## Applied Category Theory at NIST (Part 2)

18 April, 2018

Here are links to the slides and videos for most of the talks from this workshop:

Applied Category Theory: Bridging Theory & Practice, March 15–16, 2018, NIST, Gaithersburg, Maryland, USA. Organized by Spencer Breiner and Eswaran Subrahmanian.

They give a pretty good picture of what went on. Spencer Breiner put them up here; what follows is just a copy of what’s on his site.

Unfortunately, the end of Dusko Pavlovic’s talk, as well as Ryan Wisnesky’s and Steve Huntsman’s were lost due to a technical error. You can also find a Youtube playlist with all of the videos here.

Introduction to NIST:

Ram Sriram – NIST and Category Theory

Spencer Breiner – Introduction

Invited talks:

Bob Coecke – From quantum foundations to cognition via pictures

Dusko Pavlovic – Security Science in string diagrams (partial video)

John Baez – Compositional design and tasking of networks (part 1)

John Foley – Compositional design and tasking of networks (part 2)

David Spivak – A higher-order temporal logic for dynamical systems

Lightning Round Talks:

Ryan Wisnesky – Categorical databases (no video)

Steve Huntsman – Towards an operad of goals (no video)

Bill Regli – Disrupting interoperability (no slides)

Evan Patterson – Applied category theory in data science

Brendan Fong – data structures for network languages

Stephane Dugowson – A short introduction to a general theory of interactivity

Michael Robinson – Sheaf methods for inference

Cliff Joslyn – Seeking a categorical systems theory via the category of hypergraphs

Emilie Purvine – A category-theoretical investigation of the type hierarchy for heterogeneous sensor integration

Helle Hvid Hansen – Long-term values in Markov decision processes, corecursively

Alberto Speranzon – Localization and planning for autonomous systems via (co)homology computation

Josh Tan – Indicator frameworks (no slides)

Breakout round report

## Applied Category Theory 2018 Schedule

13 April, 2018

Here’s the schedule of the ACT2018 workshop:

Click to enlarge!

They put me on last, either because my talk will be so boring that it’s okay everyone will have left, or because my talk will be so exciting that nobody will want to leave. I haven’t dared ask the organizers which one.

On the other hand, they’ve put me on first for the “school” which occurs one week before the workshop. Here’s the schedule for the ACT 2018 Adjoint School:

## Applied Category Theory Course: Ordered Sets

7 April, 2018

My applied category theory course based on Fong and Spivak’s book Seven Sketches is going well. Over 250 people have registered for the course, which allows them to ask question and discuss things. But even if you don’t register you can read my “lectures”.

Here are all the lectures on Chapter 1, which is about adjoint functors between posets, and how they interact with meets and joins. We study the applications to logic – both classical logic based on subsets, and the nonstandard version of logic based on partitions. And we show how this math can be used to understand “generative effects”: situations where the whole is more than the sum of its parts!

I’m finding this course a great excuse to put my thoughts about category theory into a more organized form, and it’s displaced most of the time I used to spend on Google+ and Twitter. That’s what I wanted: the conversations in the course are more interesting!

## Applied Category Theory Course

26 March, 2018

It just became a lot easier to learn about applied category theory, thanks to this free book:

• Brendan Fong and David Spivak, Seven Sketches in Compositionality: An Invitation to Applied Category Theory.

I’ve started an informal online course based on this book on the Azimuth Forum. I’m getting pretty sick of the superficial quality of my interactions on social media. This could be a way to do something more interesting.

The idea is that you can read chapters of this book, discuss them, try the exercises in the book, ask and answer questions, and maybe team up to create software that implements some of the ideas. I’ll try to keep things moving forward. For example, I’ll explain some stuff and try to help answer questions that people are stuck on. I may also give some talks or run discussions on Google Hangouts or similar software—but only when I have time: I’m more of a text-based guy. I may get really busy some times, and leave the rest of you alone for a while. But I like writing about math for at least 15 minutes a day, and more when I have time. Furthermore, I’m obsessed with applied category theory and plan to stay that way for at least a few more years.

If this sounds interesting, let me know here—and please visit the Azimuth Forum and register! Use your full real name as your username, with no spaces. I will add spaces and that will become your username. Use a real working email address. If you don’t, the registration process may not work.

