## October 19, 2009

### Math Overflow

#### Posted by David Corfield

The math-blogosphere is abuzz with interest in the new Math Overflow, a mathematics questions and answers site. Already we at the Café have been helped with the answer to a query on the Fourier transform of a certain kernel, and there are some juicy questions for us to answer there too, including

You can read a discussion on Math Overflow, and a debate concerning its advantages relative to $n$Lab.

Posted at October 19, 2009 9:01 AM UTC

TrackBack URL for this Entry:   http://golem.ph.utexas.edu/cgi-bin/MT-3.0/dxy-tb.fcgi/2087

### Re: Math Overflow

I looked at this because it seemed interesting. I found that the text is a bit too hard, too faint, to read. Maybe this is set by default and my eyes are too old. The solution to making the contrast more readable can be found at nLab.

“This nifty script fixes the eyestrain caused by poorly designed blog contrasts. 15 seconds to install, works for Firefox only.”

Posted by: Stephen Harris on October 19, 2009 9:15 PM | Permalink | Reply to this

### Re: Math Overflow

On the subject of matrix mechanics, Zoran asks what rig is involved in the Matsubara formalism. Has this something to do with John’s gnomic utterance that ‘temperature lives on the Riemann sphere’?

Following the link to the sci.math discussion, though, I see John alludes to KMS states, which Wikipedia has as an alternative approach to thermal quantum field theoty.

Posted by: David Corfield on October 20, 2009 12:10 PM | Permalink | Reply to this

### Re: Math Overflow

I’m very enthusiastic about Math Overflow right now. There are at least four excellent things about it:

1. The administrators are keeping the level of the questions very high: it’s all research-level stuff
2. There are some seriously intelligent people playing an active part there
3. People are asking really interesting questions
4. People are giving really interesting answers!

If it can keep up the momentum, it’s going to be a magnificent thing.

Posted by: Tom Leinster on October 21, 2009 3:59 AM | Permalink | Reply to this

### Re: Math Overflow

Minhyong Kim has a nice question at MO which is right up our street.

What is a really good example of a situation where keeping track of isomorphisms leads to tangible benefit?

Posted by: David Corfield on May 12, 2010 2:53 PM | Permalink | Reply to this

### Re: Math Overflow

Group theory?

Posted by: Tom Leinster on May 12, 2010 3:07 PM | Permalink | Reply to this

### don’t forget the isomorphisms

What is a really good example of a situation where keeping track of isomorphisms leads to tangible benefit?

Group theory?

Right, in this vein:

Groupoid theory?

Traditionally people don’t speak about “groupoid theory” much. The traditional answer to the question: “Why do I need to rememeber isomorphisms of my objects is?” is:

In order to understand their moduli space. Because it won’t in general be a space, but a groupoid. Aka a stack.

In fact, even more generally it won’t even be a groupoid, because you must not forget the isomorphisms between the isomorphisms, even, and so on. In general it will be an $\infty$-groupoid. For instance the “moduli space” of derived elliptic curves is a parameterized $\infty$-groupoid aka $\infty$-stack.

But forgetting isomorphisms or not remembering how to handle them puzzles people all over the place, even if the term “isomorphism” may not explicitly come to mind. For instance, in the context of the theory of general relativity there has been and still is endless puzzlement and debate about, effectively, the isomorphisms in the category of smooth manifolds. Recently we had occasion on the $n$Lab to take apart Einstein ‘s hole argument, which was the cause of some headaches back then. But there is no headache here to be had, once you know your way with isomorphisms in categories.

One can continue this way with examples from physics where lots of trouble is caused by forgetting isomorphisms or else what to do about them. The huge machinery of “BV-BRST formalism” in gauge theory is all about how to behave in the presence of isomorphisms. Only that here they are called gauge transformations.

I don’t know, one could go on forever with this. For instance the fact that one has long sequences in cohomology from short exact sequences really comes from the fact that when computing kernels, one should do it up to and remembering the isomorphisms that relate elements to 0, i.e. instead of computing pullbacks one should compute homotopy pullbacks and form fiber sequences. Kernels with remembered isomorphisms.

From that in turn endless phenomena spring forth. For instance the notion of derived loop spaces and all the magic they bring with them. These beasts consist entirely of not-forgotten isomorphisms.

Posted by: Urs Schreiber on May 12, 2010 7:25 PM | Permalink | Reply to this

### Re: don’t forget the isomorphisms

The derived loop spaces’ link has derived only as a modifier of stacks. In plain top language, what is a derived loop space?

Posted by: jim stasheff on May 13, 2010 1:22 PM | Permalink | Reply to this

### Re: don’t forget the isomorphisms

The derived loop spaces’ link has derived only as a modifier of stacks.

I think the page in fact misses to talk about “derived loop spaces” proper (though it does give the link to Ben-Zvi/Nadler). How come nobody took the time to type out more details?!

In plain top language, what is a derived loop space?

Well, it is a free loop space object. But not in the $(\infty,1)$-topos of topological spaces. But in that of $\infty$-stacks on the site $(simplicial rings)^{op}$.

Posted by: Urs Schreiber on May 13, 2010 1:52 PM | Permalink | Reply to this

### Re: don’t forget the isomorphisms

In plain top language, what is a derived loop space?

I replied

It is a free loop space object[…] in the $(\infty,1)$-topos of $\infty$-stacks on the site $(simplicial rings)^{op}$.

I should add: concretely: for $A$ a smooth algebra of finite type, and with $Spec A$ regarded as an object in $\infty$-stacks on $(simplicial algebras)^{op}$ we have that:

the derived loop space $\mathcal{L} Spec A$ of $Spec A$ is the spectrum of the Hochschild homology complex of $A$.

This is because the defining homotopy pullback

$\array{ \mathcal{L} Spec A &\to& Spec A \\ \downarrow && \downarrow \\ Spec A &\to& Spec A \times Spec A }$

in $Sh_{(\infty,1)}(sAlg^{op})$ is dually computed as the homotopy pushout

$\array{ Q &\leftarrow& A \\ \uparrow && \uparrow \\ A &\leftarrow& A \otimes A }$

in $sAlg$. That, in turn, is computed as an ordinary pushout in $sAlg$ after replacing one of the copies of $A$ with its bar complex $B_\bullet(A)$, such as to make the diagram cofibrant. Then we compute the ordinary pushout

$\array{ Q &\leftarrow& A \\ \uparrow && \uparrow \\ B_\bullet(A) &\leftarrow& A \otimes A }$

and find the complex

$Q = B_\bullet(A) \otimes_{A \otimes A} A \,.$

This is the complex that computes the Hochschild homology of $A$. So we find that this is the “derived” algebra of functions on the derived loop space of $Spec A$:

$\mathcal{L} Spec A = Spec B_\bullet(A) \otimes_{A \otimes A} A \,.$

A little bit of this is discussed further at Hochschild cohomology.

Notice the magic that is happening here, re isomorhisms: in the ordinary topos of sheaves on $(algebras)^{op}$ the space $Spec A$ has no nontrivial isomorphism between its objects. It’s just a 0-truncated object.

But after we regard the same $Spec A$ as the $\infty$-stack that it presents on the site of simplicial algebras, it suddently exhibits infinitesimal isomorphisms, of sorts. The derived loop space of $Spec A$ has as objects these infinitesimal loops in $Spec A$.

Whereas the ordinary loop space object of $Spec A$ (i.e. with $Spec A$ regarded on the site of just algebras-op) is of course trivial (equal to $Spec A$).

Posted by: Urs Schreiber on May 13, 2010 2:08 PM | Permalink | Reply to this

### Re: don’t forget the isomorphisms

Thanks
At least
this is the “derived” algebra of functions on the derived loop space of SpecA:

L SpecA=SpecB_*(A) otimes_{ A otimes A}A.

gives some hint about the derivedness.

