The Binary Octahedral Group

29 August, 2019


The complex numbers together with infinity form a sphere called
the Riemann sphere. The 6 simplest numbers on this sphere lie at points we could call the north pole, the south pole, the east pole, the west pole, the front pole and the back pole. They’re the corners of an octahedron!

On the Earth, I’d say the “front pole” is where the prime meridian meets the equator at 0°N 0°E. It’s called Null Island, but there’s no island there—just a buoy. Here it is:

Where’s the back pole, the east pole and the west pole? I’ll leave two of these as puzzles, but I discovered that in Singapore I’m fairly close to the east pole:

If you think of the octahedron’s corners as the quaternions \pm i, \pm j, \pm k, you can look for unit quaternions q such that whenever x is one of these corners, so is qxq^{-1}. There are 48 of these! They form a group called the binary octahedral group.

By how we set it up, the binary octahedral group acts as rotational symmetries of the octahedron: any transformation sending x to qxq^{-1} is a rotation. But this group is a double cover of the octahedron’s rotational symmetry group! That is, pairs of elements of the binary octahedral group describe the same rotation of the octahedron.

If we go back and think of the Earth’s 6 poles as points 0, \pm 1,\pm i, \infty on the Riemann sphere instead of \pm i, \pm j, \pm k, we can think of the binary octahedral group as a subgroup of \mathrm{SL}(2,\mathbb{C}), since this acts as conformal transformations of the Riemann sphere!

If we do this, the binary octahedral group is actually a subgroup of \mathrm{SU}(2), the double cover of the rotation group—which is isomorphic to the group of unit quaternions. So it all hangs together.

It’s fun to actualy see the unit quaternions in the binary octahedral group. First we have 8 that form the corners of a cross-polytope (the 4d analogue of an octahedron):

\pm 1, \pm i , \pm j , \pm k

These form a group on their own, called the quaternion group. Then we have 16 that form the corners of a hypercube (the 4d analogue of a cube, also called a tesseract or 4-cube):

\displaystyle{ \frac{\pm 1 \pm i \pm j \pm k}{2} }

These don’t form a group, but if we take them together with the 8 previous ones we get a 24-element subgroup of the unit quaternions called the binary tetrahedral group. They’re also the vertices of a 24-cell, which is yet another highly symmetrical shape in 4 dimensions (a 4-dimensional regular polytope that doesn’t have a 3d analogue).

That accounts for half the quaternions in the binary octahedral group! Here are the other 24:

\displaystyle{  \frac{\pm 1 \pm i}{\sqrt{2}}, \frac{\pm 1 \pm j}{\sqrt{2}}, \frac{\pm 1 \pm k}{\sqrt{2}},  }

\displaystyle{  \frac{\pm i \pm j}{\sqrt{2}}, \frac{\pm j \pm k}{\sqrt{2}}, \frac{\pm k \pm i}{\sqrt{2}} }

These form the vertices of another 24-cell!

The first 24 quaternions, those in the binary tetrahedral group, give rotations that preserve each one of the two tetrahedra that you can fit around an octahedron like this:

while the second 24 switch these tetrahedra.

The 6 elements

\pm i , \pm j , \pm k

describe 180° rotations around the octahedron’s 3 axes, the 16 elements

\displaystyle{   \frac{\pm 1 \pm i \pm j \pm k}{2} }

describe 120° clockwise rotations of the octahedron’s 8 triangles, the 12 elements

\displaystyle{  \frac{\pm 1 \pm i}{\sqrt{2}}, \frac{\pm 1 \pm j}{\sqrt{2}}, \frac{\pm 1 \pm k}{\sqrt{2}} }

describe 90° clockwise rotations holding fixed one of the octahedron’s 6 vertices, and the 12 elements

\displaystyle{  \frac{\pm i \pm j}{\sqrt{2}}, \frac{\pm j \pm k}{\sqrt{2}}, \frac{\pm k \pm i}{\sqrt{2}} }

describe 180° clockwise rotations of the octahedron’s 6 opposite pairs of edges.

Finally, the two elements

\pm 1

do nothing!

