  We introduce an abstract concept of quantum field theory on categories
fibered in groupoids over the category of spacetimes. This provides us with a
general and flexible framework to study quantum field theories defined on
spacetimes with extra geometric structures such as bundles, connections and
spin structures. Using right Kan extensions, we can assign to any such theory
an ordinary quantum field theory defined on the category of spacetimes and we
shall clarify under which conditions it satisfies the axioms of locally
covariant quantum field theory. The same constructions can be performed in a
homotopy theoretic framework by using homotopy right Kan extensions, which
allows us to obtain first toy-models of homotopical quantum field theories
resembling some aspects of gauge theories.