Posted by: jim stasheff on May 14, 2010 1:21 PM | Permalink | Reply to this

### Re: Math Overflow

I would turn the question around:

What’s a good example of a situation where identifying isomorphic objects doesn’t get you into trouble?

Posted by: Eugene Lerman on May 12, 2010 9:37 PM | Permalink | Reply to this

### Re: Math Overflow

‘What’s a good example of a situation where identifying isomorphic objects doesn’t get you into trouble?’

Having thought about it on the train, I thought I’d reply more strongly now: Most situations.

It’s like in this Whitehead quote:

‘Civilization advances by extending the number of important operations which we can perform without thinking about them.’

Identifying isomorphic objects is such an operation. It increases efficiency most of the time.

Posted by: Minhyong Kim on May 13, 2010 9:57 AM | Permalink | Reply to this

### Re: Math Overflow

All these answers are good. But the intent of my question was to elicit examples where the delicate distinctions we all like to draw really come close to being evident and essential. To answer Eugene’s question, for example, I would argue that plenty of the time, it’s a good idea to say that the symmetry group of the icosahedren *is* the alternating group on five letters, even though I don’t see any letters on the object in question. Obviously, we came up with a certain amount of abstract language exactly because it’s helpful to forget differences sometimes. As I mentioned in the MO question, sometimes I do this deliberately in an undergraduate course in order to encourage certain intuitive associations. But then, in some other situations, it’s crucial to be careful. I don’t expect a careful treatise now on when one mode is more useful than the other. But I would like examples that might be substantially convincing even to someone with a degree of skepticism.

In fact, for this hard-core readership, I could continue my point above by issuing a very concrete challenge: Can you come up with a convincing situation where I really benefit by staying aware that a certain group is merely isomorphic to $A_5$, and not the same?

Posted by: Minhyong Kim on May 12, 2010 11:03 PM | Permalink | Reply to this

### Re: Math Overflow

Can you come up with a convincing situation where I really benefit by staying aware that a certain group is merely isomorphic to $A_5$, and not the same?

No, because these are non-interesting isomorphisms in that they are unique:

if you choose one isomorphism from the abstract group to $A_5$ and I choose another, then our two choices are related by a unique automorphism of $A_5$.

The thing is that here we are not looking the group $Aut(A_5)$ or some such, but at the undercategory of isomorphisms out of our abstract group, whose objects are identifications and whose morphisms changes of identifications. This undercategory is contractible, equivalent to the trivial groupoid, even if $Aut(A_5)$ is nontrivial.

And isomorphisms in contractible groupoids: these you may safely forget for the most part. You only have to remember that whenever you say “the group” you add a link to the $n$Lab-entry generalized the. ;-)

Posted by: Urs Schreiber on May 12, 2010 11:25 PM | Permalink | Reply to this

### Re: Math Overflow

Good. Then let’s just change the group to $\mathbb{Z}/3$.

Posted by: Minhyong Kim on May 12, 2010 11:28 PM | Permalink | Reply to this

### Re: Math Overflow

Oops, I take back the above. I misunderstood what you wrote. In fact, I don’t really understand what you wrote:

‘if you choose one isomorphism from the abstract group to A 5 and I choose another, then our two choices are related by a unique automorphism of A 5.’

This sentence seems to indicate that for *any* group, there no point in distingishing equality from isomorphism.

Posted by: Minhyong Kim on May 12, 2010 11:32 PM | Permalink | Reply to this

### Re: Math Overflow

I take it back yet again. Your sentence is true for isomorphisms between *anything at all*.

Posted by: Minhyong Kim on May 12, 2010 11:37 PM | Permalink | Reply to this

### Re: Math Overflow

Your sentence is true for isomorphisms between anything at all.

Yes. I was about to say this, but you were quicker than me.

If in any category (be it that of groups or something else) you somehow obtain an object $x$ and prefer to think of it in terms of an isomorphic object $y$, nothing interesting has happened yet. Interesting things happen when there are nontrivial automorphisms of $x$, and these translate to an equivalent group of nontrivial automorphisms of $y$.

Posted by: Urs Schreiber on May 12, 2010 11:50 PM | Permalink | Reply to this

### Re: Math Overflow

Great! This sounds like a ‘small’ enough situation to yield perhaps the kind of example I’d like. So we want two distinct objects $x$ and $y$ such that

1. $x\simeq y$;

2. leading to an isomorphism $Aut(x)\simeq Aut(y)$;

3. and there is a real danger of thinking $x=y$;

4. but remembering $x\simeq y$ leads to definite new insight.

As I write, I’m getting glimmerings of my own example, but I’ll wait for yours first.

Posted by: Minhyong Kim on May 12, 2010 11:58 PM | Permalink | Reply to this

### Re: Math Overflow

I have been thinking of determinants recently, so an example from multilinear algebra might help.

Given an $n$-dimensinonal vector space $V$, the exterior products $\Lambda^0 V$ and $\Lambda^n V$ are both 1-dimensional, so are isomorphic. Moreover there is a natural isomorphism between $End(\Lambda^0 V)$ and $End(\Lambda^n V)$.

However, the maps that $A \in End(V)$ induces on $\Lambda^0 V$ and $\Lambda^n V$ are different, $I$, the identity, in the former case, and $(det A) I$ in the latter.

Perhaps what you’re looking for is not a case when keeping track of the isomorphisms matters, but when relationships with external constructions differ despite the fact that two things are isomorphic when considered in isolation.

Posted by: Tom Ellis on May 13, 2010 2:17 AM | Permalink | Reply to this

### Re: Math Overflow

This is a nice example! You should post it on the MO site.

Posted by: Minhyong Kim on May 13, 2010 9:45 AM | Permalink | Reply to this

### Re: Math Overflow

Thanks, I have done.

Posted by: Tom on May 13, 2010 9:59 AM | Permalink | Reply to this

### Re: Math Overflow

One way in which this example is nice is that it’s very elementary, but eventually leads to a distinction important in more advanced mathematics, namely, that between functions and top-degree forms. So somehow, this process of passing from the local to the global seems to come up repeatedly when considering the importance of retaining isomorphisms. That is, a distinction that appears pedantic when considered locally, becomes unavoidable as we globalize.

Posted by: Minhyong Kim on May 13, 2010 9:51 AM | Permalink | Reply to this

### Re: Math Overflow

I agree. In fact, I was going to offer this class of examples as an answer to your question:

You cannot glue isomorphism classes of bundes, only actual bundles.

There are lots of things in geometry that are “the same” (isomorphic) locally but not globally. So if you don’t keep track of isomorphisms, you can’t pass from local descriptions to global objects. Or from maps defined locally to maps defined globally; gluing morphisms if you like.

I believe this point has already been made by Urs, perhaps using slightly different words.

Posted by: Eugene Lerman on May 13, 2010 3:57 PM | Permalink | Reply to this

### Re: Math Overflow

Yes, this is a rather ubiquitous phenomenon. Actually, this class of examples was presented as motivation in my MO question having, in fact, to do with determinants.

Posted by: Minhyong Kim on May 13, 2010 6:31 PM | Permalink | Reply to this

### Re: Math Overflow

This is a nice example!

But let’s make explicit what it is an example of:

this is an example of two objects being isomorphic in one categeory but not in another.

The two $End(V)$-modules $\Lambda^0 V$ and $\Lambda^n V$ are not isomorphic in the category of $End(V)$-modules. Under the forgetful functor down to the category of vector spaces, they do become isomorphic.

There is no lack of this kind of examples, of course.

Posted by: Urs Schreiber on May 13, 2010 12:42 PM | Permalink | Reply to this

### Re: Math Overflow

‘There is no lack of this kind of examples, of course.’

