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May 28, 2018

Applied Category Theory: Ordered Sets

Posted by John Baez

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 links to 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 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”.

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!

Posted at May 28, 2018 7:39 PM UTC

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Re: Applied Category Theory: Ordered Sets

Hi John. It’s great that you’re doing this course, and that registration figure is impressive!

However… there is something near the very start of your course that I really strenuously disagree with. You’ve already heard some version of what I’m about to say, because I said it to everyone at the Applied Category Theory meeting we both attended in Leiden. But at the time I didn’t realize we differed on this.

Here it is. Lecture 2, “What is Applied Category Theory?”, begins with the words

“Applied category theory” is fairly new.

and then a bit later you say that the term “applied category theory”

also mainly means applications outside computer science.

You might mean it that way, but I don’t, and I think the usage you’re encouraging is an awful mistake.

It also seems unnecessarily rude to those who apply category theory to computer science. That’s a set of applications at least as deserving of the term as any other. And as you mention, applications of category theory to computer science have been important for decades. That stuff is applied category theory in any reasonable sense of the phrase, and it’s not new. And honestly, if that was my research area — which it’s not — then I’d be pretty pissed off to hear someone influential saying that applied category theory was new, or that the applications of category theory I work on don’t count.

(I know you have abundant respect for the applications of category theory to computer science, because I know you personally. But I think your restrictive usage of the term “applied category theory” could easily be construed as disrespectful, even though you surely don’t intend it to be.)

Let’s step back and think about the more general term “applied mathematics”. For historical reasons, most mathematicians’ default interpretation of “applied mathematics” is much narrower than it could be. It’s usually understood to mean something like “methods of analysis applied to physical problems”. Someone who studies the Navier–Stokes equations will typically be called an applied mathematician, even if they’ve never touched a fluid in their research life. But someone who applies number theory to cryptographic systems, or knot theory to problems of genetic recombination, or category theory to the design of programming languages, will usually not be. Whether someone is called an “applied mathematician”, or is said to do “applied mathematics”, has little to do with their proximity to any actual application. It’s just a product of how language and culture have evolved.

I hope mathematicians and other scientists hurry up and realize that there’s a glittering array of applications of mathematics in which non-traditional areas of mathematics are applied to non-traditional problems. It does no one any favours to keep using the term “applied mathematics” in its current overly narrow sense. People coming up with exciting and unconventional applications of mathematics (including exactly the kinds of things you’re lecturing about) fully deserve the credit for doing applied mathematics. We should fight for the term “applied mathematics” to be used in the broadest and most inclusive way possible.

For much the same reasons, I’m dismayed (and actually a bit shocked) to see you declaring that applied category theory “mainly means applications outside computer science”. Why exclude them? It doesn’t make sense as a use of language, it’s potentially insulting, and it gives a false impression. After all, when we’re talking about applications of category theory outside mathematics, applications to computer science are the most conspicuous success story of all.

Of course, you and others are free to use “applied category theory” to mean anything you want, and if for some reason you want to exclude computer science, you can. But that would be like me declaring:

“Californians” mainly means people from California who aren’t called John.

We get to decide how to use language. You’re giving a course to hundreds of people; you’ll have an influence on how the term “applied category theory” comes to be used. Why would you want to exclude applications to one particular field? Obviously you can exclude applications to computer science from your lectures — that’s not the issue — but why would you want to exclude them from the term “applied category theory” itself?

Posted by: Tom Leinster on June 2, 2018 4:36 PM | Permalink | Reply to this

Re: Applied Category Theory: Ordered Sets

Tom wrote:

However… there is something near the very start of your course that I really strenuously disagree with. You’ve already heard some version of what I’m about to say, because I said it to everyone at the Applied Category Theory meeting we both attended in Leiden. But at the time I didn’t realize we differed on this.

I didn’t understand what you were getting at, back then. Now I think I do, because you’ve explained it more clearly.

