Azimuth Project News

It’s time to update you on the Azimuth Project. This project started out in 2010 when I moved to Singapore, had more time to think thanks to a great job at the Centre for Quantum Technologies, and decided to do something about climate change—or more broadly, the Anthropocene.

But do what? You can read my very first thoughts here. I rounded up some interested people, many of them programmers from outside academia, and we started a wiki to compile relevant scientific information. We thought a lot and wrote a lot about the huge problems confronting our civilization. We did some interesting stuff like making simple climate models—purely for educational purposes, not for trying to predict anything! We also recapitulated a network-based attempt to predict El Niños.

But it soon became clear to me that my own strengths lay not in climate science, and certainly not in leading a group of people outside academia trying to accomplish something practical. I got more and more interested in using category theory to study networks—and more generally in getting category theorists interested in practical things. I figured that category theory could really transform how we think about complex systems made of interacting parts.

I understand a bit about what motivates academics, and how to get them working on things. So, once I put my mind to it, I managed to speed up the trend toward applied category theory, which by now has its own annual conference. I’m on the steering committee of that conference, but luckily there are so many energetic people involved that I don’t have to do much. By now I can barely keep up with the progress in applied category theory, which is visible on the Category Theory Community Server, a forum set up by my student Christian Williams.

Indeed, part of how academia works is that if you get really good students, they go off and do things much better than you could do yourself!

For example, my former student Brendan Fong is an order of magnitude better at organizing things than I am. Together with Joshua Tan and Nina Otter he started the journal Compositionality, which has a strong emphasis on applied category theory, though it’s also open to other ways of thinking about compositionality (the study of how complex things can be assembled out of simpler parts). But even more importantly, Brendan now leads the Topos Institute, which brings together applied category theorists and people developing new technologies for the betterment of humanity. I’ll get back to that later.

Another amazingly successful student of mine is Nina Otter, now at Queen Mary University. At least I’ll gladly count her as a student, because she did a master’s thesis with me, on operads and the tree of life. But then she switched to topological data analysis, and she’s now using that to study weather regimes.

A big part of the Azimuth project’s focus on networks has always been studying Petri nets: a general formalism for studying chemical reactions, population biology and many other things.

A bunch of blog articles on Petri nets, written at the Centre for Quantum Technologies with Jacob Biamonte, eventually turned into our book Quantum Techniques for Stochastic Mechanics. But a new direction came when Brendan Fong developed decorated cospans, a general technique for studying open systems. My student Blake Pollard and I used these to study ‘open Petri nets’, which we called open reaction networks.

Later, my student Jade Master made the theory of open Petri nets really beautiful using structured cospans, a simplified version of Brendan’s decorated cospans developed by my student Kenny Courser.

Meanwhile something big was brewing. Two fresh PhDs named James Fairbanks and Evan Patterson came up with AlgebraicJulia, a software system that aims to “create novel approaches to scientific computing based on applied category theory”. And among many other things, they grabbed ahold of structured cospans and turned them into something you could write programs with!

In October 2020, together with Micah Halter, they used AlgebraicJulia to redo part of the UK’s main COVID model using open Petri nets. At the time I wrote:

This is a wonderful development! Micah Halter and Evan Patterson have taken my work on structured cospans with Kenny Courser and open Petri nets with Jade Master, together with Joachim Kock’s whole-grain Petri nets, and turned them into a practical software tool!

Then they used that to build a tool for ‘compositional’ modeling of the spread of infectious disease. By ‘compositional’, I mean that they make it easy to build more complex models by sticking together smaller, simpler models.

Even better, they’ve illustrated the use of this tool by rebuilding part of the model that the UK has been using to make policy decisions about COVID19.

All this software was written in the programming language Julia.

I had expected structured cospans to be useful in programming and modeling, but I didn’t expect it to happen so fast!

Here’s a video about these ideas, from 2020:

Later Evan got a job at the Topos Institute and this work blossomed into the following paper:

• Sophie Libkind, Andrew Baas, Micah Halter, Evan Patterson and James Fairbanks, An algebraic framework for structured epidemic modeling, Philosophical Transactions of the Royal Society A 380 (2022), 20210309.

I should have blogged about this, but things are happening so fast I never got around to it! This illustrates why I’ve lost interest in the Azimuth Project as originally formulated, with this blog as the main communication hub and the wiki as the information depot: academics with their own modes of communication have been pushing things forward in their own ways too fast for me to blog about it all!