Indeed of course. Most examples will be of this form. But what I’m searching for are simple examples that make a certain abstract viewpoint compelling to the non-believer. I still wish you would come up with a situation of the simplest sort, where the conflation of a single isomorphism $X\simeq Y$ with the equality $X=Y$ is simply bad. Or rather, in constructive form, where the refinement of a single equality $X=Y$ into an underlying isomorphism $X \simeq Y$ yields a huge payoff.

Posted by: Minhyong Kim on May 13, 2010 2:37 PM | Permalink | Reply to this

### Re: Math Overflow

I still wish you would come up…

Oh, I didn’t know you were waiting for me!

…with a situation of the simplest sort, where the conflation of a single isomorphism $X \simeq Y$ with the equality $X = Y$ is simply bad.

Back in the young days of this blog, we had a long discussion on a simple every-day kind of example of the kind that you might be looking for: that of units.

Suppose for the purpose of the following example that we are happy with the approximation that the 3-dimensional space that we inhabit is a 3-dimensional Euclidean vector space $X$.

If we take this to be pointed at the point at which we live, the the definition of the meter (or the foot or inch or fathom or what is it you count ) serves to establish an isomorphism

$\mathbb{R}^n \stackrel{\simeq}{\to} X$

The image of $[0,1] \subset \mathbb{R} \subset \mathbb{R}^3$ under this isomorphism is a physical distance of length one meter.

Once we agree on this isomorphism, I can tell you how to get from here to the train station. Namely go 500 meters straight and then turn right.

Okay, that works. Next comes an isomorphism sceptic to me and asks for the way to the train station. So I say: “Oh, you just go 500 straight and then turn right.”

That won’t help him much. I could tell him that for not more than 7 he can get a map in the next store that would show him the way. But as he does not believe in isomorphisms, he probably won’t belive in maps either.

There is a fun short story by Umberto Eco, called “The map 1:1.” Or similar. It is about a city whose inhabitants decide that their city is so important that it needs a map of scale 1:1. They print it on a huge sheet and it covers the entire city and ever since then they live in the darkness under this map. That’s the price to pay if one does not believe in isomorphhisms, but only in identities.

Posted by: Urs Schreiber on May 13, 2010 4:17 PM | Permalink | Reply to this

### Re: Math Overflow

That’s a lovely example. Is there anywhere we can record that on the nLab? Maybe at isomorphism?

Posted by: Mike Shulman on May 13, 2010 6:08 PM | Permalink | Reply to this

### Re: Math Overflow

I like your example, but I’m afraid I’m still not satisfied. Please ignore me if this is becoming too annoying. You see, I’m getting the feeling that so many of these examples are assuming a skeptic who is basically a straw man. Take a map. *No one* will ever forget that it’s not an identity, whether or not they understand category theory or know any mathematics at all. There’s a reason that city you mention appears only in Eco’s fantasy.

This is not entirely true. We may argue that all kinds of social ills arise from arranging people in a well-ordered line and thinking of the numbers as people. But at least at the moment, I don’t think this is the gist of the distinction we are concerned with.

Remember, the skeptic I have in mind is the reasonably-educated mathematician or student who is asking the kinds of questions I brought up with Tom: Why should I think of a group as a category, and so on.

I think I’m boring everyone with repetition, but once again, to be interesting, there has be a fairly common tendency for the isomorphism in question to be thought of casually as an identity.

Posted by: Minhyong Kim on May 13, 2010 6:27 PM | Permalink | Reply to this

### up to isomorphism

You see, I’m getting the feeling that so many of these examples are assuming a skeptic who is basically a straw man.

Woops, sorry for that. I didn’t mean to erect a straw man. I tried to seriously follow your request for a simple example. After we had – I think – listed quite a few good non-simple examples in the discussion here.

You know, I keep thinking that I do not quite understand what you are looking for.

For instance when you now say

Remember, the skeptic I have in mind is the reasonably-educated mathematician or student who is asking the kinds of questions I brought up with Tom: Why should I think of a group as a category, and so on.

The question why it may be useful to think of a group in terms of its delooping groupoid seems to me to be a rather different question than what I thought we had been discussing so far.

Maybe one problem with this discussion here is the following: I think it is one of the examples of concepts that, when one hasn’t thought about them sound mysetrious, but once one has thought about them become entirely trivial. After one has understood them.

It is unlike understanding the proof of, say, the snake lemma, where it is possible to understand it half-way, find some steps more mysterious then others, etc. but more like, say, learning to walk, learning bicycle, etc. Once you know how to do it, it is very hard to try to imagine how one could fail doing it.

I would like to ask of you the following: if you think my example about units and identifying space with $\mathbb{R}^3$ sounds like a straw-man argument, go to the library and open some book on general relativity that was written before, say, the 1970s. Wade through all the discussion that you’ll see there of “the principle of equivalence”, “general covariance”, the “hole paradox” and whatnot, and realize in each case that what is filling these pages is nothing but what today we realize is a slow, slow acquaintance with the abstract notion of manifold, a painful struggle to realize that space is not equal to $\mathbb{R}^3$, as had silently been assumed since Déscartes, but just isomorphic. (I mean, of course it is just locally isomorphic even, but already apart from this, just this concept of identifying up to isomorphism is the cause of most of the puzzlement expressed in these textbooks.)

But I can build more and more examples of precisely the type I gave, but that sound more sophisticated.

For instance:

when you teach students linear algebra, the point comes where you need to convince them of the use of the abstract notion of finite dimensional vector space. Typically they will ask why they need to bother with it if they can just identify everything with $\mathbb{R}^n$. A standard example to get them going here is to let them write out a basis for the space of polynomials of degree 4 and write out the structure constants in that basis. Lo and behold, one student gets different structure constants than his neighbour, because they thought of different “obvious” bases.

I don’t know, that may still sound like a straw man, but this does happen in practice, as I suppose you know as well as I do.

From there of course it continues. How do you teach the formula for basis transformations in linear algebra to young homo sapiens? I teach them by drawing this picture to the board

$\array{ && V \\ & {}_{B_1}\nearrow && \nwarrow^{B_2} \\ \mathbb{R}^n &&\stackrel{}{\to}&& \mathbb{R}^n }$

and tell them how we tranfer from the isomorphism on the left to the one on the right.

This is precisely the same picture in all these examples here. In my previous example it had been

$\array{ && X \\ & {}_{meters}\nearrow && \nwarrow^{feet} \\ \mathbb{R}^3 &&\stackrel{}{\to}&& \mathbb{R}^3 } \,.$

So I could continue listing examples at will. Just name any category, and we have a class of examples. One part of me thinks this is what you are asking for. But the other part thinks that you maybe mean something else.

Posted by: Urs Schreiber on May 13, 2010 7:52 PM | Permalink | Reply to this

### Re: up to isomorphism

Well, what I wanted was something that was accessible, but striking enough to convince a reasonable skeptic to think categorically, or even $n$-categorically (at least for $n=2$). But I suppose I’m just being difficult. Thanks for all your examples, anyways. They certainly make up plenty of food for thought.

Posted by: Minhyong Kim on May 13, 2010 10:35 PM | Permalink | Reply to this

### Re: up to isomorphism

It sounds to me like you may be looking for something to convince an unreasonable skeptic! I feel like all these examples ought to be enough to convince a reasonable one… (-:

Posted by: Mike Shulman on May 14, 2010 3:34 AM | Permalink | Reply to this

### Re: up to isomorphism

I suppose it’s obvious by now that I’m using a specific request to drive home the need for ‘small but striking examples’ in favor of category theory.

Last fall, Eugenia Cheng told me of a visit to some university to give a colloquium talk. The host greeted her with the observation that he doesn’t regard category theory as a field of research. OK, he was probably a bit extreme, but milder versions of that view are quite common. Now, one possible response is to regard all such people as unreasonable and talk just to friends (who of course are the reasonable people!). This is not entirely bad, because that might be a way to buy time and gain enough stability to eventually prove the earth-shattering result that will show everyone! Another way is to take up the skepticism as a constructive challenge. This I suppose is what everyone here is doing at some level, anyways.

