• Brendan Fong, *The Algebra of Open and Interconnected Systems*, Ph.D. thesis, University of Oxford, 2016. (Blog article here.)

Kenny Courser is starting to look at the next thing: how one network can turn into another. For example, a network might change over time, or we might want to simplify a complicated network somehow. If a network is morphism, a process where one network turns into another could be a ‘2-morphism’: that is, a morphism between morphisms. Just as categories have objects and morphisms, bicategories have objects, morphisms *and 2-morphisms*.

So, Kenny is looking at bicategories. As a first step, Kenny took Brendan’s setup and souped it up to define ‘decorated cospan bicategories’. I want to tell you a bit about his paper:

• Kenny Courser, Decorated cospan bicategories, to appear in *Theory and Applications of Categories*.

In this paper, he shows that these bicategories are often ‘symmetric monoidal’. This means that you can not only stick networks together end to end, you can also set them side by side or cross one over the other—and similarly for processes that turn one network into another! A symmetric monoidal bicategory is a somewhat fearsome structure, so Kenny used some clever machinery developed by Mike Shulman to get the job done […]

]]>• G. Darbo, Aspetti algebrico-categoriali della teoria dei dispotivi, *Symposia Mathematica* **IV** (1970), Istituto Nazionale di Alta Matematica, 303–336.

It’s closely connected to Brendan Fong’s thesis, but also different—and, of course, much older. According to Grandis, Darbo was the first person to work on category theory in Italy. He’s better known for other things, like ‘Darbo’s fixed point theorem’, but this piece of work is elegant, and, it seems to me, strangely ahead of its time.

]]>• Brendan Fong, *The Algebra of Open and Interconnected Systems*. (Blog article here.)

But he went further: to understand the *externally observable behavior* of an open system we often want to simplify a decorated cospan and get another sort of structure, which he calls a ‘decorated corelation’. His talk here explains decorated corelations and what they’re good for:

Then the system is just closed,

As I also tried to explain below, I would prefer to say the system is closed with respect to its environment.

]]>If it is in fact impossible to find means to interface the system with its environment, then the system is just closed. It could be turned into an open system later when the means are developed, but that’s then considered a different system.

]]>Being mathematician I think of closed systems as a special class of open systems: closed systems are those where the influence of the “environment” is zero.

I don’t know who came up with this definition but I think it is misleading. And other mathematicians would eventually come up with a different definition.

First a system usually also has an influence on the respective environment.

Moreover as I tried to point out above it is rather that a system is closed *with respect to* an environment, than just closed “as a system”. Everbody knows that if you put “systems” in a different environment than “influences” might change.

There is a precise way to say that in terms of surjective submersions. I am not sure if you’d like to see that.

I don’t have much time anymore for trying to understand lenghty math constructs and let’s put it this way – if the definition had already missed out those (what I think) important aspects than the probability that there are more problemes lurking in the construct is not too small.

]]>Being mathematician I think of closed systems as a special class of open systems: closed systems are those where the influence of the “environment” is zero. There is a precise way to say that in terms of surjective submersions. I am not sure if you’d like to see that.

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