Tittel: A classifying space for principal $G$-bundles with connection, in the category of simplicial presheaves

Veileder: Gereon Quick

Sammendrag: In this master’s thesis, we study principal $G$-bundles on smooth manifolds with connection, and how to make universal objects to classify them, similar to Grassmannians for vector bundles. The constructions take us into the category of simplicial presheaves on smooth manifolds, and result in a theorem that states that the only natural differential forms associated to the connection on principal $G$-bundles are the ones constructed in the Chern-Weil homomorphism. This was first obtained in this way by Freed and Hopkins in [7]. In the last chapter, we take a short look at what would happen if the manifolds considered were complex instead of smooth.