Other than the derived loop space, which is not exactly small, Urs’ examples are all of the simple subtle sort that can, over time, contribute to a really important change in scientific outlook and maybe even the infrastructure of a truly glorious theory. For example, I agree wholeheartedly about the horrors of the old tensor formalism. But it’s not unreasonable to ask for more striking accessible evidence of utility when it comes to the current state of category theory.

The importance of small insights and language that gradually accumulate into the edifice of a coherent and powerful theory is the usual interpretation of Grothendieck’s ‘rising sea’ philosophy. However, the process is hardly ever smooth along the way, especially the question of acceptance by the community. I’m not a historian, but I’ve studied arithmetic geometry long enough to have some sense of the changing climate surrounding etale cohomology theory, for example, over the last several decades. The full proof of the Weil conjectures took a while to come about, as you know. Acceptance came slowly with many bits and pieces sporadically giving people the sense that all those subtleties and abstractions are really worthwhile. Fortunately, the rationality of the zeta function was proved early on. However, there was a pretty concrete earlier proof of that as well using $p$-adic analysis, so I doubt it would have been the big theorem that convinced everyone. One real breakthrough came in the late sixties when Deligne used etale cohomology to show that Ramanujan’s conjecture on his tau function could be reduced to the Weil conjectures. There was no way to do this without etale cohomology and the conjecture in question concerned something very precise, the growth rate of natural arithmetic functions. This could even be checked numerically, so impressed people in the same way that experimental verification of a prediction does in physics. Clearly something deep was going on. Of course there were many other indications. The construction of entirely new representations of the Galois group of $\mathbb{Q}$ with very rich properties, the unification of Galois cohomology and topological cohomology, a clean interpretation of arithmetic duality theorems that gave a re-interpretation of class field theory, and so on.

For myself, being a fan of you folks here, I believe this kind of process is going on in category theory. But I don’t think you have to be too unreasonable to doubt it. In a similar vein, I don’t agree with Andrew Wiles’ view that physics will be irrelevant for number theory, but also think his pessimism is perfectly sensible.

I suppose I’m trying to make the obvious point that the presence of pessimists can be very helpful to the development of a theory, in so far as the optimists interact with them in constructive ways. I haven’t been coming to this site much lately, because the bit of internet time I have tends to be absorbed by Math Overflow. But I did catch David’s recent post on Frank Quinn’s article, which ended up as a catalyst for my MO question.

At the Boston conference following the proof of Fermat’s last theorem, Hendrik Lenstra is said to have said something like this: ‘When I was young, I knew I wanted to solve Diophantine equations. I also knew I didn’t want to represent functors. Now I have to represent functors to solve Diophantine equations!’ So should we conclude that he was foolish to avoid representable functors for so long? I wouldn’t.

This response to the MO question brings up the importance of knowing the specific isomorphism between some Hilbert spaces given by the Fourier transform. This is an excellent example, especially when we consider how it relates to the different realizations of the representations of the Heisenberg group and the attendant global issues, say as you vary over a family of polarizations. But I couldn’t resist recalling Irving Segal’s insistence that ‘There’s only *one* Hilbert space!’ Obviously, he knew, among many other things, the different realizations of the Stone-Von-Neumann representation as well as anyone, so you can take your own guess as to the reasoning behind that proclamation. He certainly may have lost something through that kind of philosophical intransigence. But I suspect that he, and many around him, gained something as well.

Posted by: Minhyong Kim on May 14, 2010 10:49 AM | Permalink | Reply to this

### Re: up to isomorphism

Last fall, Eugenia Cheng told me of a visit to some university to give a colloquium talk. The host greeted her with the observation that he doesn’t regard category theory as a field of research.

Aaaaaggggghhhhh!!!!

Well, okay… on the one hand, who cares what this guy thinks. On the other, I’m always stunned by how openly rude some people are to category theorists. And hard to bear when it takes place in an interview or discussion bearing on funding or defunding.

I’d like to hear more about such experiences and how people responded to them. Obviously the high-minded thing to do is try to engage the interlocutor in constructive dialogue, but not all of us are saints, you know. :-)

Somewhere John wrote down some possible rejoinders to try out, e.g., “Good! That means I’ll make much faster progress than you in my research!” Never tried that either!

Posted by: Todd Trimble on May 14, 2010 2:55 PM | Permalink | Reply to this

### Re: up to isomorphism

Not being a category theorist (or anything remotely close to it) I haven’t encountered the kind of ill treatment of category theory that is sometimes reported here first-hand; but I’ve seen enough dismissive treatment of fields I work in to find it easy to believe. I’m frankly amazed that mathematicians, of all people, don’t take more of an attitude that we all live in glass houses, where the obviousness of the value of our work is concerned.

Posted by: Mark Meckes on May 14, 2010 3:11 PM | Permalink | Reply to this

### Re: up to isomorphism

Regarding Irving Segal, I think it’s actually possible he was expressing a view not unlike yours, although not in the way you would express it. What I mean is, by emphasizing that all (separable infinite-dimensional) Hilbert spaces are isomorphic, maybe he was highlighting the need to pay attention to the other apparatus at hand, whatever that may be?

Or maybe not. Maybe he really was insistently decategorifying, which as John pointed out many weeks ago, may actually start out “as a stroke of mathematical genius”.

Posted by: Mark Meckes on May 14, 2010 3:20 PM | Permalink | Reply to this

### Re: up to isomorphism

I find it odd that your example of a mathematical subject which took a while to get accepted (etale cohomology) is one to which category theory is indispensable. If people have by now come around to accept “etale cohomology” but somehow still reject “category theory,” then it seems to me that they’re not only unreasonable but incorrigible. The process isn’t “going on” in category theory—it’s been here and long past gone.

Posted by: Mike Shulman on May 14, 2010 3:41 PM | Permalink | Reply to this

### Re: up to isomorphism

Oh yes, I definitely agree that one can take that view. That’s what makes the issue somewhat difficult to discuss with clarity. It’s also the reason I drew the parallel a while ago with the difference between logic as language of mathematics and as an area of research.

But once again, I do not disagree with your sentiment, except the bit about ‘unreasonable and incorrigible.’

Posted by: Minhyong Kim on May 14, 2010 3:52 PM | Permalink | Reply to this

### Re: up to isomorphism

…I think I have mentioned it before on this blog: When I was an undergraduate in Göttingen someone told me that Car Ludwig Siegel used to dismiss arguments using category theory as “hom hom humming”. When I first learned about category theory, my first impression was, too, that it is mainly about rephrasings, but not adding any content to what I already knew.

Posted by: Tim van Beek on May 14, 2010 4:06 PM | Permalink | Reply to this

### Re: up to isomorphism

The thing I love about category theory is that it allows me to remove content from what I already know, leaving just the bare essential bones that help me understand why things are the way they are.

Posted by: Tom Ellis on May 14, 2010 7:13 PM | Permalink | Reply to this

### Re: up to isomorphism

Sure, I was playing the advocatus diaboli, like Minhyong: My first impression was that category theory introduces a lot of concepts to express some things more abstractly than necessary :-)

Well, that is not quite true, my first impression was “Wow, we get the ‘set of all sets’ as a mathematical object and dodge all the paradoxes”.

But I don’t see that special short convincing example of a statement that requires category theory, like “to understand analysis you need to study complex analysis, example: why does the Taylor series of $\frac{1}{1+x^2}$ around zero have the convergence radius 1 and not 2 or $\infty$? Because there is a singularity at i”.