Over 70 people have registered so far, so this process will take a while.

The main advantage of the Forum over this blog is that you can initiate new threads and edit your comments. Like here you can write equations in LaTeX. Like here, that ability is severely limited: for example you can’t define macros, and you can’t use TikZ. (Maybe someone could fix that.) But equations are better typeset over there—and more importantly, the ability to edit comments makes it a lot easier to correct errors in your LaTeX.

Please let me know what you think.

What follows is the preface to Fong and Spivak’s book, just so you can get an idea of what it’s like.

### Preface

Category theory is becoming a central hub for all of pure mathematics. It is unmatched in its ability to organize and layer abstractions, to find commonalities between structures of all sorts, and to facilitate communication between different mathematical communities. But it has also been branching out into science, informatics, and industry. We believe that it has the potential to be a major cohesive force in the world, building rigorous bridges between disparate worlds, both theoretical and practical. The motto at MIT is mens et manus, Latin for mind and hand. We believe that category theory—and pure math in general—has stayed in the realm of mind for too long; it is ripe to be brought to hand.

#### Purpose and audience

The purpose of this book is to offer a self-contained tour of applied category theory. It is an invitation to discover advanced topics in category theory through concrete real-world examples. Rather than try to give a comprehensive treatment of these topics—which include adjoint functors, enriched categories, proarrow equipments, toposes, and much more–we merely provide a taste. We want to give readers some insight into how it feels to work with these structures as well as some ideas about how they might show up in practice.

The audience for this book is quite diverse: anyone who finds the above description intriguing. This could include a motivated high school student who hasn’t seen calculus yet but has loved reading a weird book on mathematical logic they found at the library. Or a machine learning researcher who wants to understand what vector spaces, design theory, and dynamical systems could possibly have in common. Or a pure mathematician who wants to imagine what sorts of applications their work might have. Or a recently-retired programmer who’s always had an eerie feeling that category theory is what they’ve been looking for to tie it all together, but who’s found the usual books on the subject impenetrable.

For example, we find it something of a travesty that in 2018 there seems to be no introductory material available on monoidal categories. Even beautiful modern introductions to category theory, e.g. by Riehl or Leinster, do not include anything on this rather central topic. The basic idea is certainly not too abstract; modern human intuition seems to include a pre-theoretical understanding of monoidal categories that is just waiting to be formalized. Is there anyone who wouldn’t correctly understand the basic idea being communicated in the following diagram?

Many applied category theory topics seem to take monoidal categories as their jumping off point. So one aim of this book is to provide a reference—even if unconventional—for this important topic.

We hope this book inspires both new visions and new questions. We intend it to be self-contained in the sense that it is approachable with minimal prerequisites, but not in the sense that the complete story is told here. On the contrary, we hope that readers use this as an invitation to further reading, to orient themselves in what is becoming a large literature, and to discover new applications for themselves.

This book is, unashamedly, our take on the subject. While the abstract structures we explore are important to any category theorist, the specific topics have simply been chosen to our personal taste. Our examples are ones that we find simple but powerful, concrete but representative, entertaining but in a way that feels important and expansive at the same time. We hope our readers will enjoy themselves and learn a lot in the process.

#### How to read this book

The basic idea of category theory—which threads through every chapter—is that if one pays careful attention to structures and coherence, the resulting systems will be extremely reliable and interoperable. For example, a category involves several structures: a collection of objects, a collection of morphisms relating objects, and a formula for combining any chain of morphisms into a morphism. But these structures need to cohere or work together in a simple commonsense way: a chain of chains is a chain, so combining a chain of chains should be the same as combining the chain. That’s it!

We will see structures and coherence come up in pretty much every definition we give: “here are some things and here are how they fit together.” We ask the reader to be on the lookout for structures and coherence as they read the book, and to realize that as we layer abstraction on abstraction, it is the coherence that makes everything function like a well-oiled machine.

Each chapter in this book is motivated by a real-world topic, such as electrical circuits, control theory, cascade failures, information integration, and hybrid systems. These motivations lead us into and through various sorts of category-theoretic concepts.

We generally have one motivating idea and one category-theoretic purpose per chapter, and this forms the title of the chapter, e.g. Chapter 4 is “Collaborative design: profunctors, categorification, and monoidal categories.” In many math books, the difficulty is roughly a monotonically-increasing function of the page number. In this book, this occurs in each chapter, but not so much in the book as a whole. The chapters start out fairly easy and progress in difficulty.

The upshot is that if you find the end of a chapter very difficult, hope is certainly not lost: you can start on the next one and make good progress. This format lends itself to giving you a first taste now, but also leaving open the opportunity for you to come back at a later date and get more deeply into it. But by all means, if you have the gumption to work through each chapter to its end, we very much encourage that!

We include many exercises throughout the text. Usually these exercises are fairly straightforward; the only thing they demand is that the reader’s mind changes state from passive to active, rereads the previous paragraphs with intent, and puts the pieces together. A reader becomes a student when they work the exercises; until then they are more of a tourist, riding on a bus and listening off and on to the tour guide. Hey, there’s nothing wrong with that, but we do encourage you to get off the bus and make contact with the natives as often as you can.

## Hypergraph Categories of Cospans

11 March, 2018

Two students in the Applied Category Theory 2018 school wrote a blog article about Brendan Fong’s theory of decorated cospans:

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

Jonathan Lorand is a math grad student at the University of Zurich working on symplectic and Poisson geometry with Alberto Cattaneo. Fabrizio Genovese is a grad student in computer science at the University of Oxford, working with Bob Coecke and Dan Marsden on categorical quantum mechanics, quantum field theory and the like.

Brendan was my student, so it’s nice to see newer students writing a clear summary of some of his thesis work, namely this paper:

• Brendan Fong, Decorated cospans, Theory and Applications of Categories 30 (2015), 1096–1120.

I wrote a summary of it myself, so I won’t repeat it here:

• John Baez, Decorated cospans, Azimuth, 1 May 2015.

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!

## An Upper Bound on Reidemeister Moves

9 March, 2018

Graham’s number is famous for being the largest number to have ever shown up in a proof. The true story is more complicated, as I discovered by asking Graham. But here’s a much smaller but still respectable number that showed up in knot theory:

$2 \uparrow \uparrow (10 \uparrow 1,000,000)$

It’s 2 to the 2 to the 2 to the 2… where we go on for 101,000,000 times. It appears in a 2011 paper by Coward and Lackenby. It shows up in their upper bound on how many steps it can take to wiggle around one picture of a link until you get another picture of the same link.

This upper bound is ridiculously large. But because this upper bound is computable, it follows that we can decide, in a finite amount of time, whether two pictures show the same link or not. We know when to give up. This had previously been unknown!

Here’s the paper:

• Alexander Coward and Marc Lackenby, An upper bound on Reidemeister moves, American Journal of Mathematics 136 (2014), 1023–1066.

Let me spell out the details a tiny bit more.

A link is a collection of circles embedded in 3-dimensional Euclidean space. We count two links as ‘the same’, or ‘ambient isotopic’, if we can carry one to another by a smooth motion where no circle ever crosses another. (This can be made more precise.) We can draw links in the plane:

and we can get between any two diagrams of the same link by distorting the plane and also doing a sequence of ‘Reidemeister moves’. There are 3 kinds of Reidemeister moves, shown above and also here:

Coward and Lackenby found an upper bound on how many Reidemeister moves it takes to get between two diagrams of the same link. Let n be the total number of crossings in both diagrams. Then we need at most 2 to the 2 to the 2 to the 2 to the 2… Reidemeister moves, where the number of 2’s in this tower is cn, where c = 101,000,000.

It’s fun to look at the paper and see how they get such a terrible upper bound. I’m sure they could have done much better with a bit of work, but that wasn’t the point. All they wanted was a computable upper bound.

Subsequently, Lackenby proved a polynomial upper bound on how many Reidemeister moves it takes to reduce a diagram of the unknot to a circle, like this:

If the original diagram has n crossings, he proved it takes at most (236n)11 Reidemeister moves. Because this is a polynomial, it follows that recognizing whether a knot diagram is a diagram of the unknot is in NP. As far as I know, it remains an open question whether this problem is in P.