So, we can have a lot of fun with the idea that a sphere has 6 poles.


Interview

1 January, 2019

Happy New Year! People like to ponder grand themes each time the Earth completes another orbit around the Sun, so let’s give that a try.

Maria Mannone is a musician who studies the relation between mathematics, music and the visual arts. We met at a conference on The Philosophy and Physics of Noether’s Theorems. Later she decided to interview me for the blog Math is in the Air. There’s a version in English and one in Italian.

She let me reprint the interview here… so with no further ado, here it is!

MM: You are one of the pioneers in using the internet and blogs for scientific education, with ‘This Week’s Finds.’ Which words would you use to feed the enthusiasm of young minds towards abstract mathematics?

JB: It seems only certain people are drawn to mathematics, and that’s fine: there are many wonderful things in life and there’s no need for everyone explore all of them. Mathematics seems to attract people who enjoy patterns, who enjoy precision, and who don’t want to remember lists of arbitrary facts, like the names of all 206 bones in the human body. In math everything has a reason and you can understand it, so you don’t really need to remember much. At first it may seem like there’s a lot to remember – for examples, lists of trig identities. But as you go deeper into math, and understand more, everything becomes simpler. These days I don’t bother to remember more than a couple of trigonometric identities; if I ever need them I can figure them out.

But the really surprising thing is that as you go deeper and deeper into mathematics, it keeps revealing more beauty, and more mysteries. You enter new worlds full of profound questions that are quite hard to explain to nonmathematicians. As the Fields medalist Maryam Mirzakhani said, “The beauty of mathematics only shows itself to more patient followers.”

MM: I love the reference to patterns, and the beauty to find. Thus, we can say that mathematical beauty is not ‘all out there’ as the beauty of a flower can be. Or, that some beautiful geometry present in nature can give a hint or can embody some mathematical beauty, but people have to work hard to find more of it—at least they have to learn how to look at things, and thus, how to mathematically think of them.

In the common opinion, a rose, or a water lily is beautiful (and it is!), but a bone is not ‘beautiful’ per se. Personally, each time I find patterns, regularities, hierarchical structures, I get excited and things seem to be at least mathematically interesting. I would like to ask you how would you relate the beauty in the natural world, both visible and ‘to discover,’ and the beauty of math. I’m wondering if they should be considered as two separate sets with occasional, random intersections, or as two displays of a generalized ‘beauty,’ as two different perspectives. Or, maybe, if the first can guide our search into math, or if math can teach us ‘how to look at things and finding beauty.’

JB: I think all forms of beauty are closely connected, and I think almost anything can be beautiful if it’s not the result of someone being heedless to their environment or deliberately hurtful.

It’s not surprising that flowers are very easy to find beautiful, since they evolved precisely to be attractive. Not to humans, at first, but to pollinators like birds and bees. It’s imaginable that what attracts those animals would not be attractive to us. But in fact there’s enough commonality that we enjoy flowers too! And then we bred them to please us even more; many of them are now symbiotic with us.

Something like a bone only becomes beautiful if you examine it carefully and think about how complex it is and how admirably it carries out its function.


From: http://acrmed.com/press-releases/

Bones are initially scary or ‘disgusting’ because when they’re doing their job they are hidden: we usually see them only when an animal is seriously injured or dead. So, you have to go past that instinctive reaction—which by the way serves a useful purpose—to see the beauty in a bone.

Mathematics is somewhere between a rose and a bone. Underlying all of nature there are mathematical patterns – but normally they are hidden from view, like bones in a body. Perhaps to some people they seem harsh or even disgusting when first revealed, but in fact they are extremely elegant. Even those who love mathematics find its patterns austere at first—but as we explore it more deeply, we see they connect in complicated delicate patterns that put the petals of a rose to shame.

MM: Thus, there seems to be an intimate dialogue between nature, both visible and hidden, and mathematical thinking. About nature and environment: in your Twitter image, there is a sketch of you as a superhero saving the planet, with the mathematical symbol ‘There is one and only one’ applied to our planet Earth. Can you tell the readers something about the way you combine your research in mathematics with your engagement for the environment?

