Types of quantum field thories
In the context of mechanics (broadly construed), one distinguishes between kinematics and dynamics:
Kinematics concerns (only) the physical fields, states and observables, including the spaces and algebras (such as a phase space or Hilbert space of (pure) states and an appropriate algebra of observables) into which they are organised;
Dynamics additionally treats the evolution of the system in time or even spacetime: as given by a Lagrangian and action functional and as given by the action of Hamiltonian quantum observables on physical states.
In the Schrödinger picture, we think of the states as evolving, while the observables evolve in the Heisenberg picture. In the interaction picture we think of the states as evolving with respect to a given time evolution and the observables to evolve, too, with respect to a perturbation of this time evolution.
with that givenm the dynamics of is , the action of the functor on morphisms.
to from a (infinity,n)-category of cobordisms with -structure to some symmetric monoidal (infinity,n)-category the dichotomy between kinematics and dynamics may be regarded as being blurred a bit: we can regard the action on objects as the genuine kinematics and the action on n-morphisms as the genuine dynamics, and then the actions as interpolating between these two notions.
So in terms of the background field data we have: