• 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

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**

]]>

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:

]]>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!

• Lecture 1 – Introduction

• Lecture 2 – What is Applied Category Theory?

• Lecture 3 – Chapter 1: Preorders

• Lecture 4 – Chapter 1: Galois Connections

• Lecture 5 – Chapter 1: Galois Connections

• Lecture 6 – Chapter 1: Computing Adjoints

• Lecture 7 – Chapter 1: Logic

• Lecture 8 – Chapter 1: The Logic of Subsets

• Lecture 9 – Chapter 1: Adjoints and the Logic of Subsets

• Lecture 10 – Chapter 1: The Logic of Partitions

• Lecture 11 – Chapter 1: The Poset of Partitions

• Lecture 12 – Chapter 1: Generative Effects

• Lecture 13 – Chapter 1: Pulling Back Partitions

• Lecture 14 – Chapter 1: Adjoints, Joins and Meets

• Lecture 15 – Chapter 1: Preserving Joins and Meets

• Lecture 16 – Chapter 1: The Adjoint Functor Theorem for Posets

• Lecture 17 – Chapter 1: The Grand Synthesis

If you want to discuss these things, please visit the Azimuth Forum and register! **Use your full real name as your username, with no spaces, and use a real working email address.** If you don’t, I won’t be able to register you. Your email address will be kept confidential.

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!

]]>• 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.

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.

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.

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.

]]>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!

]]>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:

It’s 2 to the 2 to the 2 to the 2… where we go on for 10^{1,000,000} 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 *c*^{n}, where *c* = 10^{1,000,000}.

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 (236*n*)^{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

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.

]]>• 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 onto some smaller set 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 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 and ‘objects’, and ‘vertical 1-morphisms’, and ‘horizontal 1-cells’, and 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 with:

- finite sets as objects,
- maps between finite sets as vertical 1-morphisms,
- open Markov processes as horizontal 1-cells,
- 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 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 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 to another double category, called , which has:

- finite-dimensional real vector spaces as objects,
- linear maps as vertical 1-morphisms from to ,
- linear relations as horizontal 1-cells from to ,
- squares

obeying as 2-morphisms.

Here a ‘linear relation’ from a vector space to a vector space is a linear subspace . Linear relations can be composed in the usual way we compose relations. The double category becomes symmetric monoidal using direct sum as the tensor product, but unlike 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

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 and from the symmetric monoidal double categories and . 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 is a finite set of vertices or ‘states’, is a finite set of edges or ‘transitions’ between states, the functions give the source and target of each edge, and gives the rate constant for each transition. We explained how from this data one can extract a matrix of real numbers called the ‘Hamiltonian’ of the Markov process, with two properties that are familiar in this game:

• if ,

• for all .

A matrix with these properties is called ‘infinitesimal stochastic’, since these conditions are equivalent to being stochastic for all .

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 together with an infinitesimal stochastic matrix . This allows us to work more directly with the Hamiltonian and the all-important ‘master equation’

which describes the evolution of a time-dependent probability distribution

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.

]]>I just read something cool:

• Joel David Hamkins, Nonstandard models of arithmetic arise in the complex numbers, 3 March 2018.

Let me try to explain it in a simplified way. I think all cool math should be known more widely than it is. Getting this to happen requires a lot of explanations at different levels.

Here goes:

The Peano axioms are a nice set of axioms describing the natural numbers. But thanks to Gödel’s incompleteness theorem, these axioms can’t completely nail down the structure of the natural numbers. So, there are lots of different ‘models’ of Peano arithmetic.

These are often called ‘nonstandard’ models. If you take a model of Peano arithmetic—say, your favorite ‘standard’ model —you can get other models by throwing in extra natural numbers, larger than all the standard ones. These nonstandard models can be countable or uncountable. For more, try this:

• Nonstandard models of arithmetic, Wikipedia.

Starting with any of these models you can define integers in the usual way (as differences of natural numbers), and then rational numbers (as ratios of integers). So, there are lots of nonstandard versions of the rational numbers. Any one of these will be a field: you can add, subtract, multiply and divide your nonstandard rationals, in ways that obey all the usual basic rules.

Now for the cool part: *if your nonstandard model of the natural numbers is small enough, your field of nonstandard rational numbers can be found somewhere in the standard field of complex numbers!*

In other words, your nonstandard rationals are a subfield of the usual complex numbers: a subset that’s closed under addition, subtraction, multiplication and division by things that aren’t zero.

This is counterintuitive at first, because we tend to think of nonstandard models of Peano arithmetic as spooky and elusive things, while we tend to think of the complex numbers as well-understood.

However, the field of complex numbers is actually very large, and it has room for many spooky and elusive things inside it. This is well-known to experts, and we’re just seeing more evidence of that.

I said that all this works if your nonstandard model of the natural numbers is small enough. But what is “small enough”? Just the obvious thing: your nonstandard model needs to have a cardinality smaller than that of the complex numbers. So if it’s countable, that’s definitely small enough.

This fact was recently noticed by Alfred Dolich at a pub after a logic seminar at the City University of New York. The proof is very easy if you know this result: any field of characteristic zero whose cardinality is less than or equal to that of the continuum is isomorphic to some subfield of the complex numbers. So, unsurprisingly, it turned out to have been repeatedly discovered before.

