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September 20, 2007

The Catsters’ Latest Hit: “Adjunctions”

Posted by John Baez

Who knew category theory could be so popular? The Catster’s videos on Monads have taken Youtube by storm! As of this moment, the first episode has garnered 3758 viewings, prompting one disgruntled teenager to suggest that the Catsters are cheating in the battle for high ratings:

> Its really funny that your most of
> your subscribers have no videos, are
> subscribed to only you, and have only your
> videos as favorites. Do I sense a youtube
> cheater?

Of course, we Catsters fans know it ain’t so — they’re just satisfying a hitherto unnoticed craving for math lectures on YouTube!

And the Catsters are on a roll. Now they’ve tackled adjoint functors, also known as ‘adjunctions’:

Their edgy, low-budget production makes for gripping cinema. This is the real stuff.

For more on monads and adjunctions, try old episodes of The Tale of n-Categories (which will soon be made into a movie starring Emma Watson and Orlando Bloom). In week83 I described adjunctions in an arbitrary 2-category; in week89 I did the same for monads, and in week92 I described how adjunctions give monads.

Posted at September 20, 2007 5:34 PM UTC

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Re: The Catsters’ Latest Hit: “Adjunctions”

Their edgy, low-budget production makes for gripping cinema.

Yes, there’s an electrifying moment which occurs at 9:27 in Adjunctions 2. :-)

Eugenia is really good at making these nice categorical points in under 10 minutes (the usual time limit on Youtube). Good sense of dramatic timing too; perhaps her experience as a musical performer has helped. (Although I didn’t recognize the tune she hums to the accompaniment of the pasting diagram equations for adjunctions!)

Keep ‘em coming, Catsters!

Posted by: Todd Trimble on September 20, 2007 8:26 PM | Permalink | Reply to this

Re: The Catsters’ Latest Hit: “Adjunctions”

I was on the edge of my seat by the end of Adjunctions 3.

“It’s a naturality square! Stop the clock! Stop the clock!”

That’s what cinema is all about. What a fantastic performance. Keep ‘em coming!

Posted by: Tim Silverman on September 20, 2007 11:12 PM | Permalink | Reply to this

Re: The Catsters’ Latest Hit: “Adjunctions”

Eugenia is really good at making these nice categorical points in under 10 minutes

You know, this 10-minute limit could give rise to a whole new genre — the minilecture! If you’re trying to learn something, but you can’t grok it from your textbook, what better than to have a bite-size 10-minute minilecture available, where somebody explains at writing speed exactly what’s going on, perhaps with a list of prerequisite minilectures that you might have to watch first.

If you ask me, universities should get together and put money into producing fantastic, free learning reasources like this. It wouldn’t take much cash, and it would be fantastically effective. But the people in charge are too old-fashioned! I’ll never forget my old GR lecturer, who said he wouldn’t make his powerpoint slides available, because then nobody would come to the lectures :). Perhaps that’s because spending an hour a day furiously writing down insanely complicated tensor equations, desperately trying to keep pace with the lecturer’s ability to hit the ‘next page’ key on his laptop, is not a good way to learn a subject! People can have surprisingly closed minds about teaching styles.

I have a feeling that being an undergraduate in 10 years time is going to be a lot less frustrating than it is these days. And I hope initiatives like the Catsters and SciVee.tv will get the ball rolling.

Posted by: Jamie Vicary on September 21, 2007 1:14 AM | Permalink | Reply to this

Re: The Catsters’ Latest Hit: “Adjunctions”

If we could convince Danica McKellar to explain percolation theory, then minilectures would really take off!

Well, you’d get instantly Slashdotted, which might not be the same thing as success.

Speaking in somewhat more practical terms, starting a “snowball” and making online minilectures a common thing to do would probably require starting with material which would instantly capture the attention of many teachers. Unfortunately, the material in, say, first-year quantum mechanics probably feels like such “old hat” that few professors would get fired up about doing it.

Posted by: Blake Stacey on September 21, 2007 8:12 PM | Permalink | Reply to this

Re: The Catsters’ Latest Hit: “Adjunctions”

Allan wrote:

Unfortunately, the material in, say, first-year quantum mechanics probably feels like such “old hat” that few professors would get fired up about doing it.

I’ve never taught first-year quantum mechanics, so I’d have an absolute blast teaching it — and I’m a real hambone, so I’d love getting videotaped doing it!

Unfortunately, it would be so much work figuring out how to teach first-year quantum mechanics that I can’t imagine doing it without some external motivation: actualy teaching it in a physics department, say.

And unfortunately, the professors in our physics department haven’t encouraged me to teach quantum — that’s one of the courses they like to teach. I’d have a lot easier time getting to teach basic classical mechanics… and I’d enjoy teaching that too.

(I’m getting a bit burnt out on teaching lower-level math courses: I like it, but I’ve just done them all too many times over the 15 years.)