I’ll change my usage of “applied category theory”, or maybe stop using that term.

As you know, I crank out lots of prose. That means I often don’t think very carefully about the ramifications of what I’m saying. I’m pretty careful to qualify statements about mathematical or scientific facts, to indicate how certain I am and why, etc. Mathematicians are trained to do that! But I’m not always careful to calibrate the social consequences of what I say. Often if something sounds good at the time I’ll just say it, without any very good reason. Sometimes I have to clean up messes later, like now.

For much the same reasons, I’m dismayed (and actually a bit shocked) to see you declaring that applied category theory “mainly means applications outside computer science”. Why exclude them?

When I wrote those words in Lecture 2, it was because a lot of people registering for the course said they were Haskell programmers wanting to understand monads and such. I was afraid they’d be disappointed, because the course is not about that at all. That was the main reason for saying this.

I’ll try to rewrite the lecture so it talks more about this course and less about some general concept of “applied category theory”, which is not anything I really want to define.

There are a few other issues, which I’ll mention in other comments.

Posted by: John Baez on June 3, 2018 8:00 PM | Permalink | Reply to this

Re: Applied Category Theory: Ordered Sets

For a while I’ve been using “applied category theory” to refer to some new wave of applications outside the already well-established applications to:

  1. certain portions of mathematics
  2. certain portions of computer science
  3. certain portions of mathematical physics

I now agree it’s dumb to use the term this way. But anyway, one of my main quests these days is to extend the applications of category theory to new subjects. And I often find myself wanting some term for this quest, preferably something less scary than “completing the takeover”.

Maybe there shouldn’t be any term for it. A name would be helpful for rallying people, but you could argue that any “rallying” tends to create an “in crowd” that excludes others as the “out crowd”. I don’t know.

I have one particular line of work that I call “network theory”, which applies category theory to study networks. This is specific enough and clear enough to be fairly unproblematic, I hope. Admittedly other people study networks in very different ways, and talk about “network theory” in a way that talks about graphs a lot but doesn’t mention categories. But I think this is okay: ultimately the study of networks will use whatever tools are helpful, and I’m just doing the usual thing of working within some field, advocating the use of certain tools, and seeing if anyone is interested.

The issue becomes more complicated when I find myself wanting to ally myself with other people who are applying category theory in related but different ways, like Bob Coecke and David Spivak and Spencer Breiner. We have a lot of interests in common, so we found ourselves needing some name for this common territory. So far it’s been “applied category theory”. For example, someone started a website:

and a bunch of us put together that workshop and school you went to:

and now we’re planning a repeat, which so far has the obvious name “Applied Category Theory 2019”.

But Bob Coecke, for example, dislikes the term “applied category theory” — for reasons that seem quite different from yours, Tom. So, the journal we’re starting didn’t get that name; it’s called

And of course, a bunch of people dislike that title too!

My personal solution may be to steer clear of this naming business in the future. But workshops and conferences and journals need names, so while I can dodge it, the problem won’t go away. Luckily, some people enjoy making up names for things and arguing about them.

Posted by: John Baez on June 3, 2018 8:45 PM | Permalink | Reply to this

Re: Applied Category Theory: Ordered Sets

When I talk about that field (in which I partially work as well) to other mathematicians, I usually refer to it as “categorical network theory”, or sometimes as “categorical complexity (theory)”. I find it the most descriptive, at least to other mathematicians.

I don’t know if it is any good for all the other reasons (rightfully) pointed here by you and Tom. But I just thought I’d mention that :)

Posted by: Paolo Perrone on June 4, 2018 12:21 AM | Permalink | Reply to this

Re: Applied Category Theory: Ordered Sets

What about just something like “New Applications of Category Theory?”

Of course, that has the problem mentioned elsewhere of not specifying whether you mean “applied outside math” versus “applied inside math”, but so does “applied category theory”.

Posted by: Mike Shulman on June 4, 2018 5:20 AM | Permalink | Reply to this

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