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John Baez

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Applied Category Theory - An Online Course

Two weeks ago I started teaching an online course based on this free book:

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

It's eating up the time I used to spend on G+. But I'm happy with that, because over 250 people have registered, and a bunch of them are very energetic. It's exciting!

Four days I week I write short "lectures" on the book. If you take the course you can read those lectures, read the book, try the exercises in the book and the puzzles I create, and discuss everything with me and the other students! The best part of the course, in my opinion, is the conversations. People are starting to dream up projects to work on together.

If this sounds interesting, go here and register in the box at upper left:

Use your full real name as your username, with no spaces. I will get back to you, so use a working email address. You can move through the course at your own pace: all the discussions can go on indefinitely.

Brendan Fong was my grad student; now he's doing a postdoc at MIT with David Spivak. They're at the cutting edge of applied category theory, and I"m having a lot of fun working through their book by teaching it. Here is the preface to their 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.

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 diagram below?

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.
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Sets of sets of sets of sets of sets of sets

If you want to read my posts about random stuff, they're on Twitter these days. Anyway, here's something too long for Twitter. It's about monads.

I don't have the energy to explain monads now, so if you don't know what those are... I'm sorry. You're missing lots of fun! Watch the Catsters videos:

The covariant power set functor P : Set → Set can be made into a monad whose multiplication m_X: P(P(X)) → P(X) turns a subset of the set of subsets of X into a subset of X by taking their union. Algebras of this monad are complete semilattices.

But what about powers of the power set functor? Yesterday on Twitter Jules Hedges pointed out this paper:

• Bartek Klin and Julian Salamanca, Iterated covariant powerset is not a monad,

The authors prove that the nth power of P cannot be made into a monad for n ≥ 2.

I’ve mainly looked at their proof for the case n = 2. I haven’t completely worked through it, but it focuses on the unit of any purported monad structure for P^2, rather than its multiplication. Using a cute Yoneda trick they show there are only four possible units, corresponding to the four elements of P(P(1)). Then they show that none of these can work.

Their argument uses sets of sets of sets of sets, like the one below.

As far as I’ve seen, they don’t address the following question:

Question. Does there exist an associative multiplication m: P^2 P^2 ⇒ P^2 ?

As far as I can tell, my question is fairly useless. Nonetheless I’m curious.

If there were a positive answer, we’d have a natural way to take a set of sets of sets of sets and turn it into a set of sets in such a way that the two most obvious resulting ways to turn a set of sets of sets of sets of sets of sets into a set of sets agree!

And how can anyone resist wanting to know about that?

More discussion may emerge here:
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Applied Category Theory: Resource Theories

My online course on applied category theory has just finished the second chapter of Fong and Spivak's book Seven Sketches. This chapter is about "resource theories". Resource theories help us answer questions like this:

• Given what I have, is it possible to get what I want?
• Given what I have, how much will it cost to get what I want?
• Given what I have, how long will it take to get what I want?
• 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! 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:

You can read all these posts without registering. But if you want to discuss these things with us, which could be lots of fun, please visit the Azimuth Forum and register:

Please 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.

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Applied Category Theory: Ordered Sets

My applied category theory course based on Fong and Spivak's book Seven Sketches is going well. Over 300 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 with us, please visit the Azimuth Forum and register:

Please 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.
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Faith is gaining new life across America every day (?)

Vice-President Pence writes:

“The percentage of Americans who live out their religion on a weekly basis — praying, going to church, reading and believing in the Bible — has remained remarkably consistent over the decades, even as the population of the United States has grown by leaps and bounds. I mean, think about it, today, relative to the population, four times as many Americans go to church on a regular basis than at the time of our nation’s founding. Religion in America isn’t receding. It’s just the opposite. Faith is gaining new life across America every day.”

Below is the main graph from the study he cites to back up his claim.

An author of that study, Landon Schnabel, says that Pence missed the key point about his study:

Pence’s language is “imprecise because it chooses one thing to highlight while leaving out an equally important part of the story of American religious change: Although intense religion persists, moderate religion is declining quickly,” Schnabel told The Fact Checker. “It may in fact be because of the intensity of American religion, and the perceived partisan polarization of American religion, that moderately religious people are leaving it behind.”

Schnabel added that one aspect of Pence’s remarks was

“wrong by more than omission: Weekly attendance is declining. It’s only more-than-weekly attendance that persists.” Once-a-week attendance has dropped from 28.5 percent in 1972 to 17.5 percent in 2014, according to the Association of Religion Data Archives.