• Marc Lackenby, A polynomial upper bound on Reidemeister moves, Annals of Mathematics 182 (2015), 491–564.

As a challenge, can you tell if this diagram depicts the unknot?

If you get stuck, read Lackenby’s paper!

To learn more about any of the pictures here, click on them. For example, this unknotting process:

showed up in this paper:

• Louis Kauffman and Sofia Lambropoulou, Hard unknots and collapsing tangles, in Introductory Lectures On Knot Theory: Selected Lectures Presented at the Advanced School and Conference on Knot Theory and Its Applications to Physics and Biology, 2012, pp. 187–247.

I bumped into Coward and Lackenby’s theorem here:

• Evelyn Lamb, Laura Taalman’s Favorite Theorem, Scientific American, 8 March 2018.

It says:

Taalman’s favorite theorem gives a way to know for sure whether a knot is equivalent to the unknot, a simple circle. It shows that if the knot is secretly the unknot, there is an upper bound, based on the number of crossings in a diagram of the knot, to the number of Reidemeister moves you will have to do to reduce the knot to a circle. If you try every possible sequence of moves that is at least that long and your diagram never becomes a circle, you know for sure that the knot is really a knot and not an unknot. (Say that ten times fast.)

Taalman loves this theorem not only because it was the first explicit upper bound for the question but also because of how extravagant the upper bound is. In the original paper proving this theorem, Joel Haas and Jeffrey Lagarias got a bound of

$2^{n 10^{11}}$

where n is the number of crossings in the diagram. That’s 2 to the n hundred billionth power. Yikes! When you try to put that number into the online calculator Wolfram Alpha, even for a very small number of crossings, the calculator plays dead.

Dr. Taalman also told us about another paper, this one by Alexander Coward and Marc Lackenby, that bounds the number of Reidemeister moves needed to show whether any two given knot diagrams are equivalent. That bound involves towers of powers that also get comically large incredibly quickly. They’re too big for me to describe how big they are.

So, I wanted to find out how big they are!

If you want a more leisurely introdution to the Haas–Lagarias result, try the podcast available at Eveyln Lamb’s article, or this website:

• Kevin Knudson, My favorite theorem: Laura Talman, Episode 14.

## Coarse-Graining Open Markov Processes

4 March, 2018

Kenny Courser and I have been working hard on this paper for months:

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

It may be almost done. So, it would be great if people here could take a look and comment on it! It’s a cool mix of probability theory and double categories. I’ve posted a similar but non-isomorphic article on the n-Category Café, where people know a lot about double categories. But maybe some of you here know more about Markov processes!

‘Coarse-graining’ is a standard method of extracting a simple Markov process from a more complicated one by identifying states. 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.

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.

But before you dive into the paper, let me explain all this stuff a bit more….

Very roughly speaking, a ‘Markov process’ is a stochastic model describing a sequence of transitions between states in which the probability of a transition depends only on the current state. But the only Markov processes talk about are continuous-time Markov processes with a finite set of states. These can be drawn as labeled graphs:

where the number labeling each edge describes the probability per time of making a transition from one state to another.

An ‘open’ Markov process is a generalization in which probability can also flow in or out of certain states designated as ‘inputs’ and outputs’:

Open Markov processes can be seen as morphisms in a category, since we can compose two open Markov processes by identifying the outputs of the first with the inputs of the second. Composition lets us build a Markov process from smaller open parts—or conversely, analyze the behavior of a Markov process in terms of its parts.

In this paper, Kenny extend the study of open Markov processes to include coarse-graining. ‘Coarse-graining’ is a widely studied method of simplifying a Markov process by mapping its set of states $X$ onto some smaller set $X'$ in a manner that respects the dynamics. Here we introduce coarse-graining for open Markov processes. And we show how to extend this notion to the case of maps $p: X \to X'$ that are not surjective, obtaining a general concept of morphism between open Markov processes.

Since open Markov processes are already morphisms in a category, it is natural to treat morphisms between them as morphisms between morphisms, or ‘2-morphisms’. We can do this using double categories!

Double categories were first introduced by Ehresmann around 1963. Since then, they’ve used in topology and other branches of pure math—but more recently they’ve been used to study open dynamical systems and open discrete-time Markov chains. So, it should not be surprising that they are also useful for open Markov processes..

