## March 22, 2011

### Higher Gauge Theory, Division Algebras and Superstrings

#### Posted by John Baez

I’m giving two talks at Hong Kong University this week:

These are roughly the first talk of my new life, and the last of my old. We’re chatting about talk 1 over on Azimuth, here and here. But the n-Café is the right place for chatting about talk 2!

I’m giving my talk Higher gauge theory, division algebras and superstrings as part of the Workshop on Geometry and Lie Groups organized by various folks including my old friend Jiang-Hua Lu. It’ll be great to see her again. I met her back in the early 1990’s. She was doing at postdoc at MIT. I was on leave from U. C. Riverside, teaching for 2 years at Wellesley College to be closer to Lisa Raphals (who later came to Riverside and married me).

Lu was a student of Alan Weinstein, the guru of groupoids and classical mechanics. Around the time I met her, she did some great work on Hopf algebroids obtained by quantizing Poisson Lie groupoids. So, it’s quite fitting that this workshop features a talk by Peter Bouwknegt on ‘Leibniz algebroids’. Leibniz algebras are like Lie algebras without the antisymmetry of the bracket. They show up a lot more often than you might think. So if Lie algebras get to have Lie algebroids, Leibniz algebras deserve to be oidified too.

I’ll also be happy to see my old grad school friend Mathai Varghese, who was a student of Quillen and whose thesis became the Mathai-Quillen formalism. He’s giving a talk on fractional index theory.

And I’ll be glad to see David Vogan, who was a professor at MIT back when Mathai and I were in grad school… and who still is! He’s one of the people who mapped E8—as the journalists rather vaguely put it. As a student I was terrified of him because he knew infinitely more about Lie groups than I did. Somehow I’ve lost that fear, even though he still knows infinitely more about Lie groups. I guess I realized at some point that smart people don’t actually emit a ‘death ray’ that kills anyone stupid within a 3-meter radius.

But enough gossip! What’s my talk about?

It’s all about John Huerta’s thesis. I love this thesis because it takes three old dreams of mine and shows how they fit together:

• Higher gauge theory: This is the dream that all the wonderful mathematics associated to gauge theory generalizes from point particles to higher-dimensional objects (strings and branes). This wonderful mathematics includes Lie groups, Lie algebras, fiber bundles, connections on fiber bundles, and much more. The generalization proceeds by taking all the sets in sight, and replacing them by categories, 2-categories and so on. A theory that simultaneously generalizes all this wonderful mathematics in such a natural way would be a magnificent thing even if it had no applications to physics—but a few people actually think the world might be made of higher-dimensional objects like strings and branes!
• Division algebras: It’s a stark and stunning fact that there are only four finite-dimensional normed division algebras: the real numbers, the complex numbers, the quaternions (which are noncommutative) and the octonions (which are also nonassociative). The first two of these are fundamentally important throughout mathematics, the third is useful in 3-dimensional and 4-dimensional geometry, and the fourth… who ordered that? A lot of ‘exceptional’ objects in mathematics, most notably all the exceptional Lie groups, are related to the octonions. But that doesn’t really answer the question of “what are the octonions good for?” It just shows that they can’t be brushed aside as an isolated curiosity. The dream is that the octonions are somehow relevant to our universe, and that their quirkiness somehow explains some of the quirky features of its fundamental laws.
• Superstrings: As a physicist I was never very fond of superstring theory, because there’s no evidence for “supersymmetry”, the idea that every boson has a partner fermion. But as a mathematician, I couldn’t help but want to learn a bit about it. It has a eerie way of making powerful connections between disparate ideas. And I couldn’t help being curious about one of the basic quirky facts about superstring theory: it works best in 10 dimensions. This is a very rough statement, which needs a lot of qualifications to become precise. But there’s something true about it, and part of why it’s true is that 10 = 8+2, with 8 dimensions corresponding to the octonions and 2 corresponding to the 2 dimensions of a string’s worldsheet. This fact, known for a long time, is something I’ve been itching to understand better, ever since I heard about it.