Anyway: with a little luck you may see me on some videos soon. I’m not going to let the Catsters have all the fun!

Posted by: John Baez on September 22, 2007 6:24 PM | Permalink | Reply to this

Re: The Catsters’ Latest Hit: “Adjunctions”

I’m a real hambone

A real hambone? I don’t know what that means, but I’m sure it’s not pukka.

Posted by: Jamie Vicary on September 22, 2007 11:13 PM | Permalink | Reply to this

Re: The Catsters’ Latest Hit: “Adjunctions”

Jamie wrote:

A real hambone? I don’t know what that means, but I’m sure it’s not pukka.

I’m not using it in any of the senses that appear in the Urban Dictionary, that’s for sure — those are seriously un-pukka.

In my world, a ‘hambone’ or ‘ham’ is someone who likes to ham it up. Get me teaching math to an audience of undergrads and I’ll start telling jokes, bouncing around and imitating mathematical objects, like a tangent plane or a hyperboloid of revolution. The bigger the audience, the more I ham it up. Talk to me alone in person and I’ll be incredibly dull — you’ll see soon enough.

Posted by: John Baez on September 22, 2007 11:44 PM | Permalink | Reply to this

Re: The Catsters’ Latest Hit: “Adjunctions”

I am not a mathematician, but without any hint of irony I have to say I think Category theory is one of the most exciting subjects around… though to be honest, I’m not sure I could explain why I think that. I just drop by occasionally to gawp in amazement (and incomprehension).

On finding the Catsters on YouTube I thought “Aha! Enlightenment awaits!”, only to discover that having only the rusty maths of a very old Physics-degree, I am not one of the intended audience - which is fair enough.

But while you and they are boldly going, there are those still stuck in a mathematical sub-basement somewhere, groping for the light switch, who would nonetheless like a nice safe tour flight.

Don’t stop the videos for starship pilots on how to perform an emergency orbital insertion at warp speed, but if there is any chance of a Flight Safety Video (“to fasten your seatbelt join the ends like this…”) I would be the first to book..

And speaking of books, I read “Gauge Fields, Knots, and Gravity” some years back and thought it the of the most comprehensible book containing advanced mathematics that I have ever read.

I had wished (and still do) for something similar on Category Theory but now maybe YouTube could provide an alternative.

Category Theory for Dummies - or Cat Trek perhaps?

Posted by: Julian Moore on September 23, 2007 4:04 PM | Permalink | Reply to this

Re: The Catsters’ Latest Hit: “Adjunctions”

You’re right — so far, the Catsters’ videos start by assuming you’re comfortable with categories, functors, and natural transformations, as well as some other math. It takes quite a while to get comfy with that material.

I hope that later the Catsters will do some more elementary videos, which would surely have a wider audience and a bigger impact on the universe overall.

I can easily imagine having fun creating some videos explaining math and physics at an elementary level. I’m trying to get the technology up and running to host such videos on my website — the 10-minute limit imposed by YouTube is too confining for a blabbermouth like me. If I succeed, the first videos will be of the course on ‘Geometric Representation Theory’ that I’m about to start co-teaching with James Dolan. These will be aimed at people who want to see some mind-bogglingly new mathematics. If I succeed in getting the bugs worked out of the technology, I might branch out and make videos on topics of broader interest. Stay tuned.

(I’ve also long dreamt of writing pop-level books on math and physics, but writing books takes a lot of time, which is the one thing I don’t have enough of now.)

Posted by: John Baez on September 24, 2007 2:23 AM | Permalink | Reply to this

Re: The Catsters’ Latest Hit: “Adjunctions”

John Baez wrote,

I’m trying to get the technology up and running to host such videos on my website — the 10-minute limit imposed by YouTube is too confining for a blabbermouth like me.

You can also try Google Video, which purports to have no length or file-size limit. (Some people report having issues with files larger than a gigabyte, but I’ve definitely seen hour-long lectures hosted there.) In addition, there is the Internet Archive, which was around first but (at least for me) seems to have a buggier video-embedding feature.

Posted by: Blake Stacey on September 24, 2007 5:42 PM | Permalink | Reply to this

Re: The Catsters’ Latest Hit: “Adjunctions”

Julian, you’re exactly the sort of person for whom I write the main expository line on my own weblog. You might try reading my series on category theory, which currently starts here. I will get it better organized soon enough, but for now you can read through it in reverse-chronological order from the beginning.

Posted by: John Armstrong on September 24, 2007 2:49 AM | Permalink | Reply to this

Re: The Catsters’ Latest Hit: “Adjunctions”

I’m trying to go even more slowly than John on my blog, also on wordpress. It starts here:

http://reperiendi.wordpress.com/2007/09/12/monoids/

(Of the nineteen posts that turn up on the first page of hits for the tag “category theory” on wordpress, thirteen are John’s, four are mine, and two are by people I don’t know.)

