27 March 2017:

1) On Friday, Blake and John Foley and I finished off two essays for our [Complex Adaptive Composition and Design Environment](https://johncarlosbaez.wordpress.com/2016/10/02/complex-adaptive-system-design-part-1/) project.

The first is called "Compositional Tasking: an operad-based approach to adaptive behaviors for distributed systems-of-systems and planning under uncertainty." This is a general overview of our plan to use operads to design and "task" (boss around) networks while moving up and down levels of abstraction as desired. I find this really exciting! With luck the folks at Metron will create some software to illustrate these ideas

The second, called "Compositional Tasking", has more of the mathematical details. Joseph played a key role in this by checking that the Grothendieck construction can create the operads we need. I hope we continue to improve this essay and publish it in one or more papers, which I hope to write this summer.

2) Prakash Panangaden, an expert in categories and computer science at McGill University in Montreal, has come out with a paper which takes the category-theoretic characterization of relative entropy that Tobias Fritz and I found and extends it from finite sets to more general measurable spaces (sets with a sigma-algebra of subsets):

> **Abstract**. The inspiration for the present work comes from two recent developments. The first is the beginning of a categorical understanding of Bayesian inversion and learning, the second is a categorical reconstruction of relative entropy. The present paper provides a categorical treatment of entropy in the spirit of Baez and Fritz in the setting of Polish spaces, thus setting the stage to explore the role of entropy in learning.

"Polish spaces" are a nice class of measurable spaces, loved by the analysts who lived in Poland before the Nazis invaded.

3) Prakash has also written a paper about a bicategory where the morphisms are open Markov processes. This is based on a paper that Brendan, Blake and I wrote:

> **Abstract**. We construct bicategories of Markov processes where the objects are input and output sets, the morphisms (one-cells) are Markov processes and the two-cells are simulations. This builds on the work of Baez, Fong and Pollard, who showed that a certain kinds of finite-space continuous-time Markov chain (CTMC) satisfying a detailed-balance condition can be viewed as morphisms in a category. This view allows a compositional description of their CTMCs. Our contribution is to develop a notion of simulation between processes and construct a bicategory where the two-cells are simulation morphisms. Our version is for processes that are essentially probabilistic transition systems with discrete time steps and which do not satisfy a detailed balance condition. We have also extended the theory to continuous space processes.

In short, Prakash is moving in and starting to offer competition in our field of work. A lot of people in computer science respect his work, so they will start to read our stuff and explore similar ideas. This means we can't laze around when it comes to publishing ideas we have... but it's basically good, because it means more people will be inclined to hire my grad students!