## August 15, 2020

### Open Systems: A Double Categorical Perspective (Part 1)

#### Posted by John Baez

Kenny Courser’s thesis has hit the arXiv:

He’s been the driving force behind a lot of work on open systems and networks at U. C. Riverside. By the way, he’s looking for a job, so if you think you know a position that’s good for someone who can teach all kinds of math and also strong on applied category theory, give him or me a shout.

But let me describe his thesis.

His thesis is big! It lays out a general approach to open systems—systems that can interact with their environment. In this approach, you can attach open systems in series to form larger open systems, so they act like morphisms in a category:

But you can also study 2-morphisms between open systems. These describe ways to include a little system in a bigger one, or simplify a big complicated system down to smaller one:

To handle all this, Courser uses double categories, which he explains.

His formalism also lets you set two open systems side by side ‘in parallel’ and get a new open system:

To handle this, he uses symmetric monoidal double categories. He explains what these are, and how to get them. And he illustrates his setup with examples:

At a more technical level, Courser explains the problems with Brendan Fong’s and my work on decorated cospans and shows how to fix in not just one but two ways: using structured cospans, and using a new improved version of decorated cospans. He also shows that these two approaches are equivalent under fairly general conditions.

His thesis unifies a number of papers:

The last introduces the new improved decorated cospans and proves their equivalence to structured cospans under some conditions.

I think next time I’ll explain the problems with the original decorated cospan formalism. Another nice thing about Kenny’s thesis is that it goes over a bunch of papers that were afflicted by these problems, and it shows how to fix them.

• Part 1: an overview of Courser’s thesis and related papers.

• Part 2: problems with the original decorated cospans.

• Part 3: the new improved decorated cospans.

Posted at August 15, 2020 1:51 AM UTC

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### Re: Open Systems: A Double Categorical Perspective (Part 1)

So if structured cospans are equivalent to improved decorated cospans, which one should we be using from now on? Are the “fairly general conditions” for this equivalence general enough that it makes no difference? Or are there important examples falling outside these general conditions?

Posted by: Mike Shulman on August 15, 2020 4:00 AM | Permalink | Reply to this

### Re: Open Systems: A Double Categorical Perspective (Part 1)

I find structured cospans simpler, but I think there are some examples that work with decorated cospans but not structured cospans. Also, decorated cospans feed into Brendan Fong’s theory of decorated corelations, which are good for other things: for example, in A recipe for black box functors, he and Maru Sarazola used them to give a more high-level construction of the black box functor for reaction networks (in chemistry), which Blake Pollard and I had constructed ‘by hand’.

I guess the final verdict is not clear yet. It may become a bit clearer in later posts in this series, when I describe what’s going on.

Posted by: John Baez on August 16, 2020 2:11 AM | Permalink | Reply to this

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