You can see already from my discussion that these three dreams fit together. That’s been known for quite a while, thanks to the work of many people. But John Huerta put the pieces together in a way that’s so clear that even I can understand it. Briefly:

Abstract: Classically, superstrings make sense when spacetime has dimension 3, 4, 6, or 10. It is no coincidence that these numbers are two more than 1, 2, 4, and 8, which are the dimensions of the normed division algebras: the real numbers, complex numbers, quaternions and octonions. We sketch an explanation of this already known fact and its relation to “higher gauge theory”. Just as gauge theory describes the parallel transport of supersymmetric particles using Lie supergroups, higher gauge theory describes the parallel transport of superstrings using “Lie 2-supergroups”. Recently John Huerta has shown that we can use normed division algebras to construct a Lie 2-supergroup extending the Poincaré supergroup when spacetime has dimension 3, 4, 6 or 10. He also used them to construct a Lie 3-supergroup when spacetime has dimension 4, 5, 7 or 11. The 11-dimensional case is related to 11-dimensional supergravity, and thus presumably to “M-theory”.

For a more detailed but still very gentle tour, try the slides of my talk!

If the talk is too technical for you, wait until May. Then you can read a Scientific American article that John Huerta and I wrote about the same subject. It’ll be called “The strangest numbers in string theory”.

(In case you’re wondering, the author doesn’t get to pick the title. We’d unimaginatively suggested “The octonions”, but to most people that’s about as alluring as “The szymidh”. Probably even less.)

A note for the experts: my talk slides don’t mention the fact that a somewhat more sophisticated Lie 2-supergroup extension of the Poincaré supergroup seems required for anomaly cancellation, as Urs pointed out here. I’ll say that out loud, though…

Posted at March 22, 2011 7:21 AM UTC

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### Re: Higher Gauge Theory, Division Algebras and Superstrings

As a physicist I was never very fond of superstring theory, because there’s no evidence for “supersymmetry”,

That does not sound like good reason not to be fond of superstring theory. The generic superstring vacuum gives an effective theory (which we might observe) that has no global supersymmetry. But the fact that the superstring is locally supersymmetric on its worldsheet implies that whatever the effective theory is, it will contain fermionic fields (not so for the bosonic string) and that is certainly something that there is strong evidence for.

Posted by: Urs Schreiber on March 22, 2011 6:01 PM | Permalink | Reply to this

### Re: Higher Gauge Theory, Division Algebras and Superstrings

I’d like to hear a bit more about Mathai Varghese’s talk, if you get a chance to report back. The abstract is about trying to make sense of the index theorem for manifolds which aren’t spin. His coauthors are Melrose and Isadore Singer.

I’m interested in that topic. I’d like to know how their approach relates to the gerbe-ish approach of Michael Murray and Michael Singer, who showed that you could interpret all the usual “Clifford module” stuff in index theory quite nicely via the “spin gerbe”.

Posted by: Bruce Bartlett on March 23, 2011 6:50 PM | Permalink | Reply to this

### Re: Higher Gauge Theory, Division Algebras and Superstrings

I could ask him if he has slides or notes he is willing to share. But from my experience Mathai works from his own hand-written notes, so it’s a long shot. Being there would be the best option. :)

Posted by: David Roberts on March 24, 2011 3:01 AM | Permalink | Reply to this

### Re: Higher Gauge Theory, Division Algebras and Superstrings

Ok, thanks. Well, maybe I’ll be lucky and you’ll give a brief summary of his talk afterwards…

Posted by: Bruce Bartlett on March 24, 2011 10:43 AM | Permalink | Reply to this

### Re: Higher Gauge Theory, Division Algebras and Superstrings

Well, I won’t be there either, so I’ll have to rely on the kindness others to let me know what it was about.

In other news, Mathai was just elected to be a fellow of the Australian Academy of Science. This is one of the highest honours available to mathematicians in Australia.

Posted by: David Roberts on March 25, 2011 2:58 AM | Permalink | Reply to this

### Re: Higher Gauge Theory, Division Algebras and Superstrings

Bruce wrote:

I’d like to hear a bit more about Mathai Varghese’s talk, if you get a chance to report back.

I’ve been too busy here to read email until now. Mathai’s talk in in 7 minutes!

