You are cordially invited to the Topology Seminar organized by the Department of Mathematics.
Speaker: David Jaz Myers (Topos Institute)
“Double categorical right modules as the algebra of coupled dynamical systems”
Abstract: Open dynamical systems whose dynamics depend on free parameters and which expose some variables of their state may be coupled by setting their parameters as functions of the exposed variables of other systems. Together with their parallel (cartesian) product, these systems constitute a lax symmetric monoidal functor from a category of interfaces (consisting of parameter and exposed variable sets) and coupling laws (often expressed as wiring diagrams) to the category of sets — that is, we have a symmetric monoidal right module of systems over the symmetric monoidal category of interfaces and coupling laws. Schultz, Spivak and Vasilakopoulou show that the behavior of these systems may be expressed as a morphism of lax symmetric monoidal functors from this module of systems to a module of time-varying sets — sheaves on the interval domain of the real line.
In this talk, we’ll see that the SSV behavior functors — and many others similar behavior functors — are in fact representable when seen not as concerning right modules of categories, but as concerning right modules over double categories. We will develop the theory of (loose) modules between double categories using an approach inspired by Joyal’s “barrels” (joint work with Sophie Libkind), and describe the cartesian pseudo-functoriality of restriction of loose right modules which allows for the pseudo-functorial construction of symmetric monoidal loose right modules of open dynamical systems from an abstract notion of “tangent bundle category”. By expanding the definition of “tangent bundle” in this way, we include all sorts of generalized Moore machines (including not only systems of ordinary differential equations, but also partially observable Markov processes and various sorts of non-deterministic automata).
We’ll then see a general result (joint work with Matteo Capucci) giving conditions under which discrete opfibration classifiers in a 2-category K can be lifted to the 2-category of algebras and lax morphisms for a 2-monad T on K. We will use this result to show that a certain symmetric monoidal loose right module of spans is a discrete opfibration classifier among symmetric monoidal loose right modules, and conclude by showing that a variety of behavior functors for open dynamical systems are covariantly representable. Time permitting, we will also see that system safety and stability properties are often themselves contravariantly representable via the representability of Lyapunov and control barrier functions by functions into simple systems.
Date: April 28, Monday, 2025
Time: 13:30 UTC+3
Place: ZOOM
To request the event link, please send a message to cihan.okay@bilkent.edu.tr