Double Categories of Open Systems: The Cospan Approach

Abstract This is an overview of double categories of ‘open systems’: systems that can interact with their environment. We focus on the variable sharing paradigm, where we compose open systems by identifying variables. This paradigm is often implemented using structured or decorated cospans. We explain this approach using three main examples: open Petri nets, open dynamical systems, and open Petri nets with rates. We compare the virtues of structured and decorated cospan double categories, and study their common features. We show that any symmetric monoidal structured or decorated cospan double category comes with maps from two simpler double categories: its ‘exoskeleton’ and its ‘outer shell’. Finally, we study the concept of ‘hypergraph double category’, a kind of double category that should subsume structured and decorated cospans in a common framework for studying open systems in the variable sharing paradigm.