### Applied Category Theory Papers

#### Posted by John Baez

In preparation for the Applied Category Theory special session at U.C. Riverside this weekend, my crew dropped three papers on the arXiv.

My student Adam Yassine has been working on Hamiltonian and Lagrangian mechanics from an ‘open systems’ point of view:

Adam Yassine, Open systems in classical mechanics.

**Abstract.**Using the framework of category theory, we formalize the heuristic principles that physicists employ in constructing the Hamiltonians for open classical systems as sums of Hamiltonians of subsystems. First we construct a category where the objects are symplectic manifolds and the morphisms are spans whose legs are surjective Poisson maps. Using a slight variant of Fong’s theory of decorated cospans, we then decorate the apices of our spans with Hamiltonians. This gives a category where morphisms are open classical systems, and composition allows us to build these systems from smaller pieces.

He also gets a functor from a category of Lagrangian open systems to this category of Hamiltonian systems.

Kenny Courser and I have been continuing my work with Blake Pollard and Brendan Fong on open Markov processes, bringing 2-morphisms into the game. It seems easiest to use a double category:

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

Abstract.Coarse-graining is a standard method of extracting a simple Markov process from a more complicated one by identifying states. Here we extend coarse-graining to open Markov processes. An ‘open’ Markov process is one where probability can flow in or out of certain states called ‘inputs’ and ‘outputs’. One can build up an ordinary Markov process from smaller open pieces in two basic ways: composition, where we identify the outputs of one open Markov process with the inputs of another, and tensoring, where we set two open Markov processes side by side. In previous work, Fong, Pollard and the first author showed that these constructions make open Markov processes into the morphisms of a symmetric monoidal category. Here we go further by constructing a symmetric monoidal double category where the 2-morphisms are ways of coarse-graining open Markov processes. We also extend the already known ‘black-boxing’ functor from the category of open Markov processes to our double category. Black-boxing sends any open Markov process to the linear relation between input and output data that holds in steady states, including nonequilibrium steady states where there is a nonzero flow of probability through the process. To extend black-boxing to a functor between double categories, we need to prove that black-boxing is compatible with coarse-graining.

Finally, the Complex Adaptive Systems Composition and Design Environment project with John Foley of Metron Scientific Solutions and my students Joseph Moeller and Blake Pollard has finally given birth to a paper! I hope this is just the first; it starts laying down the theoretical groundwork for designing networked systems. John is here now and we’re coming up with a bunch of new ideas:

- John Baez, John Foley, Joseph Moeller and Blake Pollard, Network models.

Abstract.Networks can be combined in many ways, such as overlaying one on top of another or setting two side by side. We introducenetwork modelsto encode these ways of combining networks. Different network models describe different kinds of networks. We show that each network model gives rise to an operad, whose operations are ways of assembling a network of the given kind from smaller parts. Such operads, and their algebras, can serve as tools for designing networks. Technically, a network model is a lax symmetric monoidal functor from the free symmetric monoidal category on some set to Cat, and the construction of the corresponding operad proceeds via a symmetric monoidal version of the Grothendieck construction.

I blogged about this last one here:

- Complex adaptive systems (part 6),
*Azimuth*.