BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:David Jaz Myers (Topos Institute) DTSTART:20250502T103000Z DTEND:20250502T113000Z DTSTAMP:20260422T140128Z UID:BilTop/115 DESCRIPTION:Title: Double categorical right modules as the algebra of coupled dynamical syst ems\nby David Jaz Myers (Topos Institute) as part of Bilkent Topology Seminar\n\nLecture held in SA 141.\n\nAbstract\nOpen dynamical systems who se dynamics depend on free parameters and which expose some variables of t heir state may be coupled by setting their parameters as functions of the exposed variables of other systems. Together with their parallel (cartesia n) product\, these systems constitute a lax symmetric monoidal functor fro m a category of interfaces (consisting of parameter and exposed variable s ets) and coupling laws (often expressed as wiring diagrams) to the categor y of sets --- that is\, we have a symmetric monoidal right module of syste ms over the symmetric monoidal category of interfaces and coupling laws. S chultz\, Spivak and Vasilakopoulou show that the behavior of these systems may be expressed as a morphism of lax symmetric monoidal functors from th is module of systems to a module of time-varying sets --- sheaves on the i nterval domain of the real line.\n\nIn 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 categorie s\, but as concerning right modules over double categories. We will develo p the theory of (loose) modules between double categories using an approac h inspired by Joyal's "barrels" (joint work with Sophie Libkind)\, and des cribe the cartesian pseudo-functoriality of restriction of loose right mod ules which allows for the pseudo-functorial construction of symmetric mono idal loose right modules of open dynamical systems from an abstract notion of "tangent bundle category". By expanding the definition of "tangent bun dle" in this way\, we include all sorts of generalized Moore machines (inc luding not only systems of ordinary differential equations\, but also part ially observable Markov processes and various sorts of non-deterministic a utomata).\n\nWe'll then see a general result (joint work with Matteo Capuc ci) giving conditions under which discrete opfibration classifiers in a 2- category K can be lifted to the 2-category of algebras and lax morphisms f or a 2-monad T on K. We will use this result to show that a certain symmet ric monoidal loose right module of spans is a discrete opfibration classif ier among symmetric monoidal loose right modules\, and conclude by showing that a variety of behavior functors for open dynamical systems are covari antly representable. Time permitting\, we will also see that system safety and stability properties are often themselves contravariantly representab le via the representability of Lyapunov and control barrier functions by f unctions into simple systems.\n LOCATION:https://researchseminars.org/talk/BilTop/115/ END:VEVENT END:VCALENDAR