January 18, 2021Luidnel Maignan (LACL)
On one hand, one often contrasts open dynamical systems with “ordinary/closed” dynamical systems, while knowing that, formally, open dynamical systems are a particular case of dynamical systems in many instances. On the other hand, a closed dynamical system is essentially a (collection of) function from a set X to itself. How to formalize “open” functions from a set X to another set Y ? This building block is necessary when designing a system as a composition of many modules. Also, how to formalize “open” functors from a category C to another category D. This building block is necessary when the dynamical system has some notion of “locality” and “space”. These questions arose in the study of so-called Global Transformations, dynamical systems where the space might be arbitrary and evolving concurrently and synchronously. In this talk, we begin by formalizing the matter as directly as possible with respect to the intuition. Then we connect the obtained construction to existing frameworks via three points of view. This might be seen as a (pedagogical ?) entry point to many categorical ideas starting from the simple ground of cellular automata. This is based on works in progress with Alexandre Fernandez and Antoine Spicher.