Power Series in Synthetic Differential Geometry

Davorin Lesnik
Friday, July 26, 2013 - from 16:30 to 16:55
Synthetic differential geometry (SDG) is an approach to differential geometry (and analysis in general), where the axioms are changed in such a way that the smooth structure (and in particular topology) becomes an intrinsic part of sets, rather than something that is artificially affixed to them. This generally makes definitions and proofs simpler, and one can show that in a suitable sense synthetic theorems imply the corresponding classical ones. In SDG nontrivial infinitesimals exist, which in particular implies the failure of classical logic. Models of SDG are certain kind of topoi (categories which allow the interpretation of higher-order intuitionistic logic). Limit arguments cannot be used in SDG, and are replaced by suitable axioms (essentially replacing analytical $\varepsilon\delta$-arguments with algebraic ones). It is well known, what the axioms which enable differentiation and integration are. In this talk we propose an axiom which allows us to evaluate power series. We observe that in classical mathematics the embedding of analytical functions into smooth ones is a differential algebra morphism which --- when parametrized by the radius of convergence --- becomes a natural transformation. We find a characterization of this natural transformation which does not involve limit arguments and works constructively. We then show that postulating the existence of a map satisfying this characterization in SDG enables us to utilize power series, as desired (in particular, for defining functions and solving certain differential equations).