I’ve been talking with him a lot in the last few days, not about index theory, but about nonassociative tori and T-duality. “Nonassociative torus” is actually a mildly misleading name for an associative algebra living in some deformation of the usual monoidal category of representations of a torus. The associator has been tweaked, so these algebras look nonassociative “from the outside”, but inside the monoidal category they look perfectly nice and associative.

Of course the “tweaking” consists of an element of $H^3(\widehat{T}, U(1))$ where $\widehat{T} \cong \mathbb{Z}^n$ is the dual of the torus $T \cong U(1)^n$.

Posted by: John Baez on March 26, 2011 2:59 AM | Permalink | Reply to this

### Re: Higher Gauge Theory, Division Algebras and Superstrings

Here are the slides for Mathai’s talk:

He described a concept of a ‘projective’ differential or pseudodifferential operator on a vector bundle, which allowed him to give a topological interpretation of the $A$-hat genus for manifolds that aren’t spin.

I asked him if one of these ‘projective’ operators could be thought of as an operator on a vector bundle twisted by some gerbe. He said yes, and his slides certainly seem to point toward such a viewpoint, but I didn’t get a strong sense that this viewpoint has been explicitly worked out yet.

If you want to know, Bruce, you could either ask him or get me to do it. (It seems like he and I might write a little paper on nonassociative tori.)

Posted by: John Baez on March 27, 2011 3:20 PM | Permalink | Reply to this

### Re: Higher Gauge Theory, Division Algebras and Superstrings

I asked him if one of these `projective’ operators could be thought of as an operator on a vector bundle twisted by some gerbe.

That sounds like something I’d be interested in. But I can’t get to the slides. I think your link is broken.

Posted by: Chris Rogers on March 27, 2011 8:36 PM | Permalink | Reply to this

### Re: Higher Gauge Theory, Division Algebras and Superstrings

Hi Chris, try this link to Mathai’s slides (JB was missing the .pdf)

Posted by: David Roberts on March 27, 2011 9:22 PM | Permalink | Reply to this

### Re: Higher Gauge Theory, Division Algebras and Superstrings

Thanks David!

Posted by: Chris Rogers on March 27, 2011 10:14 PM | Permalink | Reply to this

### Re: Higher Gauge Theory, Division Algebras and Superstrings

I fixed the link. Thanks for catching that.

Posted by: John Baez on March 28, 2011 11:56 AM | Permalink | Reply to this

### Re: Higher Gauge Theory, Division Algebras and Superstrings

I wonder if Bernd Sturmfel’s paper Can biology lead to new theorems? is of relevance to the “Magnitude programme” of Simon Willerton and Tom Leinster? I’m thinking about the “Dress-Bandelt split decomposition” theorem mentioned in that paper, which shows that every metric on a finite set decomposes into a product of prime metrics, in some sense.

Posted by: Bruce Bartlett on March 23, 2011 7:32 PM | Permalink | Reply to this

### Re: Higher Gauge Theory, Division Algebras and Superstrings

The constructions in Sturmfels’ paper are disappointing to me. I mean the examples of the kind, if I make such and such crude model of biological reality than it will have such and such very special cool features. Modeling in general has lots of cute special cases which can be toy of their own. But once you add more complexity to the modeling you get out of special circumstances. Biology asks for very robust mathematics to be and stay relevant. In physics there are some simple equations which are really essential and hold in some sense at a very fundamental level. They are really laws of nature in some sense (even if not a theory of everything they appear in some well defined limits), and the deeper you go into studying them mathematically can pay back in understanding the physics. The examples in this paper are good mathematics, but there are tens of thousands of similar special models in mathematics; if I would to attach their special importance because they are from biology, I would expect a model which will not become obsolete with modeler’s wish to include one more parameter or aspect of the story into consideration.

Posted by: Zoran Skoda on March 24, 2011 12:53 AM | Permalink | Reply to this

### Re: Higher Gauge Theory, Division Algebras and Superstrings

I found it was a nicely written paper, but it seemed to go in a strange direction. He started out asking if biology can lead to new theorems, giving me the impression that we were were going to see the direction biology -> maths, but instead it seems the results are more like some nice theorems from pure mathematics which are of relevance to biology, so it seems to go maths -> biology.

Posted by: Bruce Bartlett on March 24, 2011 10:47 AM | Permalink | Reply to this

### Re: Higher Gauge Theory, Division Algebras and Superstrings

Looking in our existing math toolkit for tools to attack problems in biology may be missing the boat.

After all, calculus etc were invented to attack problems in physics.

Posted by: jim stasheff on March 24, 2011 2:37 PM | Permalink | Reply to this

### Re: Higher Gauge Theory, Division Algebras and Superstrings

Sturmfels’ paper asks not “what mathematics can do for biology” but “what biology can do for mathematics”. This way of posing the question tends to let mathematicians continue thinking about concepts they already like… and find connections to biology here and there. But it’s possible that ultimately biology will do more for mathematics if mathematicians take a more “submissive” role and try very hard to understand biology. It may not work, but it may lead to something truly new. Something like this is what led mathematicians thinking about physics to invent calculus. Newton’s Principia did contain new theorems about Euclidean geometry, but that’s not what we remember it for.

Thinking about this gave me new respect for Gauss’ line about mathematics being the “queen and handmaiden of the sciences”. The submissive role of handmaiden may be inseparable from the majestic role of queen.

Posted by: John Baez on March 28, 2011 12:10 PM | Permalink | Reply to this

### Re: Higher Gauge Theory, Division Algebras and Superstrings

Related to this point, rather than the main thread questions, A Symbolic Computational Approach to a Problem Involving Multivariate Poisson Distributions by Eduardo Sontag and Doron Zeilberger is an interesting paper tackling some chemical/biological network coefficient computation. Now, it certainly fits Bruce’s point above that it’s really still taking existing maths and making minor extensions to fit it to a model class, rather than taking a model class one is genuinely interested in and figuring out some mathematical properties of it, but it raises an interesting question of whether what’s in the paper is of “not purely being polite” mathematical interest?

Posted by: dave tweed on March 28, 2011 4:45 PM | Permalink | Reply to this

### Re: Higher Gauge Theory, Division Algebras and Superstrings

Thanks for pointing out the paper A symbolic computational approach to a problem involving multivariate Poisson distributions.

It’s about situations where you have a bunch of independent Poisson random variables $X_j$ with known Poisson parameters, and you want to know the distribution of one conditioned on the values of some linear combinations of these variables.

It give some ways to efficiently compute this distribution using ‘Wilf-Zeilberger theory’ and the ‘Apagodu-Zeilberger algorithm’. The authors wrote a Maple package that does the job.

Since I don’t know Wilf-Zeilberger theory or the Apagodu-Zeilberger algorithm, it’s hard for me to get deeply interested in this material.

However, Section 6, on biochemical applications, is fascinating to me! It’s about ‘chemical reaction networks’. Over on Azimuth, I’m just about to start explaining the connection between chemical reaction networks and quantum field theory. So, this is a topic I’m eager to hear more about.

The connection is that chemical reaction networks satisfying a certain technical condition have stable equilibrium states where the number $X_j$ of molecules of the $j$th type is a Poisson random variable, and these random variables are independent. So, the work in this paper applies.

But what I really like is that Section 6 is a self-contained detailed introduction to chemical reaction networks, with lots of examples, and references to a bunch of the most important papers on the subject!

Posted by: John Baez on March 29, 2011 1:58 AM | Permalink | Reply to this

### Re: Higher Gauge Theory, Division Algebras and Superstrings

In case I was a bit vague above, my point was that other than a couple of theorems that justify some problem transformations, it’s basically about symbolic algorithmic techniques that produce some recurrences and initial conditions for any given system. They look sufficiently complicated that I think most mathematicians would abandon this approach if forced to do it by hand. It’s clearly rigorous and useful if one is interested in results for a given network, but is that kind mathematics the kind of thing that will have an effect within “mathematics for it’s own intrinsic interest” (I don’t want to say pure maths, because that’s not quite what I mean)?

