Functorial Evolution of Quantum Fields

We present a compositional algebraic framework to describe the evolution of quantum fields in discretised spacetimes. show how familiar notions from Relativity and causality can be recovered purely order-theoretic way causal order events spacetime, with no direct mention analysis or topology. formulate theory-independent over orders compositional, functorial way. draw strong connection Algebraic Quantum Field Theory (AQFT), using sheaf-theoretical approach our definition spaces states regions spacetime. introduce symmetry cellular automata, which we subsume existing definitions Cellular Automata (QCA) previous literature. Given extreme flexibility constructions, propose that used as starting point for new developments AQFT, QCA more generally Theory.