• Brandon Coya, *Circuits, Bond Graphs, and Signal-Flow Diagrams: A Categorical Perspective*, Ph.D. thesis, U. C. Riverside, 2018.

It’s about networks in engineering. He uses category theory to study the diagrams engineers like to draw, and functors to understand how these diagrams are interpreted.

His thesis raises some really interesting pure mathematical questions about the category of corelations and a ‘weak bimonoid’ that can be found in this category. Weak bimonoids were invented by Pastro and Street in their study of ‘quantum categories’, a generalization of quantum groups. So, it’s fascinating to see a weak bimonoid that plays an important role in electrical engineering!

However, in what follows I’ll stick to less fancy stuff: I’ll just explain the basic idea of Brandon’s thesis, say a bit about circuits and ‘bond graphs’, and outline his main results.

]]>• John Baez, Brandon Coya and Franciscus Rebro, Props in network theory.

[…]

In Section 5 we discuss presentations of props. Proposition 19, proved in Appendix A.2 using a result of Todd Trimble, shows that the category of props is monadic over the category of **signatures**, This lets us work with props using generators and relations. We conclude by recalling a presentation of due to Lack and a presentation of due to Coya and Fong:

• Steve Lack, Composing PROPs, *Theory and Applications of Categories* **13** (2004), 147–163.

• Brandon Coya and Brendan Fong, Corelations are the prop for extraspecial commutative Frobenius monoids, *Theory and Applications of Categories* **32** (2017), 380–395. (Blog article here.)

• 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.

The paper’s title translates as ‘Algebraic-categorical aspects of the theory of devices’, and its main concept is that of a ‘universe of devices’: *a collection of devices of some kind that can be hooked up using wires to form more devices of this kind*. Nowadays we might study this concept using operads—but operads didn’t exist in 1970, and Darbo did quite fine without them.

The key is the category which has finite sets as objects and ‘corelations’ as morphisms. I explained corelations here:

• Corelations in network theory, 2 February 2016.

]]>[3] V.-E. Cazanescu, Gh. Stefanescu: Classes of finite relations as initial abstract data types I. Discrete Mathematics 90: 233–265 (1991)

I am very excited by the novel applications and connections that you are pushing, and looking forward to more results in the future.

]]>I became a big fan of the term because it taught me something: 1) read carefully before thinking someone’s wrong and 2) beware of “co”s in category theory!

]]>[1] R. Bruni, F. Gadducci: Some algebraic laws for spans. RelMiS 2001, ENTCS 44(3), 175-193

[2] R. Bruni, F. Gadducci, U. Montanari: Normal forms for algebras of connections. TCS 286(2): 247-292 (2002)

[1] discusses the interplay between the two monoidal operators and the span/cospan dichotomy over Set, while [2] basically discusses the same situation, yet wrt. Graph.

More to the point, it seems to me that in [1] we call equivalence relations what you call corelations, and we have the same extra and special laws. Check e.g. Table 1, axioms 1 and 5.

I will try and better understand your results, in order to trace a precise comparison. Best, Fabio

]]>The most important partitions that occur in my work are partitions of individual organisms into sets called ‘species’. I often think of these partitions in terms of trees. In the diagram of the lattice of partitions of a set, each path from any node X up to the top represents a binary tree whose tips are the subsets in the partition at X.

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