Posted by: dave tweed on March 29, 2011 9:07 AM | Permalink | Reply to this

### Re: Higher Gauge Theory, Division Algebras and Superstrings

Dave wrote:

They look sufficiently complicated that I think most mathematicians would abandon this approach if forced to do it by hand. It’s clearly rigorous and useful if one is interested in results for a given network, but is the kind of thing that will have an effect within “mathematics for its own intrinsic interest” […]?

Maybe you know this already, but:

Doron Zeilberger, one of the coauthors of that paper, is famous for using computers to prove complicated identities involving hypergeometric series. He credits his computer as a coauthor on many of his papers under the name Shalosh B. Ekhad. He argues that programming is even more fun than proving, and more importantly it gives as much, if not more, insight and understanding. He also argues that meta-understanding will replace understanding:

Is is high time that we, mathematicians, give up that obsession to follow all the details. Mathematics, even before computers, got so complicated, that it is hopeless to try and “understand” (in the local sense) all the results that one uses. It is fair game to use a proved theorem, approved by experts, as a macro (or “black box”), as long as one understands the statement, and the conditions under which it holds, without feeling obligated to (locally) understand its proof.

So even with human-generated mathematics, it is no longer possible to have perfect “local” “understanding. With computer-generated mathematics, it is entirely hopeless. We should learn to trust the output of the computer as a “black box”. Of course, computers, at present, are still programmed by those feeble-minded creatures called humans, so they should not be blindly trusted. But with the right “architecture”, both for hardware and software, and lots of quality-control tests, we should learn to trust computers more and more.

We also have to change our notion of “beauty”. Dear Uncle Paul Erdös, I have bad news for you. Most theorems do not have proofs in your divine book, even if their statements are “beautiful” (e.g. the Four Color Theorem). You probably already knew it, since I am sure that Kurt Gödel told you that there exist short (i.e. “pretty” in your eyes) statements with arbitrarily long proofs (i.e. “ugly” to your eyes), but I bet that you dismissed it as “metamathematical nonsense”. So indeed, if the definition of a “beautiful” proof is a “proof from the book”, i.e. that you can read and understand all the details of in five minutes, then, luckily, most results are not pretty, since, if you, a mere human, can understand it so well, it can’t be very deep.

[You may retort that an easy-to-follow proof is not necessarily easy-to-find, and you would be right, but if its proof is short, it is, at any rate, a posteriori trivial, even if it not a priori so.]

But, we humans should not despair. We just have to adjust, and learn to live with, computer-generated mathematics, and tweak both our notions of “understanding” and “beauty”. As for understanding, we should give up on “local”, micro-managing, understanding, and be happy with the global kind, i.e. trade understanding with meta-understanding, and if necessary, even with meta-meta-understanding. Understanding the algorithm that generated the proof, or even merely understanding the meta-algorithm that generated the algorithm that generated the proof.

So, it’s possible that what mathematicians will find most interesting about this paper is whether it advances Zeilberger’s controversial vision of mathematics.

I don’t think we can draw any general conclusions about math inspired by biology or chemistry from this case. The only real link between this paper and those subjects is the ‘chemical master equation’. And as I hope to show soon on Azimuth, this equation can also serve as the basis for mathematical reflections much more congenial to lovers of elegance as traditionally defined: we’ll see how it’s related to quantum field theory, category theory, algebraic geometry, and the like.

Posted by: John Baez on March 29, 2011 4:05 PM | Permalink | Reply to this

### Re: Higher Gauge Theory, Division Algebras and Superstrings

I might be inadvertantly coming across as sceptical in these comments, whereas I’m really trying to think about what it would look like for a different field (say biology) to lead to new mathematics. As mentioned, one of the interesting thing about this paper is the way that, despite being perfectly precise and rigorous, it doesn’t look like your typical mathematics paper (threre’s no new organising concepts, no significant theorems/classifications, etc). It would be interesting to look back and see what the assimilation-into-general-mathematics process for papers similar to this will turn out to be in the coming years.

(Personally, I’m both fascinated and inspired by Zeilberger and collaborators stuff, but my interests are influenced as much by interest in computational issues as mainstream mathematical views.)