And the result I just mentioned follows from this: any two algebraically closed fields of characteristic zero that have the same uncountable cardinality must be isomorphic. So, say someone hands you a field F of characteristic zero whose cardinality is smaller than that of the continuum. You can take its algebraic closure by throwing in roots to all polynomials, and its cardinality won’t get bigger. Then you can throw in even more elements, if necessary, to get a field whose cardinality is that of the continuum. The resulting field must be isomorphic to the complex numbers. So, F is isomorphic to a subfield of the complex numbers.

To round this off, I should say a bit about *why* nonstandard models of Peano arithmetic are considered spooky and elusive. **Tennenbaum’s theorem** says that for any countable non-standard model of Peano arithmetic there is no way to code the elements of the model as standard natural numbers such that either the addition or multiplication operation of the model is a computable function on the codes.

We can, however, say some things about what these countable nonstandard models are like as ordered sets. They can be linearly ordered in a way compatible with addition and multiplication. And then they consist of one copy of the standard natural numbers, followed by a lot of copies of the standard integers, which are packed together in a dense way: that is, for any two distinct copies, there’s another distinct copy between them. Furthermore, for any of these copies, there’s another copy before it, and another after it.

I should also say what’s good about algebraically closed fields of characteristic zero: they are **uncountably categorical**. In other words, any two models of the axioms for an algebraically closed field with the same cardinality must be isomorphic. (This is not true for the countable models: it’s easy to find different countable algebraically closed fields of characteristic zero. They are not spooky and elusive.)

So, any algebraically closed field whose cardinality is that of the continuum is isomorphic to the complex numbers!

For more on the logic of complex numbers, written at about the same level as this, try this post of mine:

• The logic of real and complex numbers, *Azimuth* 8 September 2014.

• Daniel Cicala and Jules Hedges, Cartesian bicategories, The *n*-Category Café, 19 February 2018.

Jules Hedges is a postdoc in the computer science department at Oxford who is applying category theory to game theory and economics. Daniel Cicala is a grad student working with me on a compositional approach to graph rewriting, which is about stuff like this:

This picture shows four ‘open graphs’: graphs with inputs and outputs. The vertices are labelled with operations. The top of the picture shows a ‘rewrite rule’ where one open graph is turned into another: the operation of multiplying by 2 is replaced by the operation of adding something to itself. The bottom of the picture shows one way we can ‘apply’ this rule: this takes us from open graph at bottom left to the open graph at bottom right.

So, we can use graph rewriting to think about ways to transform a computer program into another, perhaps simpler, computer program that does the same thing.

How do we formalize this?

A computer program wants to be a morphism, since it’s a process that turns some input into some output. Rewriting wants to be a 2-morphism, since it’s a ‘meta-process’ that turns some program into some other program. So, there should be some bicategory with computer programs (or labelled open graphs!) as morphisms and rewrites as 2-morphisms. In fact there should be a bunch of such bicategories, since there are a lot of details that one can tweak.

Together with my student Kenny Courser, Daniel has been investigating these bicategories:

• Daniel Cicala, Spans of cospans, *Theory and Applications of Categories* **33** (2018), 131–147.

Abstract.We discuss the notion of a span of cospans and define, for them, horizonal and vertical composition. These compositions satisfy the interchange law if working in a topos C and if the span legs are monic. A bicategory is then constructed from C-objects, C-cospans, and doubly monic spans of C-cospans. The primary motivation for this construction is an application to graph rewriting.

• Daniel Cicala, Spans of cospans in a topos, *Theory and Applications of Categories* **33** (2018), 1–22.

Abstract.For a topos T, there is a bicategory MonicSp(Csp(T)) whose objects are those of T, morphisms are cospans in T, and 2-morphisms are isomorphism classes of monic spans of cospans in T. Using a result of Shulman, we prove that MonicSp(Csp(T)) is symmetric monoidal, and moreover, that it is compact closed in the sense of Stay. We provide an application which illustrates how to encode double pushout rewrite rules as 2-morphisms inside a compact closed sub-bicategory of MonicSp(Csp(Graph)).

This stuff sounds abstract and esoteric when they talk about it, but it’s really all about things like the picture above—and it’s an important part of network theory!

Recently Daniel Cicala has noticed that some of the bicategories he’s getting are ‘cartesian bicategories’ in the sense of this paper:

• Aurelio Carboni and Robert F. C. Walters, Cartesian bicategories I, *Journal of Pure and Applied Algebra* **49** (1987), 11–32.

And that’s the paper he’s blogging about now with Jules Hedges!

]]>• Marc Rigby, Insect population decline leaves Australian scientists scratching for solutions, *ABC Far North*, 23 February 2018.

I’ll quote the start:

A global crash in insect populations has found its way to Australia, with entomologists across the country reporting lower than average numbers of wild insects.

University of Sydney entomologist Dr. Cameron Webb said researchers around the world widely acknowledge that insect populations are in decline, but are at a loss to determine the cause.