A 2-morphism in a double category looks like this:

While a mere category has only objects and morphisms, here we have a few more types of things. We call $A, B, C$ and $D$ ‘objects’, $f$ and $g$ ‘vertical 1-morphisms’, $M$ and $N$ ‘horizontal 1-cells’, and $\alpha$ a ‘2-morphism’. We can compose vertical 1-morphisms to get new vertical 1-morphisms and compose horizontal 1-cells to get new horizontal 1-cells. We can compose the 2-morphisms in two ways: horizontally by setting squares side by side, and vertically by setting one on top of the other. The ‘interchange law’ relates vertical and horizontal composition of 2-morphisms.

In a ‘strict’ double category all these forms of composition are associative. In a ‘pseudo’ double category, horizontal 1-cells compose in a weakly associative manner: that is, the associative law holds only up to an invertible 2-morphism, the ‘associator’, which obeys a coherence law. All this is just a sketch; for details on strict and pseudo double categories try the paper by Grandis and Paré.

Kenny and I construct a double category $\mathbb{M}\mathbf{ark}$ with:

1. finite sets as objects,
2. maps between finite sets as vertical 1-morphisms,
3. open Markov processes as horizontal 1-cells,
4. morphisms between open Markov processes as 2-morphisms.

I won’t give the definition of item 4 here; you gotta read our paper for that! Composition of open Markov processes is only weakly associative, so $\mathbb{M}\mathbf{ark}$ is a pseudo double category.

This is how our paper goes. In Section 2 we define open Markov processes and steady state solutions of the open master equation. In Section 3 we introduce coarse-graining first for Markov processes and then open Markov processes. In Section 4 we construct the double category $\mathbb{M}\mathbf{ark}$ described above. We prove this is a symmetric monoidal double category in the sense defined by Mike Shulman. This captures the fact that we can not only compose open Markov processes but also ‘tensor’ them by setting them side by side.

For example, if we compose this open Markov process:

with the one I showed you before:

we get this open Markov process:

But if we tensor them, we get this:

As compared with an ordinary Markov process, the key new feature of an open Markov process is that probability can flow in or out. To describe this we need a generalization of the usual master equation for Markov processes, called the ‘open master equation’.

This is something that Brendan, Blake and I came up with earlier. In this equation, the probabilities at input and output states are arbitrary specified functions of time, while the probabilities at other states obey the usual master equation. As a result, the probabilities are not necessarily normalized. We interpret this by saying probability can flow either in or out at both the input and the output states.

If we fix constant probabilities at the inputs and outputs, there typically exist solutions of the open master equation with these boundary conditions that are constant as a function of time. These are called ‘steady states’. Often these are nonequilibrium steady states, meaning that there is a nonzero net flow of probabilities at the inputs and outputs. For example, probability can flow through an open Markov process at a constant rate in a nonequilibrium steady state. It’s like a bathtub where water is flowing in from the faucet, and flowing out of the drain, but the level of the water isn’t changing.

Brendan, Blake and I studied the relation between probabilities and flows at the inputs and outputs that holds in steady state. We called the process of extracting this relation from an open Markov process ‘black-boxing’, since it gives a way to forget the internal workings of an open system and remember only its externally observable behavior. We showed that black-boxing is compatible with composition and tensoring. In other words, we showed that black-boxing is a symmetric monoidal functor.

In Section 5 of our new paper, Kenny and I show that black-boxing is compatible with morphisms between open Markov processes. To make this idea precise, we prove that black-boxing gives a map from the double category $\mathbb{M}\mathbf{ark}$ to another double category, called $\mathbb{L}\mathbf{inRel}$, which has:

1. finite-dimensional real vector spaces $U,V,W,\dots$ as objects,
2. linear maps $f : V \to W$ as vertical 1-morphisms from $V$ to $W$,
3. linear relations $R \subseteq V \oplus W$ as horizontal 1-cells from $V$ to $W$,
4. squares

obeying $(f \oplus g)R \subseteq S$ as 2-morphisms.