Posted by: dave tweed on March 29, 2011 5:47 PM | Permalink | Reply to this

### Re: Higher Gauge Theory, Division Algebras and Superstrings

I enjoyed that paper of Sturmfels, and like you, Bruce, it led me to discover the work of Dress and co on finite metric spaces. It’s particularly striking that the complete bipartite graph $K_{3, 2}$ (with nodes $a, b, c, A, B$, and and edge from each small letter to each big one) appears as a minimal counterexample both there and in work on magnitude.

Specifically: interpret $K_{3, 2}$ as a five-point metric space, where distances are shortest path-lengths and a single edge has length $t$. Then for a suitable choice of $t$, this is a minimal example of a metric space without magnitude.

I spent a while leafing through the papers by Dress and co, but didn’t put enough time into it to understand the decomposition theorem you mention.

Posted by: Tom Leinster on March 24, 2011 6:51 PM | Permalink | Reply to this

### Re: Higher Gauge Theory, Division Algebras and Superstrings

Tom (or indeed anyone else), one thing that has been niggling me for some time, but I’ve not got round to thinking seriously about, is whether or not the notion of “tight span” or “injective hull” of metric spaces, as described in the above papers of Dress et al., generalizes to enriched categories. I feel sure that Lawvere would have considered such a thing but can’t pin it down in any of his papers that I’ve got. Any thoughts?

Posted by: Simon Willerton on April 7, 2011 2:25 PM | Permalink | Reply to this

### Re: Higher Gauge Theory, Division Algebras and Superstrings

I think that’s an excellent question. I’ve looked through Isbell’s original paper on injective hulls, and began to get an understanding of them, but didn’t get to thinking about possible generalizations to enriched categories. And I’m not aware that Lawvere discusses them anywhere.

A mystery that might be related is the question I posed to the categories list a month or two ago, and to which I got very little by way of answer. (A couple of people sent me interesting emails in private.) That was as follows. Let $A$ be a small category. Isbell conjugacy gives an adjunction between $Set^{A^{op}}$ and $(Set^A)^{op}$. Like all adjunctions, it restricts canonically to an equivalence between full subcategories, both of which I’ll call $I(A)$. What can you say about $I(A)$?

It’s fairly easy to see that $I(A)$ contains the Cauchy-completion of $A$, but in general it’s bigger. In the case of posets, it’s the Dedekind–MacNeill completion. At first people thought that $I(A)$ might be the category obtained by adjoining to $A$ all small limits and colimits non-freely, i.e. preserving all limits and colimits that are already there. But Isbell showed that it’s not that. And of course, one can ask the same question in the enriched setting.

I don’t know if these two questions are really related. It might be that all they have in common is the name of Isbell.

Posted by: Tom Leinster on April 7, 2011 8:09 PM | Permalink | Reply to this

### Re: Higher Gauge Theory, Division Algebras and Superstrings

There was some abstract description of injective hulls here. Don’t know if that could help.

Posted by: David Corfield on April 7, 2011 8:27 PM | Permalink | Reply to this

### Re: Higher Gauge Theory, Division Algebras and Superstrings

My current feeling is that the tight-span is related to the full subcategory of objects “satisfying Isbell duality” in the “Isbell envelope” of the metric space. I got these phrases really from the nlab entry on Isbell envelope, but there’s no references there. Andrew Stacey wrote the entry, it’s tied in with his paper on Comparative Smoothology, and I know he learnt about the terminology “Isbell envelope” from the categories mailing list (it was previously folklore, it seems), but I don’t know if the idea of objects “satisfying Isbell duality” is due to him or not. I’m hoping he will pop along in a little while and tell us.

Posted by: Simon Willerton on April 13, 2011 1:57 PM | Permalink | Reply to this

### Re: Higher Gauge Theory, Division Algebras and Superstrings

My enquiries on the category mailing list led back to Lawvere’s thesis as the origin of the basic idea. Apart from that, whilst there was a fair bit of folkloric stuff, there were very few actual references. Indeed, in the published version of Comparative Smootheology then I simply refer to Lawvere’s thesis.

The stuff on the nLab page on Isbell envelope was mostly me working out the details for Froelicher spaces (see the nLab page on http://ncatlab.org/nlab/show/Fr%C3%B6licher+spaces+and+Isbell+envelopes) but in a more general context. I make no claim to originality, but I also did not have any particular sources for that.

