There and back again  -  On the functorial foundation of algebraic quantum field theory and algebraic varieties

Recent developments in the formulation of algebraic quantum field theory led Fredenhagen et al. to promote to formulate quantum field theory as a functor from globally hyperbolic manifolds into star operator algebras. This approach should be compared with the notion of a Tannakian representation category as used for example in the work of Connes-Marcolli. I will give arguments that these two notions form a pair of adjoint functors (Tannaka-Krein duality). To exemplify this claim, we will study combinatorial representation theory and recall that an algebraic variety can be defined as a functor. This approach also provides the notion of an algebraic group scheme and implements the Hopf algebraic structures which finally pup up (dualized) in the quantum field theory. The notion of adjointness imposes restrictions to the source and target categories, which may help to identify (non-commutative) geometries which can carry a quantum field theory and so complements a functorial approach to quantum gravity as proposed recently by Verch and Paschke.