Posted by: Tim van Beek on May 14, 2010 8:11 PM | Permalink | Reply to this

### Re: up to isomorphism

The thing I love about category theory is that it allows me to remove content from what I already know, leaving just the bare essential bones that help me understand why things are the way they are.

Category theory is the Zen of math.

Posted by: Urs Schreiber on May 14, 2010 8:04 PM | Permalink | Reply to this

### Re: up to isomorphism

My late teacher Serge Lang, rather famous for his argumentative temperament, was actually remarkably pluralistic for most of his life when it came to purely mathematical matters. There was some tension between Lang and Mordell/Siegel that’s been well-publicized in the AMS notices. But Lang, even as a big fan of abstract algebraic geometry, had great respect for the explicit methods of Siegel, Severi, etc. You can find this attitude expressed in his letter to Mordell, reprinted in his book on Diophantine Geometry. As far I was able to witness, the philosophy that he espouses there was completely genuine.

Posted by: Minhyong Kim on May 14, 2010 9:16 PM | Permalink | Reply to this

### Re: up to isomorphism

I guess I’m in such a state of unreasonability conc. derived algebraic geometry, as I know that it is a “natural” further step in the development which started with Grothendieck and connects to Illusie’s cotangent complex and to deformation theory somehow, but miss infos on “concrete” applications in “normal” algebraic/arithmetic geometry for being really motivated to take a closer look into it.

Posted by: Thomas on May 14, 2010 4:33 PM | Permalink | Reply to this

### Re: up to isomorphism

…the obvious point that the presence of pessimists can be very helpful to the development of a theory, in so far as the optimists interact with them in constructive ways.

Obvious or not, it seems that constructive interaction is difficult to achieve, requiring genuine openness to the idea that one’s own point of view may be partial.

I suppose with such a vast amount to learn there’s very little time to train students in mathematical criticism. I once had the thought that a nice exercise for school children would be to have them make the case for one or other side of various comparisons, e.g., between base ten and binary, or even hopeless cases like the benefits of Roman numerals in arithmetic over arabic ones. Well XIX times XI is CXC + XIX = CCIX.

Posted by: David Corfield on May 14, 2010 4:09 PM | Permalink | Reply to this

### Re: up to isomorphism

Well, in spite of my sermonizing to Urs below, I do have to say this: Unless *some* people engaged in obstinate and passionate advocacy, academia would be boring!

There’s that nice popular book on linguistics by Randy something, with a title like ‘The language wars.’ He remarks at some point that all the bitter infighting in the sixties helped to convince outsiders that something serious was going on.

Posted by: Minhyong Kim on May 14, 2010 8:44 PM | Permalink | Reply to this

### Re: up to isomorphism

By the way, David, I have to admit that I tend to view such purely technical training in criticism and argumentation as being of rather limited value.

Posted by: Minhyong Kim on May 15, 2010 12:01 AM | Permalink | Reply to this

### Re: up to isomorphism

I’ll try one more shop-worn example (but one that hasn’t made it yet into undergraduate textbooks, as far as I know). I believe it shows that there are situations where it’s natural to think of groups as one object categories.

Suppose $f:X\to Y$ is part of homotopy equivalence of topological spaces and you want to convince your students that the induced map on fundamental groups $\Pi f:\pi_1 (X,x) \to \pi_1 (Y, f(x))$ is an isomorphism of groups. Here is the way I tried it last month.

A map $f:X\to Y$ defines/extends to a map of fundamental groupoids $\Pi f:\Pi X \to \Pi Y$. A homotopy $\phi$ from $f$ to $g:X\to Y$ defines a natural isomorphism $\Pi\phi: \Pi f \Rightarrow \Pi g$. Hence if $f:X\to Y$ is part of a homotopy equivalence, then $\Pi f:\Pi X \to \Pi Y$ is part of an equivalence of categories. In particular, it’s fully faithful. Since $\pi_1 (X,x) = \Pi X (x,x)$, $\Pi f:\pi_1 (X,x) \to \pi_1 (Y, f(x))$ is a bijection.

Posted by: Eugene Lerman on May 14, 2010 4:58 PM | Permalink | Reply to this

### Re: up to isomorphism

I see. This is a nice observation. I guess your point is that a homotopy inverse needn’t preserve base-points, making it a bit tricky to define the inverse map of groups. Considering the functor of groupoids gets rid of this problem. Is this correct?

Posted by: Minhyong Kim on May 14, 2010 6:04 PM | Permalink | Reply to this

### Re: up to isomorphism

I guess your point is that a homotopy inverse needn’t preserve base-points, making it a bit tricky to define the inverse map of groups.

Yes, that’s one of the points.

The other, I guess, is a motivation for 2-categories and 2-functors in undergraduate topology (although I avoided these words in my course).

Posted by: Eugene Lerman on May 14, 2010 6:42 PM | Permalink | Reply to this

### striking category theory

I suppose it’s obvious by now that I’m using a specific request to drive home the need for ‘small but striking examples’ in favor of category theory.

That wasn’t obvious to me, as I tried to indicate a few times. It seemed to me that we were talking about why one should not conflate isomorphisms with identities.

But okay, it seems we effectively agreed that only a straw man would do that anyway! ;-)

But category theory is not what happens when we identify a category. Or an isomorphism. Much as number theory is not what happens when I find that the number of apples in my grocery bag is 3.

Category theory is what happens when the Yoneda lemma is applied, the adjoint functor theorem, the monadicity theorem, the Grothendieck construction-theorem and such things.

Of course in order to apply all that, first the categories need to be identified. It’s the observation that many different constructions and statements in math all come down to special cases of such abstract statements in category theory that we care about trying to identify categorical structures as much as possible. It’s because the whole world is divided into abstract structures and their concrete models. Category theory tries to understand the abstract structures shared by all concrete models.

Okay, so let’s shift gears and start giving small but striking examples for category theory .

Here is one that I found small and striking when I became aware of it recently:

Theorem. (Tannaka duality for permutation representations).

Let $G$ be a group and $Rep_{perm}(G)$ its category of permutation representations. Let $F : Rep_{perm}(G) \to Set$ be the forgetful functor that remembers the representing set.

Then $G$ may be recovered as the endomorophisms of this fiber functor $G \simeq End(F)$.

To prove this, apply the Yoneda lemma precisely four times in a row. And nothing else. The details are spelled out here.

This is pretty cool, if you know that Tannaka duality appears in plenty of guises throughout mathematics. That simple Yoneda-power-proof above will not work verbatim for more sophisticated examples, but it determines the basic pattern.

But there are striking examples of where that proof does go through verbatim.

Here is one: Let $X$ be a locally contractible topological space and $Sh_{(\infty,1)}(X)$ the $(\infty,1)$-topos of $\infty$-stacks over it. Let $LConst(X)$ be the $\infty$-groupoid of locally constant $\infty$-stacks on $X$. For $x \in X$ any point, there is a forgetful fiber functor $F_x : LConst(X) \to \infty Grpd$ that produces the fiber of the covering $\infty$-bundle at $X$ given by the locally constant $\infty$-stack (in immediate analogy to the familiar statement obtained by discarding all the $\infty$s here.)

Then:

Corollary ($\infty$-Galois theory) The homotopy groups of $X$ at $x$ are those of the automorphisms of the fiber functor:

$\pi_n(X,x) \simeq \pi_n \mathbf{B} Aut(F_x) \,.$

Even if it may not immediately look like it, but the proof of this is, again, given by applying the $(\infty,1)$-Yoneda lemma four times in a row. This is just using that a reprentation of the fundamental $\infty$-group of a space on covering $\infty$-bundles is a “permuation representation”. Details are at $n$Lab: homotopy groups in an $(\infty,1)$-topos – In terms of monodromy and Galois theory.