Posted by: Mike Stay on September 24, 2007 5:56 PM | Permalink | Reply to this

Re: The Catsters’ Latest Hit: “Adjunctions”

Nice posts on monoids and categories, Mike! I bet this is the sort of exposition Julian Moore is seeking — full of fun examples, full of sentences in plain English…

But, you’re supposed to be finishing that ‘Rosetta Stone’ paper, where you explain how symmetric monoidal closed categories show up in many different guises in logic, topology, and physics.

Oh, but wait — I’m supposed to be writing that paper too.

Posted by: John Baez on September 24, 2007 6:56 PM | Permalink | Reply to this

Re: The Catsters’ Latest Hit: “Adjunctions”

But, you’re supposed to be finishing that ‘Rosetta Stone’ paper, where you explain how symmetric monoidal closed categories show up in many different guises in logic, topology, and physics. grumpy

(Ducks fireball fireball) No, wait! I’ve been working on it! I understand what I’m supposed to be writing about a lot better now. For example, here’s what it means to be a multiplicative intuitionistc linear logic (MILL):

  • in linear logic, propositions are resources, like a bolt. We can’t use the same bolt to hold on the wheel of my bike and the door of a car!

  • in multiplicative logic, we can get new resources by sticking others together. For example,

    sandwich = bread \otimes peanut butter \otimes jam \otimes bread

    Note that the order matters! The empty resource is denoted II.

  • in intuitionistic logic, a proposition is true iff there’s a proof of it, and a proposition is false iff there’s a proof of its negation. So statements that were tautologies in classical logic, like P¬P,P \wedge \neg P, don’t hold any more.

    But what’s the proof of a resource!? It’s a process for taking a resource and turning it into something else:

    grind : wheat \to flour.

    There can be lots of different ways to transform a resource of one kind AA into a resource of another kind BB. The set of ways to do that is denoted AB.A \vdash B.

    A description of a process is a recipe, and a recipe is a resource itself. It’s a proposition that the set ABA \vdash B is nonempty, and is denoted AB.A \multimap B. A proof of ABA \multimap B from no assumptions is a process

    f:IABf:I \to A \multimap B

    that produces (from nothing) a recipe for converting an AA into a BB—a description of an element of the set ABA \vdash B! For each process in the set, there’s a corresponding recipe, and that’s currying.

I’ve also understood the difference between Hilbert style proofs (all the complexity in the axioms, with only one inference rule modus ponens) and Gentzen-style proofs (one axiom, hyp, and lots of inference rules).

I understand how the axioms that are usually given aren’t really axioms, they’re “axiom schemata,” which says that any process of the proper form is an axiom. The usual inference rules work the same way: for every process of a given form, we get another process of this other form.

This comes into play when we try to translate from the function symbols in the theory of a braided monoidal closed category (BMCC) into the language of MILL: they’re really specific processes, but we can encode them as inference rules that mention specific resources in the same way that the left and right unitors mention the void resource.

Also, I drew up dinatural transformation diagrams that show how the axiom hyp says that composition has left and right units, while the inference rule cut says that composition is associative.

I pared down Lambek & Scott’s definiton of a cartesian closed category (CCC) until I was left with a BMCC. Now I’m working on pictures for the string diagrams.

Posted by: Mike Stay on September 24, 2007 10:10 PM | Permalink | Reply to this

Re: The Catsters’ Latest Hit: “Adjunctions”

Mike and John, perhaps you might be interested to know that intuitionistic logic is part of Martin-Lof’s type theory which can be used to construct formal (or point-free) topology. Here is a one page web intro [1] which roughly describes what formal topology is, and then there are many papers on the subject.

I like this paper [2] for three reasons:

(1) It is a pretty good intro and survey of the literature.

(2) It contains the most fundamental definition of convergence that I have ever seen.

(3) It is written by the founding father of formal topology, Giovanni Sambin.

However, this web version of the paper’s print looks pretty lousy, but it is effectively readable if you print out the paper (though the print resolution could be better). Or you could look at the print version of the paper in Theoretical Computer Science 305 (2003) pp.347-408.

If this does interest anyone then there are also various papers about the relationships between formal topology and categories and topoi.


[1] http://www.3wftop.math.unipd.it/formal-topology.html

[2] http://www.math.unipd.it/~sambin/txt/SP.pdf

Posted by: Charlie Stromeyer Jr on September 25, 2007 12:44 AM | Permalink | Reply to this

Re: The Catsters’ Latest Hit: “Adjunctions”

Unfortunately, the material in, say, first-year quantum mechanics probably feels like such “old hat” that few professors would get fired up about doing it.

I don’t really understand what you mean here…but in any case you only need one professor to get fired up about doing it!

By the way thanks for the affirmation everyone.

Posted by: Eugenia Cheng on September 22, 2007 12:34 AM | Permalink | Reply to this

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