Also, it is often said that beauty will save the world. Do you think that mathematical beauty can save the world?

JB: I mainly think of beauty—in all its forms—-as a reason why the world is worth saving. But we are very primitive when it comes to the economics of beauty. Paintings can sell for hundreds of millions of dollars, and we have a market for them. But nobody attaches any value to this critically endangered frog, Atelopus varius:


Atelopus varius,
from https://www.iucnredlist.org/species/54560/11167883

To my mind it’s more beautiful and precious than any painting. Not the individual, of course, but the species, which has taken millions of years to evolve. We are busy destroying species like this as if they were worthless trash. Our descendants, if we have any, will probably think we were barbaric idiots.

But I digress! I switched from pure mathematics and highly theoretical physics to more practical concerns around 2010, when I spent two years at the Centre for Quantum Technologies, in Singapore. I was very lucky that the director encouraged me to think about whatever I wanted. I was wanting a change in direction, and I soon realized that mathematicians, like everyone else, need to think about global warming and what we can do about it: it’s the crisis of our time. I spent some time learning the basics of climate science and working on some projects connected to that. It became clear that to do anything about global warming we need new ideas in politics and economics. Unfortunately, I’m not especially good at those things. So I decided to do something I can actually do, namely to get mathematicians to turn their attention from math inspired by the physics of the microworld—for example string theory—toward math inspired by the visible world around us: biology, ecology, engineering, economics and the like. I’m hoping that mathematicians can solve some problems by thinking more abstractly than anyone else can.

So to finally answer your last question: I’m not sure the beauty of mathematics can save the world, but its beauty is closely connected to clear thinking, and we really need clear thinking.

MM: Yes, in a certain sense, despite culture, technology, and thousands of years of human history, people are quite primitive when it comes to evaluating beauty as detached from the economy.

You brought up an important point: the research focus of mathematicians. This is a tricky point because young researchers are kind of split between following new ideas and projects, and the search for funds, that often leads them to join existing projects or just well-funded areas and to put aside their more ‘visionary’ ideas. What would be your suggestion to find a balance?

JB: I don’t know if I can give advice here: I’ve never needed to search for funds, I get paid to teach calculus and other courses, so I always just do the best research I can. That’s already quite hard—I could talk all day about that!

I suppose if you’re struggling for funds you have to fight to remember your dreams, and try to work your way into a situation where you can pursue these dreams. I imagine this is also true for any entrepreneur with a visionary idea. Academics struggling to get grants really aren’t all that different from executives in a large corporation trying to get funding for their projects.

MM: My last question is about the theme of peace, very important to the Baez family. Many innovations are related to the military. Do you think that the needed clear thinking you mentioned, can first of all come from times, themes, and ideas of peace?

JB: We are currently in a struggle that’s much bigger, and more inspiring, than any war between human tribes. We’re struggling to come to terms with the Anthropocene: the epoch where the Earth’s ecosystems and even geology are being transformed by humans. We are used to treating our impact on nature as negligible. This is no longer true! The Arctic is rapidly melting:

And since 1970, the abundance of many vertebrate species worldwide has dropped 60%. You can see it in this chart prepared by the Worldwide Wildlife Fund:

If this were a war, and these were humans dying, this would be the worst war the world has ever seen! But these changes will not merely affect other species; they are starting to hit us too. We need to wake up. We will either deliberately change our civilization, quite quickly, or we will watch as our cities burn and drown. Isn’t it better to use that intelligence we humans love to boast about, and take action?

MM: Thank you Professor, I hope these words will enlighten many people.


Compositionality – Now Open For Submissions

24 August, 2018

Our new journal Compositionality is now open for submissions!

It’s an open-access journal for research using compositional ideas, most notably of a category-theoretic origin, in any discipline. Topics may concern foundational structures, an organizing principle, or a powerful tool. Example areas include but are not limited to: computation, logic, physics, chemistry, engineering, linguistics, and cognition.

Compositionality is free of cost for both readers and authors.



CALL FOR PAPERS

We invite you to submit a manuscript for publication in the first issue of Compositionality (ISSN: 2631-4444), a new open-access journal for research using compositional ideas, most notably of a category-theoretic origin, in any discipline.

