A coalgebraic approach to the wave equation

Abstract

I’ll explain how the discrete wave equation and other Hamiltonian dynamical systems arise coalgebraically and compose. In particular, I’ll give a dynamic traced semicategory indexed by manifolds, meaning a structure that interprets wiring diagrams which include serial composition, parallel composition, and feedback, as well as arbitrary smooth maps, but where the wiring can change through time. Here, we use a semicategory—meaning that the wiring diagrams do not include identities—because dynamical systems take nonzero time to transfer information. The wiring diagrams are Hamiltonian in the sense that they change based on a potential. Any (unchanging) serial composite of a certain very simple map recovers the discrete wave equation.