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