To submit a manuscript, please visit http://www.compositionality-journal.org/for-authors/.

SCOPE

Compositionality refers to complex things that can be built by sticking together simpler parts. We welcome papers using compositional ideas, most notably of a category-theoretic origin, in any discipline. This may concern foundational structures, an organising principle, a powerful tool, or an important application. Example areas include but are not limited to: computation, logic, physics, chemistry, engineering, linguistics, and cognition.

Related conferences and workshops that fall within the scope of Compositionality include the Symposium on Compositional Structures (SYCO), Categories, Logic and Physics (CLP), String Diagrams in Computation, Logic and Physics (STRING), Applied Category Theory (ACT), Algebra and Coalgebra in Computer Science (CALCO), and the Simons Workshop on Compositionality.

SUBMISSION AND PUBLICATION

Submissions should be original contributions of previously unpublished work, and may be of any length. Work previously published in conferences and workshops must be significantly expanded or contain significant new results to be accepted. There is no deadline for submission. There is no processing charge for accepted publications; Compositionality is free to read and free to publish in. More details can be found in our editorial policies at http://www.compositionality-journal.org/editorial-policies/.

STEERING BOARD

John Baez, University of California, Riverside, USA
Bob Coecke, University of Oxford, UK
Kathryn Hess, EPFL, Switzerland
Steve Lack, Macquarie University, Australia
Valeria de Paiva, Nuance Communications, USA

EDITORIAL BOARD

Corina Cirstea, University of Southampton, UK
Ross Duncan, University of Strathclyde, UK
Andree Ehresmann, University of Picardie Jules Verne, France
Tobias Fritz, Max Planck Institute, Germany
Neil Ghani, University of Strathclyde, UK
Dan Ghica, University of Birmingham, UK
Jeremy Gibbons, University of Oxford, UK
Nick Gurski, Case Western Reserve University, USA
Helle Hvid Hansen, Delft University of Technology, Netherlands
Chris Heunen, University of Edinburgh, UK
Aleks Kissinger, Radboud University, Netherlands
Joachim Kock, Universitat Autonoma de Barcelona, Spain
Martha Lewis, University of Amsterdam, Netherlands
Samuel Mimram, Ecole Polytechnique, France
Simona Paoli, University of Leicester, UK
Dusko Pavlovic, University of Hawaii, USA
Christian Retore, Universite de Montpellier, France
Mehrnoosh Sadrzadeh, Queen Mary University, UK
Peter Selinger, Dalhousie University, Canada
Pawel Sobocinski, University of Southampton, UK
David Spivak, MIT, USA
Jamie Vicary, University of Birmingham and University of Oxford, UK
Simon Willerton, University of Sheffield, UK

Sincerely,

The Editorial Board of Compositionality


Applied Category Theory 2018/2019

15 June, 2018

A lot happened at Applied Category Theory 2018. Even as it’s still winding down, we’re already starting to plan a followup in 2019, to be held in Oxford. Here are some notes Joshua Tan sent out:

  1. Discussions: Minutes from the discussions can be found here.
  2. Photos: Ross Duncan took some very glamorous photos of the conference, which you can find here.

  3. Videos: Videos of talks are online here: courtesy of Jelle Herold and Fabrizio Genovese.

  4. Next year’s workshop: Bob Coecke will be organizing ACT 2019, to be hosted in Oxford sometime spring/summer. There will be a call for papers.

  5. Next year’s school: Daniel Cicala is helping organize next year’s ACT school. Please contact him at if you would like to get involved.

  6. Look forward to the official call for submissions, coming soon, for the first issue of Compositionality!

The minutes mentioned above contain interesting thoughts on these topics:

• Day 1: Causality
• Day 2: AI & Cognition
• Day 3: Dynamical Systems
• Day 4: Systems Biology
• Day 5: Closing


Applied Category Theory Course: Databases

6 June, 2018

 

In my online course on applied category theory we’re now into the third chapter of Fong and Spivak’s book Seven Sketches. Now we’re talking about databases!

