Title: Field theories in synthetic differential geometry
A proper mathematical treatment of field theories requires a good understanding and handling of the involved differentiable structures. In this talk we will explore the possibility to formulate field theories in the language of synthetic differential geometry. For (nonlinear) scalar field theories this approach was suggested and carried out by Marco Benini and Alexander Schenkel. They chose the Cahier topos as a well adapted model and were able to induce a natural differentiable structure on the solution space of the theories in question. I will review some of their ideas. On the way I will explain some basics of field theory in order to give a context for those who are unfamiliar with it.
The Cahiers topos is a category of set valued sheafs. This is not the right data to describe the fields of Yang Mills theory with. Following ideas of Benini, Schenkel and Urs Schreiber we will explore how a natural formulation would involve groupoid valued presheaves satisfying a homological descent condition.