Causal Graph Dynamics and Kan Extensions

Abstract: On the one side, the formalism of Global Transformations comes with the claim of capturing any transformation of space that is local, synchronous and deterministic. The claim has been proven for different classes of models such as mesh refinements from computer graphics, Lindenmayer systems from morphogenesis modeling and cellular automata from biological, physical and parallel computation modeling. The Global Transformation formalism achieves this by using category theory for its genericity, and more precisely the notion of Kan extension to determine the global behaviors based on the local ones. On the other side, Causal Graph Dynamics describe the transformation of port graphs in a synchronous and deterministic way and has not yet being tackled. In this paper, we show the precise sense in which the claim of Global Transformations holds for them as well. This is done by showing different ways in which they can be expressed as Kan extensions, each of them highlighting different features of Causal Graph Dynamics. Along the way, this work uncovers the interesting class of Monotonic Causal Graph Dynamics and their universality among General Causal Graph Dynamics.