Posted by: Urs Schreiber on May 14, 2010 6:05 PM | Permalink | Reply to this

### Re: striking category theory

Well, I feel like repeating again that I don’t need convincing myself. For example, my main reason for traveling to Montpellier last winter was to ask Toen about results like your corollary. But meanwhile, I know plenty of arithmetic geometers who use versions of your theorem all the time (in fact, would even regard it as entirely commonplace), but who I believe would react quite negatively to the corollary. I may try to explain in more detail my own guess about such negativity, if I can find the energy. It’s somewhat like my exchange with Mike about etale cohomology. Sure it uses plenty of category theory. But I’m fairly confident that the majority of people who use etale cohomology in their everyday research are pretty skeptical of current research-level category theory. To dismiss all of them as unreasonable would be unreasonable, I think.

Posted by: Minhyong Kim on May 14, 2010 6:43 PM | Permalink | Reply to this

### Re: striking category theory

Actually, if possible, maybe I’ll retract the response above. It does seem to be too conjectural…

Posted by: Minhyong Kim on May 14, 2010 6:45 PM | Permalink | Reply to this

### Re: striking category theory

Minhyong,

you write:

arithmetic geometers […] use versions of your theorem all the time (in fact, would even regard it as entirely commonplace), […] would react quite negatively to the corollary. […]

I am a bit at a loss of what to make of this, to be honest. Sorry. Maybe you could explain what you mean. Or maybe we leave it at that.

But I’m fairly confident that the majority of people who use etale cohomology in their everyday research are pretty skeptical of current research-level category theory.

Oh, and not just them! To be sure.

To dismiss all of them as unreasonable would be unreasonable, I think.

But then the case is proven, right? Category theory is useless because many colleagues believe so, and they can’t all be unreasonable. QED.

I know this state of affairs from my physicist days: when I studied, string theory was all true because all colleages thought so. Then times changed, and now string theory is all wrong because it even says so in the newspapers. So there.

I feel differently: when I think about math and physics, I try to convince myself about what I see. If I find out for myself that category theory is the coolest thing since sliced bread, then all the king’s horses and their scepticism won’t change my insight.

Posted by: Urs Schreiber on May 14, 2010 8:00 PM | Permalink | Reply to this

### Re: striking category theory

I guess you won’t allow me to retract! Oh well. What I meant was that Tannaka duality for groups is very much bread and butter now. Stacky versions will take more time, and some event analogous to experimental evidence in physics.

And I will respond to this point:

‘But then the case is proven, right? Category theory is useless because many colleagues believe so, and they can’t all be unreasonable. QED.

I know this state of affairs from my physicist days: when I studied, string theory was all true because all colleages thought so. Then times changed, and now string theory is all wrong because it even says so in the newspapers. So there.’

at greater length once you tell me that you seriously believe this to be my position. I can’t help but fear that you’re setting up the straw man again.

In short, there’s an enormous difference between the statement:

(1) Category theory is useless;

(2) I can understand why some intelligent people are skeptical about the utility of category theory.

Forgive me descending again into by tiresome sermonizing mood: For a productive and energetic leader in category theory like yourself, I would guess it’s *mathematically* useful to have a sophisticated understanding of *why* someone who uses category theory so much, say in computing Grothendieck cohomology, might remain unimpressed by the more advanced stuff (say, $\infty$-stacks to take a somewhat random example).

Posted by: Minhyong Kim on May 14, 2010 8:38 PM | Permalink | Reply to this

### Re: striking category theory

If you will allow me another reformulation, an alternative would be find the fact that many people

1. compute Grothendieck cohomology;

2. but dislike advanced category theory,

highly interesting, in mathematical and other authentic intellectual senses. In my view, one *wrong* way would be the ironical sense of ‘interesting that people can be so stupid,’ which pundits employ so depressingly often in political discourse.

Posted by: Minhyong Kim on May 14, 2010 9:02 PM | Permalink | Reply to this

### Re: striking category theory

I would guess it’s mathematically useful to have a sophisticated understanding of why someone who uses category theory so much, say in computing Grothendieck cohomology, might remain unimpressed by the more advanced stuff

Yes, that would be useful. Unfortunately such an understanding is just what I haven’t gotten out of this discussion. I don’t feel like I understand any better what leads such people to hold a view which seems to me to be inherently contradictory.

Posted by: Mike Shulman on May 14, 2010 10:16 PM | Permalink | Reply to this

### Re: striking category theory

Of course we could gradually get into that now if you’d like. But I think it’s useful to dispense first with at least a simplistic version of idea that the conjunction of 1. and 2. is inherently contradictory. As with so many other matters, a naive sense of contradiction arises from an ‘all or nothing’ refusal to acknowledge various commonsensical *degrees* of belief or affinity. In all spheres of human activity it’s quite natural to have a taste for certain things or ideas, but still feel that some versions are too extreme, too complicated, or even misguided distortions of a good idea. To give a random example that just happens to be on my mind, George Orwell was a committed socialist, but had a distaste for Stalinists. Was that inherently contradictory? Certainly some communists thought so. In the introduction to some impenetrable book about the philosophy of history, George Lukac writes essentially that all western socialists are living at the edge of an abyss of contradiction, and so on. Anyways, I think I’m belaboring an obvious point.

Now, a careful analysis may reveal more serious contradiction. It could indeed be interesting to carry out such an analysis with an open mind (or even with a closed mind).

Posted by: Minhyong Kim on May 14, 2010 11:52 PM | Permalink | Reply to this

### Re: striking category theory

I don’t think it’s hard to explain why someone could believe both 1 and 2.

E.g. algebraic number theorists tend to be inherently conservative, with a distaste for machinery until it proves its ability to solve a concrete Diophantine problem. (Cf. Lenstra’s comment referenced by Minhyong above, which was made in light of the use of deformation theory to solve Fermat’s Last Theorem.)

Thus etale cohomology has proved its worth, but higher category theory as yet has had no applications in number theory. Thus an algebraic number theorist could easily believe both 1 and 2.

This is one big difference between people working on arithmetic aspects of the Langlands program, and those working on geometric Langlands: in the latter program, category theory (and higher category theory) plays an important role in formulating the questions, in making the arguments, and really in every aspect of the subject. This is not at all the case (yet) in arithmetic Langlands. (The closest point of contact perhaps is Ngo’s proof of the fundamental lemma, which uses geometric representation theory techniques with the goal of proving a result of the utmost importance for the most classical, arithmetic parts of the Langlands program. But at the moment one can regard Ngo’s proof as being “quarantined” from the rest of classical Langlands: one just takes the fundamental lemma as a statement and uses it; one does not bring the proof ideas along as well.)

Posted by: Matthew Emerton on May 15, 2010 7:13 AM | Permalink | Reply to this

### Re: striking category theory

Do you know of any prospects to bring geometric Langlands techniques to arithmetic Langlands?

Minhyong has some idea that a TQFT-like category might appear there.

Posted by: David Corfield on May 15, 2010 9:48 AM | Permalink | Reply to this

### Re: striking category theory

Dear David,

This is a very interesting question, which in particular some of those working on p-adic Langlands (which you shouldn’t think of as being yet another Langlands program, separate from arithmetic or geometric, but rather as a sub-branch of arithmetic Langlands, created in response to Wiles proof of Shimura–Taniyama and subsequent developments) are acutely aware of, without, on the other hand, knowing the answer.

What I do think is happening, and likely to continue to happen, is that, with p-adic Langlands, one is seeing homological techniques becoming more important. (I won’t elaborate now, except to say: where one would have intertwiners in classical local Langlands, i.e. maps between parabolic inductions labelled by Weyl group elements, in p-adic local Langlands one seems to have derived intertwiners, i.e. Exts of various degrees labelled by Weyl group elements: so one has RHoms instead of Homs, and the situation seems to be some sort of fibre product of the classical theory of intertwiners and the more geometric (though still classical) picture of Borel–Weil–Bott. Also, Paskunas’s very recent proof of the surjectivity of the p-adic local Langlands correspondence for GL_2(Q_p), which is deformation-theoretic in nature, seems to involve(at least covertly) techniques of homotopical algebra in a way that previous deformation theory arguments in number theory haven’t.)