Another example: last summer in Buffalo I helped mentor a bunch of students at a program on applied category theory run by the American Mathematical Society. This led to two very nice papers on open Petri nets and related open networks:

• Rebekah Aduddell, James Fairbanks, Amit Kumar, Pablo S. Ocal, Evan Patterson and Brandon T. Shapiro, A compositional account of motifs, mechanisms, and dynamics in biochemical regulatory networks.

• Benjamin Merlin Bumpus, Sophie Libkind, Jordy Lopez Garcia, Layla Sorkatti and Samuel Tenka, Additive invariants of open Petri nets.

I want to blog about these, and I will soon!

But at the same time, the use of category theory in epidemiological modeling keeps growing. The early work attracted the attention of a bunch of actual epidemiologists, notably my old grad school pal Nate Osgood, who now works at the University of Saskatchewan, both in computer science and also the department of community health and epidemiology. He helps the government of Canada run its main COVID models! This was a wonderful coincidence, made even sweeter by the fact that Nate was hankering to apply category theory to these tasks.

Nate explained that for modeling disease, Petri nets are less popular than another style of diagram, called ‘stock-flow diagrams’. But one can deal with open stock-flow diagrams using the same category-theoretic tricks that work for Petri nets: decorated or structured cospans. We worked this out together with Evan Patterson, Nate’s grad student Xiaoyan Li, and Sophie Libkind at the Topos Institute. And these folks—not me—converted these ideas into AlgebraicJulia code for making big models of epidemic disease out of smaller parts!

We wrote about it here:

• John Baez, Xiaoyan Li, Sophie Libkind, Nathaniel D. Osgood and Evan Patterson, Compositional modeling with stock and flow diagrams, to appear in the proceedings of Applied Category Theory 2022.

Alas, I’ve been too busy to properly blog about this paper, but I’ve given a bunch of talks about it, and you can see some on YouTube. The easiest is probably this one:

Since then we’ve made a huge amount of progress, due largely to Nate and Xiaoyan’s enthusiasm for converting abstract ideas into practical tools for epidemiologists. The current state of the art is pretty well reflected in this paper:

• John Baez, Xiaoyan Li, Sophie Libkind, Nathaniel D. Osgood and Eric Redekopp, A categorical framework for modeling with stock and flow diagrams.

In particular, Nate’s student Eric Redekopp built a graphical user interface for the software, so epidemiologists knowing nothing of category theory or the language Julia can collaboratively build disease models on their web browsers!

So, a lot of my energy that originally went into the Azimuth Project has, by a series of unpredictable events, become focused on the project of applied category theory, with the most practical application for me currently being disease models.

What happened to climate change? Well, a lot of these modeling methodologies could be applied to power grids or world economic models. In fact stock-flow diagrams were first developed for economics and business in James Forrester’s book Industrial Dynamics, and they were later used in the famous Limits to Growth model of the world economy and ecology, called World3. So there is a lot to do in this direction. But—I’ve realized—it would require finding an energetic expert who is willing to learn some category theory and teach me (or some other applied category theorist) what they know.

For now, a more instantly attractive option is working with someone I’ve known since I was a postdoc: Minhyong Kim. He’s now head of the International Center of Mathematical Sciences, and he’s dreamt up a project called Mathematics for Humanity. This will fund research workshops, conferences and courses in these areas:

A. Integrating the global research community

B. Mathematical challenges for humanity

C. Global history of mathematics

I’m hoping to coax people to run a workshop on mathematical epidemiology, but also get people together to tackle many other mathematical challenges for humanity. Minhyong has listed some examples:

The deadline to apply for funding is now June 1st, so if you know anyone who might be interested, please tell them about this—and tell me about them!

So, a lot is going on. But I’ve had very little time to do anything with the Azimuth Wiki or the Azimuth Forum (an online discussion forum for the Azimuth Project). Indeed I’ve largely ignored them for years now. David Tanzer has nobly been providing tech support for these sites. But after many conversations with him about this, I’ve decided that it’s time to close down those sites. So that’s what we plan to do on May 1st.

It’s a bit sad, but as I hope I’ve explained, the spirit of the Azimuth Project lives on. And at this moment I want to thank everyone who has been in involved with it in any way. There are many of you not mentioned above. If I tried to list all of you I’d leave some out, so please accept these collective thanks—and good luck with all your projects!

The Binary Octahedral Group (Part 1)

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 you can look for unit quaternions such that whenever is one of these corners, so is . 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 to 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 on the Riemann sphere instead of , we can think of the binary octahedral group as a subgroup of ), since this acts as conformal transformations of the Riemann sphere!

If we do this, the binary octahedral group becomes a subgroup of , 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 actually 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):

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):

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:

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

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

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

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

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

Finally, the two elements

do nothing!

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

Interview

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

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

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

 

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

 

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

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

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)

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.