“Systems of systems” is a fashionable buzzword for complicated systems that are themselves made of complicated systems, often of disparate sorts. They’re important in modern engineering, and it takes some thought to keep them from being unmanageable. Biology and ecology are full of systems of systems.
David Spivak has been working a lot on operads as a tool for describing systems of systems. Here’s a nice programmatic talk advocating this approach:
• David Spivak, Operads as a potential foundation for
systems of systems.
This was a talk he gave at the Generalized Network Structures and Dynamics Workshop at the Mathematical Biosciences Institute at Ohio State University this spring.
You won’t learn what operads are from this talk—for that, try this:
• Wikipedia, Operad.
But if you know a bit about operads, it may help give you an idea of their flexibility as a formalism for describing ways of sticking together components to form bigger systems!
I’ll probably talk about this kind of thing more pretty soon. So far I’ve been using category theory to study networked systems like electrical circuits, Markov processes and chemical reaction networks. The same ideas handle all these different kind of systems in a unified way. But I want to push toward biology. Here we need more sophisticated ideas. My philosophy is that while biology seems “messy” to physicists, living systems actually operate at higher levels of abstraction, which call for new mathematics.