To some extent this is just an excuse to (finally) introduce categories, functors, natural transformations, adjoint functors and Kan extensions. Great stuff, and databases are a great source of easy examples.

But it’s also true that Spivak helps run a company called Categorical Informatics that actually helps design databases using category theory! And his partner, Ryan Wisnesky, would be happy to talk to people about it. If you’re interested, click the link: he’s attending my course.

To read and join discussions on Chapter 3 go here:

Chapter 3

You can also do exercises and puzzles, and see other people’s answers to these.

Here are the lectures I’ve given so far:

Lecture 34 – Chapter 3: Categories
Lecture 35 – Chapter 3: Categories versus Preorders
Lecture 36 – Chapter 3: Categories from Graphs
Lecture 37 – Chapter 3: Presentations of Categories
Lecture 38 – Chapter 3: Functors
Lecture 39 – Chapter 3: Databases
Lecture 40 – Chapter 3: Relations
Lecture 41 – Chapter 3: Composing Functors
Lecture 42 – Chapter 3: Transforming Databases
Lecture 43 – Chapter 3: Natural Transformations
Lecture 44 – Chapter 3: Categories, Functors and Natural Transformations
Lecture 45 – Chapter 3: Composing Natural Transformations
Lecture 46 – Chapter 3: Isomorphisms
Lecture 47 – Chapter 3: Adjoint Functors
Lecture 48 – Chapter 3: Adjoint Functors
Lecture 49 – Chapter 3: Kan Extensions
Lecture 50 – Chapter 3: Kan Extensions
Lecture 51 – Chapter 3: Right Kan Extensions
Lecture 52 – Chapter 3: The Hom-Functor
Lecture 53 – Chapter 3: Free and Forgetful Functors
Lecture 54 – Chapter 3: Tying Up Loose Ends


Applied Category Theory Course: Resource Theories

12 May, 2018

 

My course on applied category theory is continuing! After a two-week break where the students did exercises, I’m back to lecturing about Fong and Spivak’s book Seven Sketches. Now we’re talking about “resource theories”. Resource theories help us answer questions like this:

  1. Given what I have, is it possible to get what I want?
  2. Given what I have, how much will it cost to get what I want?
  3. Given what I have, how long will it take to get what I want?
  4. Given what I have, what is the set of ways to get what I want?

Resource theories in their modern form were arguably born in these papers:

• Bob Coecke, Tobias Fritz and Robert W. Spekkens, A mathematical theory of resources.

• Tobias Fritz, Resource convertibility and ordered commutative monoids.

We are lucky to have Tobias in our course, helping the discussions along! He’s already posted some articles on resource theory here on this blog:

• Tobias Fritz, Resource convertibility (part 1), Azimuth, 7 April 2015.

• Tobias Fritz, Resource convertibility (part 2), Azimuth, 10 April 2015.

• Tobias Fritz, Resource convertibility (part 3), Azimuth, 13 April 2015.

We’re having fun bouncing between the relatively abstract world of monoidal preorders and their very concrete real-world applications to chemistry, scheduling, manufacturing and other topics. Here are the lectures so far:

Lecture 18 – Chapter 2: Resource Theories
Lecture 19 – Chapter 2: Chemistry and Scheduling
Lecture 20 – Chapter 2: Manufacturing
Lecture 21 – Chapter 2: Monoidal Preorders
Lecture 22 – Chapter 2: Symmetric Monoidal Preorders
Lecture 23 – Chapter 2: Commutative Monoidal Posets
Lecture 24 – Chapter 2: Pricing Resources
Lecture 25 – Chapter 2: Reaction Networks
Lecture 26 – Chapter 2: Monoidal Monotones
Lecture 27 – Chapter 2: Adjoints of Monoidal Monotones
Lecture 28 – Chapter 2: Ignoring Externalities
Lecture 29 – Chapter 2: Enriched Categories
Lecture 30 – Chapter 2: Preorders as Enriched Categories
Lecture 31 – Chapter 2: Lawvere Metric Spaces
Lecture 32 – Chapter 2: Enriched Functors
Lecture 33 – Chapter 2: Tying Up Loose Ends

 


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”.