This can be taken as a sign that one is moving in a geometric direction, but the situation is difficult, the phenomena are nuanced, and it is (all too!) easy to jump to conclusions. At the moment, I think one is left with the unsatisfying answer that “time will tell”.

Posted by: Matthew Emerton on May 15, 2010 4:12 PM | Permalink | Reply to this

### Re: striking category theory

Matthew certainly put it in a much more succinct and relevant way than my pseudo-philosphical ramblings!

David, I’m happy to discuss the questions you raise, but please don’t refer to the remarks at LNT as my *idea*. Those were vague murmurings of the subconscious briefly emerging. In any case, they’re also not immediately relevant to arithmetic Langlands.

In case people here don’t know it, Matthew is probably *the* most qualified person in the world at the moment to discuss the prospects David asks about.

Posted by: Minhyong Kim on May 15, 2010 10:11 AM | Permalink | Reply to this

### Re: striking category theory

Perhaps the only point in Matthew’s comment I would disagree with is that the conservatism in question is particular to number-theorists. The dialectic between hard-nosed pragmatism and visionary theorizing is omnipresent and complex, I think, even within any one individual.

Posted by: Minhyong Kim on May 15, 2010 10:17 AM | Permalink | Reply to this

### Re: striking category theory

By the way, Matthew, could I trouble you to explain the sense in which higher category theory is important to geometric Langlands? I’m not sure I have a clear sense of that. I can believe that it’s present here and there in the Beilinson-Drinfeld outlook, but perhaps not so much in the older Drinfeld approach developed by Lafforgue and then Frenkel-Gaitsgory-Vilonen and so on. I couldn’t myself point to a clear-cut relevance in either work.

In any case, isn’t it the case that it’s still hard to make even the geometric Langlands correspondence *one*-categorical?

Posted by: Minhyong Kim on May 15, 2010 10:29 AM | Permalink | Reply to this

### Re: striking category theory

Dear Minhyong,

When I wrote “geometric Langlands” I should probably have been more nuanced, because I was thinking not of Drinfeld’s original work and Lafforgue’s generalization, and nor was I thinking of Frenkel-Gaitsgory-Vilonen.

Rather, I was thinking (as you say) of the outlook of Beilinson and Drinfeld (involving categories of automorphic sheaves on moduli stacks), or recent local conjectures of Frenkel and Gaitsgory, where rather than group actions on vector spaces one has group actions on categories (which then sit insided an ambient 2-category), recent work of Gaitsgory–Lurie generalizing geometric Satake (in which higher categorical techniques are prominent), and also the recent and ongoing work of Ben-Zvi–Nadler, in which they combine higher categorical techniques with ideas from geometric Langlands to study representations of real groups.

So probably it would have been better to say contemporary work in geometric Langlands (where contemporary should be understood in a strong sense!).

Also, with regard to Ngo, perhaps his work is not particularly *higher* categorical, but it does involve in a crucial way categories of sheaves on algebraic stacks, and (perhaps just because this is distant from the kind of objects I usually work with) once I see categorical ideas and stacky ideas playing key roles simultaneously, I tend to instinctively file the work in the higher category theory section.
(Also, the geometry involved in Ngo’s work is pretty close to the geometry involved in the other, more directly higher-categorical, parts of the geometric Langlands program mentioned above.)

Posted by: Matthew Emerton on May 15, 2010 3:54 PM | Permalink | Reply to this

### Re: striking category theory

One more remark for now, related to your last question about *one*-categoricity.

My (somewhat vague) understanding is that, yes, it is difficult to competely formulate even a 1-categorical form of global geometric Langlands; in particular, local systems that are reducible give trouble. (See various work on geometric Eisenstein series for hints of this.) But my impression is that this in part related to the fact that these are the most stacky points of moduli stacks of local systems, and that
in trying to correct formulate the conjecures at such points, one is led into issues to do with categories of sheaves on stacks, dg-categories of sheaves on stacks, and related issues, which (again perhaps partly because I am viewing them from a distance) I think of as pointing in a higher categorical direction.

Posted by: Matthew Emerton on May 15, 2010 3:59 PM | Permalink | Reply to this

### Re: striking category theory

Still, one might ask for something more specific. As I’ve been exaplaining in the course of this loose debate with the people here, I have no objection to calling a wide range of work higher-categorical. But for people serious about higher category theory, it’s still reasonable to set the bar high when it comes to demonstrating the role of their theory. It is possible to view all (1-)stack work as higher categorical, since they live inside a 2-category. But it would be nice to do better, since group theory is not evidently 1-categorical just because groups form a 1-category.

Here’s one example to consider. Suppose you do intersection theory on an algebraic stack, which is, by now, a rather old story. It is actually quite explicitly 2-categorical in some sense, since the intersection product is in fact a 2-product. Nevertheless, my guess would be that someone like Angelo Vistoli, who wrote perhaps the first comprehensive work on the theory, would be very reluctant to label it as such (I can ask him). Somehow, and this would take some effort to explain, if I can do it at all, just the presence of a 2-product doesn’t seem to be convincing enough as an ‘application of higher category theory.’

So, is there some particular ingredient in the work you cite that I could trouble you to adduce as evidence in favor of an active role for higher categories?

Posted by: Minhyong Kim on May 15, 2010 5:37 PM | Permalink | Reply to this

### Re: striking category theory

The geometric Langlands program, when considered in the simplest unramified setting, already concerns harmonic analysis of derived categories of sheaves, which one can pretend is 1-categorical but to actually do harmonic analysis or representation theory you have to eventually treat them as some form of $(\infty,1)$-categories. Once you do so you can really develop analogues of many of the structures in representation theory as well as the classical theory of automorphic forms. OK, maybe like me you want to think of $(\infty,1)$-categories as really “homotopic versions of 1-categories”, but however you say it you need a perspective significantly beyond traditional category theory. The basic objects in this story are monoidal dg categories (or if you prefer their $(\infty,2)$-categories of modules), and to work with these you need the machinery developed by Lurie.

In order to really attack geometric Langlands you want to consider analogues of the local Langlands program. This is already implicit in Beilinson-Drinfeld, explicit in Frenkel-Gaitsgory, and essential to Gaitsgory-Lurie’s ongoing work. In fact this is beautifully encapsulated in the framework introduced by Kapustin-Witten, that geometric Langlands is actually about an extended 4d topological field theory. In this setting local Langlands is what happens on the circle — though that suggests an “even more local” Langlands, the theory on a point, underlying the rest. Specifically, the local geometric Langlands program is a Morita equivalence of monoidal $(\infty,1)$-categories (i.e., an equivalence of $(\infty,2)$-categories of module categories) which are roughly analogues of the Hecke algebra of a p-adic group and the group algebra of a local Galois group.

In any case the TFT perspective (which requires higher categories) says that the structures you see in the local story (on the circle) determine the adelic story. What this means in practice is that there are powerful local-to-global principles (chiral homology or the cobordism hypothesis), that will allow you to deduce the global conjectures for curves by “integrating” local versions which have a higher-categorical nature. In fact this is exactly what B-D were doing but without the structural framework in place - or more precisely, they developed much of the necessary framework en route. With current technology one should be able to go much further – I fully expect Gaitsgory-Lurie to prove a form of the geometric Langlands conjecture before very long in this way. A weaker result along these lines is in my ongoing work with Nadler (described in our Character Theory paper), which proves a “dimensionally reduced” (or “decategorified”) form of geometric Langlands using higher categorical ideas (specifically Lurie’s results on TFT). The implications for surfaces are being spelled out in work with Sam Gunningham (very roughly, relating cohomology of stacks of local systems for Langlands dual groups). Now one can pretend there’s nothing 2-categorical here, just like one can pretend Morita theory of algebras or the theory of (1-)stacks is not 1-categorical. But I firmly believe there’s no way around the $(\infty,-)$ aspect of geometric representation theory in one form or another.