Schnabel also said his research does not support Pence’s bottom-line conclusion — that “religion in America isn’t receding.”

“Our data do not support the conclusion that religion is on the rise in the United States,” Schnabel said. “Some of the coverage of our paper made that argument, including the Federalist coverage, but our study simply shows that intense religion is persistent even as moderate religion declines. Subsequently, of those who remain religious, a higher proportion are intensely religious.”

He likened it to a “container getting smaller, but more concentrated,” such as espresso vs. coffee.

For more details:

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Gauge theory, cats and gravity

They say that cats always land on their feet. But not on Monday morning, before they've had enough coffee.

The way cats land on their feet uses gauge theory, which is how physicists combine symmetry and geometry. It relies on the fact that rotations don't commute. Try rotating a book 90 degrees about the x axis and then 90 degrees about the y axis. Then do these operations in the other order. You'll get different results! It's not obvious, but this is the key to how cats rotate themselves. When floating in empty space you can't move yourself forward by wriggling around, because translations commute. But you can rotate yourself by wriggling around, because rotations don't!

For details, try this:

• Richard Montgomery, Gauge theory of the falling cat,

For a simple overview, try this - with pictures:

I thank +Charles Filipponi for pointing out this gif.

Animated Photo
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Announcing a new journal: Compositionality

Yay! Our new journal is finally here! It's all about building big things from smaller parts. Another proposed title that didn't make the cut: Applied Category Theory. For details, go here:

and for even more details, check out the links there.
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Applied Category Theory 2018 - live streaming video

You can watch our workshop now, here:

or at the link below. The videos will also be available permanently, on YouTube.

You can see the schedule here:

The times are those in the Netherlands - that is, GMT +2.
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Some of us took a break from the Applied Category Theory workshop and went to Amsterdam yesterday. We saw a great piece of art at the Stedelijk Museum.

When you walk in, you see an enormous concrete block hanging from the ceiling. Ho hum, you think: typical modern art.

But wait: it's slowly rotating in strange ways.

And wait: there are no wires - it's floating in midair!

Then the fun starts, as you try to figure out what's going on.

I had a moment of fear, where I instinctively thought: should I get out of here? If the usual laws of nature seem to vanish, and you find yourself in the same room with a hundred-ton concrete block that can freely move any way it wants, maybe you should quickly get out.

However, since we have no technology that can make such a thing possible, I immediately sought other explanations for what I was seeing.

Besides, I reassured myself, it's unlikely that the museum director would want to murder visitors in this way.

The show could have been made more intense by having the huge block occasionally make sudden threatening motions. But I think this would prompt lots of complaints, and maybe even lawsuits from museum-goers who fell over or suffered heart attacks.

This piece, called Drifter, was part of an exhibition by Studio Drift, a collective of artists who use technology in clever ways:

Check out this video:

Good stuff!

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Props in network theory

Long before the invention of Feynman diagrams, engineers were using similar diagrams to reason about electrical circuits and more general networks containing mechanical, hydraulic, thermodynamic and chemical components. We can formalize this reasoning using categories called props, which have natural numbers as objects. In this approach, each kind of network corresponds to a prop, and each network of this kind is a morphism in that prop. A network with m inputs and n outputs is a morphism from m to n. Putting networks together in series is composition, and setting them side by side is tensoring.

In this paper, two of my students and I work out the props for various kinds of electrical circuits:

• John Baez, Brandon Coya and Franciscus Rebro, Props in network theory.

We start with circuits made solely of ideal perfectly conductive wires. Then we consider circuits with passive linear components like resistors, capacitors and inductors. Finally we turn on the power and consider circuits that also have voltage and current sources.

And here’s the cool part: each kind of circuit corresponds to a prop that pure mathematicians would eventually invent on their own! So, what’s good for engineers is often mathematically natural too.

We describe the ‘behavior’ of these various kinds of circuits using morphisms between props. In particular, we give a new proof of the black-boxing theorem that I proved with +Brendan Fong. Unlike the original proof, this new one easily generalizes to circuits with nonlinear components! We also use a morphism of props to clarify the relation between circuit diagrams and the signal-flow diagrams in control theory.

For a quick tour of the main ideas, check out this blog article!
Props in Network Theory
Props in Network Theory
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