We study the applications to logic—both classical logic based on subsets, and a 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. But the real payoff comes in Chapter 2, where we discuss “resource theories”.

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+. 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.


Azimuth Backup Project (Part 5)

5 October, 2017

I haven’t spoken much about the Azimuth Climate Data Backup Project, but it’s going well, and I’ll be speaking about it soon, here:

International Open Access Week, Wednesday 25 October 2017, 9:30–11:00 a.m., University of California, Riverside, Orbach Science Library, Room 240.

“Open in Order to Save Data for Future Research” is the 2017 event theme.

Open Access Week is an opportunity for the academic and research community to learn about the potential benefits of sharing what they’ve learned with colleagues, and to help inspire wider participation in helping to make “open access” a new norm in scholarship, research and data planning and preservation.

The Open Access movement is made of up advocates (librarians, publishers, university repositories, etc.) who promote the free, immediate, and online publication of research.

The program will provide information on issues related to saving open data, including climate change and scientific data. The panelists also will describe open access projects in which they have participated to save climate data and to preserve end-of-term presidential data, information likely to be and utilized by the university community for research and scholarship.

The program includes:

• Brianna Marshall, Director of Research Services, UCR Library: Brianna welcomes guests and introduces panelists.

• John Baez, Professor of Mathematics, UCR: John will describe his activities to save US government climate data through his collaborative effort, the Azimuth Climate Data Backup Project. All of the saved data is now open access for everyone to utilize for research and scholarship.

• Perry Willett, Digital Preservation Projects Manager, California Digital Library: Perry will discuss the open data initiatives in which CDL participates, including the end-of-term presidential web archiving that is done in partnership with the Library of Congress, Internet Archive and University of North Texas.

• Kat Koziar, Data Librarian, UCR Library: Kat will give an overview of DASH, the UC system data repository, and provide suggestions for researchers interested in making their data open.

This will be the eighth International Open Access Week program hosted by the UCR Library.

The event is free and open to the public. Light refreshments will be served.


Complex Adaptive System Design (Part 4)

22 August, 2017

Last time I introduced typed operads. A typed operad has a bunch of operations for putting together things of various types and getting new things of various types. This is a very general idea! But in the CASCADE project we’re interested in something more specific: networks. So we want operads whose operations are ways to put together networks and get new networks.

That’s what our team came up with: John Foley of Metron, my graduate students Blake Pollard and Joseph Moeller, and myself. We’re writing a couple of papers on this, and I’ll let you know when they’re ready. These blog articles are kind of sneak preview—and a gentle introduction, where you can ask questions.

For example: I’m talking a lot about networks. But what is a ‘network’, exactly?

There are many kinds. At the crudest level, we can model a network as a simple graph, which is something like this:

There are some restrictions on what counts as a simple graph. If the vertices are agents of some sort and the edges are communication channels, these restrictions imply:

• We allow at most one channel between any pair of agents, since there’s at most one edge between any two vertices of our graph.

• The channels do not have a favored direction, since there are no arrows on the edges of our graph.

• We don’t allow a channel from an agent to itself, since an edge can’t start and end at the same vertex.

For other purposes we may want to drop some or all of these restrictions. There is an appalling diversity of options! We might want to allow multiple channels between a pair of agents. For this we could use multigraphs. We might want to allow directed channels, where the sender and receiver have different capabilities: for example, signals may only be able to flow in one direction. For this we could use directed graphs. And so on.

We will also want to consider graphs with colored vertices, to specify different types of agents—or colored edges, to specify different types of channels. Even more complicated variants are likely to become important as we proceed.

To avoid sinking into a mire of special cases, we need the full power of modern mathematics. Instead of separately studying all these various kinds of networks, we need a unified notion that subsumes all of them.

To do this, the Metron team came up with something called a ‘network model’. There is a network model for simple graphs, a network model for multigraphs, a network model for directed graphs, a network model for directed graphs with 3 colors of vertex and 15 colors of edge, and more.

You should think of a network model as a kind of network. Not a specific network, just a kind of network.