“On one hand it might be the widespread use of insecticides, on the other hand it might be urbanisation and the fact that we’re eliminating some of the plants where it’s really critical that these insects complete their development,” Dr Webb said.

“Add in to the mix climate change and sea level rise and it’s incredibly difficult to predict exactly what it is. It’s left me dumbfounded.”

Entomologist and owner of the Australian Insect Farm, near Innisfail in far north Queensland, Jack Hasenpusch is usually able to collect swarms of wild insects at this time of year.

“I’ve been wondering for the last few years why some of the insects have been dropping off and put it down to lack of rainfall,” Mr. Hasenpusch said.

“This year has really taken the cake with the lack of insects, it’s left me dumbfounded, I can’t figure out what’s going on.”

Mr Hasenpusch said entomologists he had spoken to from Sydney, Brisbane, Perth and even as far away as New Caledonia and Italy all had similar stories.

The Australian Butterfly Sanctuary in Kuranda, west of Cairns, has had difficulty breeding the far north’s iconic Ulysses butterfly for more than two years.

“We’ve had [the problem] checked by scientists, the University of Queensland was involved, Biosecurity Queensland was involved but so far we haven’t found anything unusual in the bodies [of caterpillars] that didn’t survive,” said breeding laboratory supervisor Tina Kupke.

“We’ve had some short successes but always failed in the second generation.”

Ms. Lupke said the problem was not confined to far north Queensland, or even Australia. “Some of our pupae go overseas from some of our breeders here and they’ve all had the same problem,” she said. “And the Melbourne Zoo has been trying for quite a while with the same problems.”

## Limited lifecycle prefaces population plummet

Dr. Webb, who primarily researches mosquitoes, said numbers were also in decline across New South Wales this year, which was indicative of the situation in other insect populations.

“We’ve had a really strange summer; it’s been very dry, sometimes it’s been brutally hot but sometimes it’s been cooler than average,” he said.

“Mosquito populations, much like a lot of other insects, rely on the combination of water, humidity and temperature to complete their lifecycle. When you mix around any one of those three components you can really change the local population dynamics.”

All this reminds me of a much more detailed study showing a dramatic insect population decline in Germany over a much longer time period:

• Gretchen Vogel, Where have all the insects gone?, *Science*, 10 May 2017.

I’ll just quote a bit of this article:

Now, a new set of long-term data is coming to light, this time from a dedicated group of mostly amateur entomologists who have tracked insect abundance at more than 100 nature reserves in western Europe since the 1980s.

Over that time the group, the Krefeld Entomological Society, has seen the yearly insect catches fluctuate, as expected. But in 2013 they spotted something alarming. When they returned to one of their earliest trapping sites from 1989, the total mass of their catch had fallen by nearly 80%. Perhaps it was a particularly bad year, they thought, so they set up the traps again in 2014. The numbers were just as low. Through more direct comparisons, the group—which had preserved thousands of samples over 3 decades—found dramatic declines across more than a dozen other sites.

It also mentions a similar phenomenon in Scotland:

Since 1968, scientists at Rothamsted Research, an agricultural research center in Harpenden, U.K., have operated a system of suction traps—12-meter-long suction tubes pointing skyward. Set up in fields to monitor agricultural pests, the traps capture all manner of insects that happen to fly over them; they are “effectively upside-down Hoovers running 24/7, continually sampling the air for migrating insects,” says James Bell, who heads the Rothamsted Insect Survey.

Between 1970 and 2002, the biomass caught in the traps in southern England did not decline significantly. Catches in southern Scotland, however, declined by more than two-thirds during the same period. Bell notes that overall numbers in Scotland were much higher at the start of the study. “It might be that much of the [insect] abundance in southern England had already been lost” by 1970, he says, after the dramatic postwar changes in agriculture and land use.

Here’s the actual research paper:

• Caspar A. Hallmann, Martin Sorg, Eelke Jongejans, Henk Siepel, Nick Hofland, Heinz Schwan, Werner Stenmans, Andreas Müller, Hubert Sumser, Thomas Hörren, Dave Goulson and Hans de Kroon, More than 75 percent decline over 27 years in total flying insect biomass in protected areas, *PLOS One*, 18 October 2017.

Abstract.Global declines in insects have sparked wide interest among scientists, politicians, and the general public. Loss of insect diversity and abundance is expected to provoke cascading effects on food webs and to jeopardize ecosystem services. Our understanding of the extent and underlying causes of this decline is based on the abundance of single species or taxonomic groups only, rather than changes in insect biomass which is more relevant for ecological functioning. Here, we used a standardized protocol to measure total insect biomass using Malaise traps, deployed over 27 years in 63 nature protection areas in Germany (96 unique location-year combinations) to infer on the status and trend of local entomofauna. Our analysis estimates a seasonal decline of 76%, and mid-summer decline of 82% in flying insect biomass over the 27 years of study. We show that this decline is apparent regardless of habitat type, while changes in weather, land use, and habitat characteristics cannot explain this overall decline. This yet unrecognized loss of insect biomass must be taken into account in evaluating declines in abundance of species depending on insects as a food source, and ecosystem functioning in the European landscape.

It seems we are heading into strange times.

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