  The goal of this work, motivated by the desire to understand causality in
classical and quantum gravity, is an in depth investigation of causality in
classical field theories with quasilinear equations of motion, of which General
Relativity is a prominent example. Several modern geometric tools (jet bundle
formulation of partial differential equations (PDEs), the theory of symmetric
hyperbolic PDE systems, covariant constructions of symplectic and Poisson
structures) and applies them to the construction of the phase space and the
algebra of observables of quasilinear classical field theories. This
construction is shown to be diffeomorphism covariant (using auxiliary
background fields if necessary) using categorical tools in a strong parallel
with the locally covariant field theory (LCFT) formulation of quantum field
theory (QFT) on curved spacetimes. In this context, generalized versions of
LCFT axioms become theorems of classical field theory, which includes a
generalized Causality property. Considering deformation quantization as the
connection to QFT, a plausible conjecture is made about the Causal structure of
quantum gravity. In the process, conal manifolds are identified as the
generalization of the causal structure of Lorentzian geometry to quasilinear
PDEs. Several important concepts and results are generalized from Lorentzian to
conal geometry. Also, the proof of compatibility of the Peierls formula for
Poisson brackets and the covariant phase space symplectic structure for
hyperbolic systems is generalized to now encompass systems with constraints and
gauge invariance.
