms. Finally, in the third part I shall sketch how one can construct well-adapted models using C-infinity rings.