Here a ‘linear relation’ from a vector space $V$ to a vector space $W$ is a linear subspace $R \subseteq V \oplus W$. Linear relations can be composed in the usual way we compose relations. The double category $\mathbb{L}\mathbf{inRel}$ becomes symmetric monoidal using direct sum as the tensor product, but unlike $\mathbb{M}\mathbf{ark}$ it is a strict double category: that is, composition of linear relations is associative.

Our main result, Theorem 5.5, says that black-boxing gives a symmetric monoidal double functor

$\blacksquare : \mathbb{M}\mathbf{ark} \to \mathbb{L}\mathbf{inRel}$

As you’ll see if you check out our paper, there’s a lot of nontrivial content hidden in this short statement! The proof requires a lot of linear algebra and also a reasonable amount of category theory. For example, we needed this fact: if you’ve got a commutative cube in the category of finite sets:

and the top and bottom faces are pushouts, and the two left-most faces are pullbacks, and the two left-most arrows on the bottom face are monic, then the two right-most faces are pullbacks. I think it’s cool that this is relevant to Markov processes!

Finally, in Section 6 we state a conjecture. First we use a technique invented by Mike Shulman to construct symmetric monoidal bicategories $\mathbf{Mark}$ and $\mathbf{LinRel}$ from the symmetric monoidal double categories $\mathbb{M}\mathbf{ark}$ and $\mathbb{L}\mathbf{inRel}$. We conjecture that our black-boxing double functor determines a functor between these symmetric monoidal bicategories. This has got to be true. However, double categories seem to be a simpler framework for coarse-graining open Markov processes.

Finally, let me talk a bit about some related work. As I already mentioned, Brendan, Blake and I constructed a symmetric monoidal category where the morphisms are open Markov processes. However, we formalized such Markov processes in a slightly different way than Kenny and I do now. We defined a Markov process to be one of the pictures I’ve been showing you: a directed multigraph where each edge is assigned a positive number called its ‘rate constant’. In other words, we defined it to be a diagram

where $X$ is a finite set of vertices or ‘states’, $E$ is a finite set of edges or ‘transitions’ between states, the functions $s,t : E \to X$ give the source and target of each edge, and $r : E \to (0,\infty)$ gives the rate constant for each transition. We explained how from this data one can extract a matrix of real numbers $(H_{i j})_{i,j \in X}$ called the ‘Hamiltonian’ of the Markov process, with two properties that are familiar in this game:

$H_{i j} \geq 0$ if $i \neq j$,

$\sum_{i \in X} H_{i j} = 0$ for all $j \in X$.

A matrix with these properties is called ‘infinitesimal stochastic’, since these conditions are equivalent to $\exp(t H)$ being stochastic for all $t \ge 0$.

In our new paper, Kenny and I skip the directed multigraphs and work directly with the Hamiltonians! In other words, we define a Markov process to be a finite set $X$ together with an infinitesimal stochastic matrix $(H_{ij})_{i,j \in X}$. This allows us to work more directly with the Hamiltonian and the all-important ‘master equation’

$\displaystyle{ \frac{d p(t)}{d t} = H p(t) }$

which describes the evolution of a time-dependent probability distribution

$p(t) : X \to \mathbb{R}$

Clerc, Humphrey and Panangaden have constructed a bicategory with finite sets as objects, ‘open discrete labeled Markov processes’ as morphisms, and ‘simulations’ as 2-morphisms. The use the word ‘open’ in a pretty similar way to me. But their open discrete labeled Markov processes are also equipped with a set of ‘actions’ which represent interactions between the Markov process and the environment, such as an outside entity acting on a stochastic system. A ‘simulation’ is then a function between the state spaces that map the inputs, outputs and set of actions of one open discrete labeled Markov process to the inputs, outputs and set of actions of another.

Another compositional framework for Markov processes was discussed by de Francesco Albasini, Sabadini and Walters. They constructed an algebra of ‘Markov automata’. A Markov automaton is a family of matrices with non-negative real coefficients that is indexed by elements of a binary product of sets, where one set represents a set of ‘signals on the left interface’ of the Markov automata and the other set analogously for the right interface.

So, double categories are gradually invading the theory of Markov processes… as part of the bigger trend toward applied category theory. They’re natural things; scientists should use them.