This is a wonderful development! Micah Halter and Evan Patterson have taken my work on structured cospans with Kenny Courser and open Petri nets with Jade Master, together with Joachim Kock’s whole-grain Petri nets, and turned them into a practical software tool!
Then they used that to build a tool for ‘compositional’ modeling of the spread of infectious disease. By ‘compositional’, I mean that they make it easy to build more complex models by sticking together smaller, simpler models.
Even better, they’ve illustrated the use of this tool by rebuilding part of the model that the UK has been using to make policy decisions about COVID19.
All this software was written in the programming language Julia.
I had expected structured cospans to be useful in programming and modeling, but I didn’t expect it to happen so fast!
For details, read this great article:
• Micah Halter and Evan Patterson, Compositional epidemiological modeling using structured cospans, 17 October 2020.
Abstract. The field of applied category theory (ACT) aims to put the compositionality inherent to scientific and engineering processes on a firm mathematical footing. In this post, we show how the mathematics of ACT can be operationalized to build complex epidemiological models in a compositional way. In the first two sections, we review the idea of structured cospans, a formalism for turning closed systems into open ones, and we illustrate its use in Catlab through the simple example of open graphs. Finally, we put this machinery to work in the setting of Petri nets and epidemiological models. We construct a portion of the COEXIST model for the COVID-19 pandemic and we simulate the resulting ODEs.
You can see related articles by James Fairbanks, Owen Lynch and Evan Patterson here:
Also try these videos:
• James Fairbanks, AlgebraicJulia: Applied category theory in Julia, 29 July 2020.
• Evan Patterson, Realizing applied category theory in Julia, 16 January 2020.
I’m biased, but I think this is really cool cutting-edge stuff. If you want to do work along these lines let me know here and I’ll get Patterson to take a look.
Here’s part of a network created using their software:
Azimuth 
Interesting
Professor Baez,
Being no expert, I don’t have a solid understanding of the Category theory. But I’ve been having the impression that the Category theory as I understand it is about a mathematical formulation or framework of the networks and can in principle be applied to anything involving a network.
My interested subject of research and study has lately been related to (Biological or Artificial) Neural Network, or distributed learning in the view of Connectionism (I’d say).
My question is, are you familiar with any research regarding Neural Networks where the idea or methods of Category Theory are used?
A paper I have found relevant in this line of research is:
The Representation Theory of Neural Networks
(https://arxiv.org/abs/2007.12213)
Marco Antonio Armenta, Pierre-Marc Jodoin
which used Quiver to formulate the network.
Anyhow, I admire your works and find them inspiring, although I have to learn more to fully appreciate them. Thank you for your contributions to community.
For actual neurobiology, try this:
• Kathryn Hess, Towards a categorical approach to neuroscience, ACT2018 talk.
I believe that Google’s Tensorflow is connected to monoidal categories:
• John Baez, Programming with data flow graphs, 5 June 2016.
but I haven’t had time to sort this out in my mind. (This is completely separate from whether there’s a graphical interface to Tensorflow; apparently there was not when I posted that article.)
Another thing worth looking at is this:
• Brendan Fong, David I. Spivak and Rémy Tuyéras, Backprop as functor: a compositional perspective on supervised learning.
There should be a lot more to say about applications of category theory to both neurobiology and artificial neural networks, but I haven’t seen enough work on this.
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