The only other reference that I can think of that might be relevant is a paper by Isbell called Adequate subcategories (details on lSpace: http://ncatlab.org/lspace/show/Adequate+subcategories). But it’s been a while since I looked at that paper so I’m not sure how relevant it will be.

I’ve not been following this discussion so I’m afraid I don’t have anything more to contribute. Is there an easy way to get up to speed? The preceding comments are full of stuff I don’t understand, and I’m unsure of the context! What little I did follow sounded interesting, and when I was looking at Isbell envelopes then I found it an attractive idea so I’d like to see if it is relevant or not.

Posted by: Andrew Stacey on April 13, 2011 7:18 PM | Permalink | Reply to this

### Re: Isbell envelope and Isbell duality

Okay, so I’ve now figured out how the thing Tom called $I(A)$ is the same as thing that Andrew called the “full subcategory of objects satisfying Isbell duality in the Isbell envelope”. In case anyone’s interested, I’ll explain it here. I hope to say what this has to do with metric spaces in another post. The relation between the two comes down, essentially, to the following category-theoretic observation – it took me a long time to spot this and is now kind of obvious to me – it’s probably some standard thing.

Theorem If $L: C\to D$ and $R: D\to C$ form an adjunction $L\dashv R$ then the following three categories are equivalent:

• the full subcategory of $C$, denoted $Fix(R L)$, consisting of the objects $P\in C$ such that the unit map $\eta_P\colon P\to R L(P)$ is an isomorphism;

• the full subcategory of $D$, denoted $Fix(L R)$, consisting of the objects $Q\in D$ such that the counit map $\epsilon_Q\colon L R(Q)\to Q$ is an isomorphism;

• the category, denoted $Dual(L,R)$, where the objects of $Dual(L,R)$ consist of a pair $P\in C$ and $Q\in D$ together a pair of isomorphisms $\alpha\colon P\stackrel\sim\to R (Q)$ and $\beta\colon L (P)\stackrel\sim\to Q$ which are adjoint, and where the morphisms of $Dual(L,R)$ from $(P,Q,\alpha,\beta)$ to $(P',Q',\alpha',\beta')$ are the obvious things [oh, well if you insist, they’re given by a pair of morphisms $f\colon P\to P'$ and $g\colon Q\to Q'$ such that $R(g)=\alpha' f \alpha^{-1}$ and $L(f)=\beta'^{-1}g\beta$].

The are several comments to make here. Firstly, I would like this to work in the enriched setting as well, so that when $C$ and $D$ are $V$-categories, the three described categories are $V$-categories as well, this means that I shouldn’t mention actual morphisms in the description of $Dual(L,R)$. This oughtn’t be a problem.

Secondly, whilst the third description looks more complicated than the first two it seems to me to sometimes be more natural in examples due to its symmetric nature. To take a very simple example, if you’re describing a Dedekind cut of the rationals, it’s intuitively easier to think of it as a partition of the rationals into two sets $P$ and $Q$ such that every element of $P$ is less than or equal to every element of $Q$, rather than as a single set $P$ with some less obvious condition on it.

Thirdly, following on from that comment, it might be useful to think of the reasonably analogous situation where expressing a vector space $P$ as isomorphic to its own double dual is essentially the same as finding another vector space $Q$ and expressing $Q$ as isomorphic to the dual of $P$ while expressing $P$ as isomorphic to the dual of $Q$.

I should say how the equivalences between the three categories in the theorem are given. The equivalence between $Fix(R L)\subset C$ and $Fix(L R)\subset D$ is just given by the restriction of $L$ and $R$ to these subcategories. The equvialence between $Fix(R L)$ and $Dual(L,R)$ is given on the one hand by $P\mapsto (P,L(P),\eta_P,id_{L(P)})$ and other hand by $(P,Q,\alpha,\beta)\mapsto P$. I will let you guess the other equivalence.

