From Lie algebras to Lie groups within synthetic differential geometry: Weil sprouts of Lie's third fundamental theorem

Weil prolongations of a Lie group are naturally Lie groups. It is not known in the theory of in nite-dimensional Lie groups how to construct a Lie group with a given Lie algebra as its Lie algebra or whether there exists such a Lie group at all. We will show in this paper how to construct some Weil prolongations of this mythical Lie group from a given Lie algebra. We will do so within our favorite framework of synthetic di erential geometry