Our team proved that for each network model G there is an operad O_G whose operations describe how to put together networks of that kind. We call such operads ‘network operads’.

I want to make all this precise, but today let me just show you one example. Let’s take G to be the network model for simple graphs, and look at the network operad O_G. I won’t tell you what kind of thing G is yet! But I’ll tell you about the operad O_G.

Types. Remember from last time that an operad has a set of ‘types’. For O_G this is the set of natural numbers, \mathbb{N}. The reason is that a simple graph can have any number of vertices.

Operations. Remember that an operad has sets of ‘operations’. In our case we have a set of operations O_G(t_1,\dots,t_n ; t) for each choice of t_1,\dots,t_n, t \in \mathbb{N}.

An operation f \in O_G(t_1,\dots,t_n; t) is a way of taking a simple graph with t_1 vertices, a simple graph with t_2 vertices,… and so on, and sticking them together, perhaps adding new edges, to get a simple graph with

t = t_1 + \cdots + t_n

vertices.

Let me show you an operation

f \in O_G(3,4,2;9)

This will be a way of taking three simple graphs—one with 3 vertices, one with 4, and one with 2—and sticking them together, perhaps adding edges, to get one with 9 vertices.

Here’s what f looks like:

It’s a simple graph with vertices numbered from 1 to 9, with the vertices in bunches: {1,2,3}, {4,5,6,7} and {8,9}. It could be any such graph. This one happens to have an edge from 3 to 6 and an edge from 1 to 2.

Here’s how we can actually use our operation. Say we have three simple graphs like this:

Then we can use our operation to stick them together and get this:

Notice that we added a new edge from 3 to 6, connecting two of our three simple graphs. We also added an edge from 1 to 2… but this had no effect, since there was already an edge there! The reason is that simple graphs have at most one edge between vertices.

But what if we didn’t already have an edge from 1 to 2? What if we applied our operation f to the following simple graphs?

Well, now we’d get this:

This time adding the edge from 1 to 2 had an effect, since there wasn’t already an edge there!

In short, we can use this operad to stick together simple graphs, but also to add new edges within the simple graphs we’re sticking together!

When I’m telling you how we ‘actually use’ our operad to stick together graphs, I’m secretly describing an algebra of our operad. Remember, an operad describes ways of sticking together things together, but an ‘algebra’ of the operad gives a particular specification of these things and describes how we stick them together.

Our operad O_G has lots of interesting algebras, but I’ve just shown you the simplest one. More precisely:

Things. Remember from last time that for each type, an algebra specifies a set of things of that type. In this example our types are natural numbers, and for each natural number t \in \mathbb{N} I’m letting the set of things A(t) consist of all simple graphs with vertices \{1, \dots, t\}.

Action. Remember that our operad O_G should have an action on A, meaning a bunch of maps

\alpha : O_G(t_1,...,t_n ; t) \times A(t_1) \times \cdots \times A(t_n) \to A(t)

I just described how this works in some examples. Some rules should hold… and they do.

To make sure you understand, try these puzzles:

Puzzle 1. In the example I just explained, what is the set O_G(t_1,\dots,t_n ; t) if t \ne  t_1 + \cdots + t_n?

Puzzle 2. In this example, how many elements does O_G(1,1;2) have?

Puzzle 3. In this example, how many elements does O_G(1,2;3) have?

Puzzle 4. In this example, how many elements does O_G(1,1,1;3) have?

Puzzle 5. In the particular algebra A that I explained, how many elements does A(3) have?

Next time I’ll describe some more interesting algebras of this operad O_G. These let us describe networks of mobile agents with range-limited communication channels!


Some posts in this series:

Part 1. CASCADE: the Complex Adaptive System Composition and Design Environment.

Part 2. Metron’s software for system design.

Part 3. Operads: the basic idea.

Part 4. Network operads: an easy example.

Part 5. Algebras of network operads: some easy examples.

Part 6. Network models.

Part 7. Step-by-step compositional design and tasking using commitment networks.

Part 8. Compositional tasking using category-valued network models.

Part 9 – Network models from Petri nets with catalysts.