• Jonathan Lorand and Fabrizio Genovese, Hypergraph categories of cospans, The *n*-Category Café, 28 February 2018.

What’s especially interesting to me is that both Jonathan and Fabrizio know some mathematical physics, and they’re part of a group who will be working with me on some problems as part of the Applied Category Theory 2018 school! Brendan and Blake Pollard and I used symplectic geometry and decorated cospans to study the black-boxing of electrical circuits and Markov processes… maybe we should try to go further with that project!

]]>• John Baez and Kenny Courser, Coarse-graining open Markov processes.

A while back, Brendan Fong, Blake Pollard and I showed that these constructions make open Markov processes into the morphisms of a symmetric monoidal category:

• A compositional framework for Markov processes, *Azimuth*, January 12, 2016.

Here Kenny and I go further by constructing a symmetric monoidal *double* category where the 2-morphisms include ways of coarse-graining open Markov processes. We also extend the previously defined ‘black-boxing’ functor from the category of open Markov processes to this double category.

The ultimate goal is to understand the nonequilibrium thermodynamics of open systems—systems where energy and maybe matter flows in and out. If we could understand this well enough, we could understand in detail how *life* works. That’s a difficult job! But one has to start somewhere, and this is one place to start.

We have a few papers on this subject:

• Blake Pollard, A Second Law for open Markov processes. (Blog article here.)

• John Baez, Brendan Fong and Blake Pollard, A compositional framework for Markov processes. (Blog article here.)

• Blake Pollard, Open Markov processes: A compositional perspective on non-equilibrium steady states in biology. (Blog article here.)

]]>However, he shows that another minimum principle holds exactly, even for steady states that are far from equilibrium. He calls this the ‘principle of minimum dissipation’.

We actually discussed the principle of minimum dissipation in an earlier paper:

• John Baez, Brendan Fong and Blake Pollard, A compositional framework for Markov processes. (Blog article here.)

But one advantage of Blake’s new paper is that it presents the results with a minimum of category theory. Of course I love category theory, and I think it’s the right way to formalize open systems, but it can be intimidating.

]]>I’m glad you like this stuff. Indeed, I’m taking concepts from topological quantum field theory and adapting them. But instead of chopping spacetime into pieces, I’m chopping ‘space’—or some abstraction thereof, like an electrical circuit—and chopping it into pieces. And instead of doing a quantum theory, and getting linear operators between Hilbert spaces of quantum states, I’m doing a classical theory, and getting linear Lagrangian relations between symplectic vector spaces describing classical boundary conditions. This seems like a good way to go…

So far we’re mainly doing statics, or *almost* statics, like ‘nonequilibrium steady states’, or problems where you use the Laplace transform to convert an ODE into a mere algebra problem. But with some 2-categories we can do true dynamics.