Now to get to the case in hand. We have a category $A$ (which Andrew calls $\mathcal{T}$) – I would really like that to be a $V$-enriched category, but haven’t quite set myself up for that, so we’ll just let it be an ordinary category for now. From that we get an adjunction, Isbell conjugation, between the category of presheaves on $A$ and the opposite category of copresheaves on $A$:

$L\colon Set^{A^{op}}\to (Set^A)^{op};\quad L(P)(a)\coloneqq Set^{A^{op}}(P,A({-},a))$ $R\colon (Set^A)^{op}\to Set^{A^{op}};\quad R(Q)(a)\coloneqq Set^{A}(Q,A(a,{-}))$

In his comment above, Tom defines $I(A)$ to be either $Fix(L R)$ or $Fix(R L)$, he doesn’t care which.

In his nlab entry, Andrew defines the Isbell duality subcategory of the Isbell envelope of $A$ to be $Dual(L,R)$. I will say a few words about this below.

By the above Theorem, this means that Tom and Andrew were talking about the same thing. And this thing is an extension of the category $A$. By the Yoneda embedding, $A$ naturally embeds fully and faithfully in $Dual(L,R)$:

$i\colon A\to Dual(L,R);\quad a\mapsto (A(-,a),A(a,-),id,id).$

Furthermore, an object $W=(P,Q,\alpha,\beta)$ of $Dual(L,R)$ is supposed to be thought of as a “new” object of the category $A$ with its homsets being given by $P$ and $Q$, so for all $b\in A$ we should think “$A(b,W)=P(b)$” and “$A(W,b)=Q(b)$”. Actually this is true in $Dual(L,R)$, namely $Dual(L,R)(i(b),W)\cong P(b)$ and $Dual(L,R)(W,i(b))\cong Q(b)$.

I will explain in another post what this has to do with metric spaces, but here I’ll just say a few words about the way Andrew defines what I’ve called $Dual(L,R)$.

Andrew says that objects in the Isbell duality subcategory of the Isbell envelope of $A$ should consist of a presheaf $P$, a copresheaf $Q$ (or rather $F$ in his notation) and a natural transformation $c\colon P\times Q\to A({-},{-})$ such that two things (specified in Definition 3) are isomorphisms. However, as Todd observes in his helpful MathOverflow answer, the set of natural transformation $Nat\bigl(P\times Q,A({-},{-})\bigr)$ is naturally identified with $Set^{A^{op}}(P,R(Q))$ and $(Set^{A})^{op}(L(P),Q)$. So Andrew’s $c$ gives rise to both my $\alpha$ and $\beta$, his condition of “satisfying Isbell duality” is precisely that $\alpha$ and $\beta$ are isomorphisms. So his category is the same as mine.

Posted by: Simon Willerton on April 26, 2011 11:48 AM | Permalink | Reply to this

### Re: Isbell envelope and Isbell duality

That looks very nice. May I copy it across to the nLab?

So my guess is that you want to apply all of this to the case of a metric space considered as an enriched category. Is that close?

Posted by: Andrew Stacey on April 28, 2011 1:15 PM | Permalink | Reply to this

### Re: Isbell envelope and Isbell duality

Sure. You are free (indeed, encouraged!) to take the mathematical ideas and modify them as much as you like. This includes verbatim copying of theorems, definitions, and proofs. But you are required to refer to the original article.

This gets me thinking, do we operate under some license here at the Café? I’ll ask this over at the other thread.

Posted by: Simon Willerton on April 28, 2011 9:22 PM | Permalink | Reply to this

### Re: Higher Gauge Theory, Division Algebras and Superstrings

I meant to ask if there’s an easy characterization of $I(A)$. It’s not immediate to me whether it is related to the thing I described, but it certainly smells similar and I should think some more about this. Is there a reference for Isbell’s result?

Posted by: Simon Willerton on April 13, 2011 6:18 PM | Permalink | Reply to this

### Re: Higher Gauge Theory, Division Algebras and Superstrings

Professor John Baez,

you often say things like ‘you’re stupid’ but are you secretly downplaying your true cleverness and intelligence and being modest?

Posted by: dr.zeuss on April 12, 2011 12:01 AM | Permalink | Reply to this

### Re: Higher Gauge Theory, Division Algebras and Superstrings

Yes.

Posted by: John Baez on April 12, 2011 1:32 AM | Permalink | Reply to this
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