Last week I reviewed how the category of affine Lagrangian relations is the natural setting for certain time-independent, non-deterministic mechanical systems, sketching its presentation as a compact closed category. In my second talk, I will discuss how the syntax and semantics can be augmented with a notion of discrete-time dynamics, using the theory of guarded and delayed traces.
There are several inequivalent ways to add discrete-time dynamics to this category. First, I will discuss how a notion of time can be added via the "state construction," using the coend calculus. Next, I will discuss how discrete-time dynamics can be added by moving to frequency space, and working over the field of rational functions. I will sketch a complete graphical language for this second approach, and I will discuss how both approaches are related to each other.
This is ongoing work with my coauthor Giovanni de Felice.