  Many systems of interest in science and engineering are made up of
interacting subsystems. These subsystems, in turn, could be made up of
collections of smaller interacting subsystems and so on. In a series of papers
David Spivak with collaborators formalized these kinds of structures (systems
of systems) as algebras over presentable colored operads. It is also very
useful to consider maps between dynamical systems, which in effect amounts to
viewing dynamical systems as objects in an appropriate category. This is the
point of view taken by DeVille and Lerman in the study of dynamics on networks.
The goal of this paper is to describe an algebraic structure that encompasses
both approaches to systems of systems. This allows us, on one hand, build new
large open systems out of collections of smaller open subsystems and on the
other keep track of maps between open systems. Consequently we obtain synchrony
results for open systems which generalize the synchrony results of Golubitsky,
Stewart and their collaborators for groupoid invariant vector fields on coupled
cell networks.