Posted by: David Ben-Zvi on May 15, 2010 9:51 PM | Permalink | Reply to this

### Re: striking category theory

Thanks very much! I hope you don’t mind if I inquire a bit more deeply. Let me say beforehand that I have an annoying habit of doggedly pursuing anyone kind enough to explain such things, so please feel free to ignore me if my questions become too persistent.

Here is the little I understand of the Kapustin-Witten framework you refer to:

I will denote by $H_G$ a moduli space of $G$-Higgs bundles on the Riemann surface $X$. So we have the Hitchin fibrations

$H_G\rightarrow A^N \leftarrow H_{G^d}.$

Starting from a $G^d$ local system on $X$ one gets a coherent sheaf on $H_{G^d}$. I only understand the case where this is the skyscraper a sheaf of a point, although I presume it can be more general? Anyways, one would like to apply mirror symmetry at this point to go from the coherent sheaf on $H_{G^d}$ to a $D$-module on $H_G$, which I understand can be done rather easily when the sheaf lives on a smooth fiber of the Hitchin fibration for $H_{G^d}$. My knowledge of this stuff is a few years old, so I have no idea what the status is of the program to extends this to singular fibers, and then, the difficult part of going from a $D$-module on $H_G$ to a $D$-module on $Bun_G$. Are these the points at which $(\infty, 1)$ and $(\infty, 2)$ stacks intervene? When you say you expect Gaitsgory and Lurie to prove a version of geometric Langlands, are you referring to a complete realization of this mirror symmetry strategy?

I have about a million questions about the local version you allude to, but I’ll put them on hold for now.

Posted by: Minhyong Kim on May 15, 2010 10:55 PM | Permalink | Reply to this

### Re: striking category theory

The picture of mirror symmetry along the Hitchin fibration is more complicated to realize mathematically, even for smooth fibers, than your summary. It is certainly true that to a point of the dual Hitchin moduli sitting on a generic fiber we can assign a Lagrangian submanifold of the Hitchin space for G. However getting from there to a D-module on Bun_G is still a serious step. There are three strategies to make this idea into mathematical reality.

1. (Kapustin-Witten picture) The Lagrangian (with a local system on it, describing the position of your original point on its Hitchin fiber) defines an A-brane – mathematically, object of a Fukaya category. KW suggest that the Fukaya category of a holomorphic cotangent bundle should be given as D-modules on the base. This partially inspired a beautiful theorem of Nadler, describing a version of the Fukaya category of a cotangent bundle as the constructible derived category of the base. This is close to what we want here, but it doesn’t yet apply to stacks or to nonexact Lagrangians (such as Hiotchin fibers), though see his more recent paper on Springer theory for a discussion of a special case when the underlying Riemann surface is a cuspidal cubic.

2. (nonabelian Hodge picture) Your sheaf on a Lagrangian in the cotangent defines a Higgs sheaf on BunG, which you can hope to turn into a D-module on BunG using an appropriate version of the nonabelian Hodge theorem. The relevant (and amazing) technical results are due to Takuro Mochizuki, who develops NAH theory with singularities in great generality. It’s still quite hard to make this dream (which was originally due to Donagi in the mid 90s) a reality - this is the subject of ongoing work of Donagi, Pantev and Simpson who understand eg GL_2 this way.

3. (deformation theory picture) In a classical limit, D-modules on the base Bun_G become O-modules on the cotangent bundle (ie Hitchin space), and one can use versions of the Fourier-Mukai transform. At this classical limit the picture you described is fully and satisfyingly worked out in the work of Donagi-Pantev. Now you want to deform back to D-module world, which is the subject of work by Arinkin but I don’t think everything has been worked out yet.

Now everything I wrote about was the “easy” case of smooth fibers. All 3 approaches get much harder with singularities. For the classical limit recent work of Arinkin is extremely promising, but much is left to be understood. (The topology of singular Hitchin fibers is becoming much better understood thanks to Ngo and Chaudouard-Laumon though.)

Posted by: David Ben-Zvi on May 16, 2010 3:31 PM | Permalink | Reply to this

### Re: striking category theory

Let me mention one place where more sophisticated categorical machinery is likely needed. As Matthew explained when we move away from irreducible local systems (ie away from cusp forms) the structure of the corresponding pieces of the automorphic category get more and more complicated - this can be seen from the fact that reducible local systems form a nontrivial derived stack, so the category of coherent sheaves on it is an interesting dg (or $\infty$-)category. This is related to the usual analytical issues in the theory of Langlands spectral decomposition and then the Arthur trace formula - there are precise categorical analogs of all of these issues, and they require $\infty$-categorical machinery to formulate and deal with. In the cuspidal unramified case one can get away with only abelian categories, but no further I think. This is related to the singularities of the Hitchin fibers but maybe more pervasive and subtle – the actual geometric Langlands theory is not really about the Hitchin space away from some nice locus. This is in perfect agreement with the physics BTW – geometric Langlands physically (following Kapustin-Witten) is NOT really about mirror symmetry (a 2d TFT phenomenon), that’s just a nice approximation in some locus, it’s about a duality of four-dimensional field theories.. that physical picture is sophisticated enough to see all the mathematical phenomena we encounter in the subject.

Again I would rather say this much more broadly - geometric Langlands is a chapter in categorified harmonic analysis, where we replace representations of group algebras and Hecke algebras with monoidal dg categories. In order to talk about direct integral decompositions of representations, traces and characters, Fourier transforms, orbital integrals, etc etc one needs some sophisticated categorical tools. The positive take is that all those tools ARE already in place and we CAN really work to imitate the classical theory of say p-adic and adelic groups, and this is what indeed is being done right now.

Posted by: David Ben-Zvi on May 16, 2010 4:22 PM | Permalink | Reply to this

### Re: striking category theory

Maybe I can sneak in one more question for now without irreversibly cluttering up the exchange. What little I understood of the old Beilinson-Drinfeld program involved $D$-modules on $Bun_G$ that were some version of *conformal blocks*. Regardless of how accurate this perception is, I always thought the technology was at the level of conformal field theory, which never looked really $n$-categorical to me. Is this correct?

Posted by: Minhyong Kim on May 15, 2010 11:19 PM | Permalink | Reply to this

### Re: striking category theory

The Beilinson-Drinfeld story with conformal blocks is closely related to the TFT and higher categorical story. Conformal blocks, or their derived version, BD’s chiral homology, is the CFT analogue of the integration performed by the cobordism hypothesis in TFT (as Lurie explains in his TFT announcement). One of the subtleties is that geometric Langlands is not strictly topologically invariant, though it appears so on some coarse level (Kapustin-Witten are really studying a “completed” topological version of geometric Langlands), hence the role of CFT. But the underlying ideas are the same.

The basic field theory idea in Beilinson-Drinfeld is to start with local field data (in their case, critical level reps of loop algebras) and integrate it over a Riemann surface. One then deduces a global duality statement (in their case, construction of Hecke eigensheaves for opers) from local results in representation theory (Feigin-Frenkel’s description of the center of Kac-Moody vertex algebras, and BD’s proof of a local Hecke eigenproperty for the resulting representations). This is the MO (not Math Overflow) in topological field theory – in some sense BD figured out how to solve the 2d cobordism hypothesis in the concrete case where what you assign to the point is controlled by a $E_2$ algebra (topologist’s form of a vertex algebra). Or rather this is a post-Lurie POV on what they did (in the less rigid context of CFT).

Post Kapustin-Witten we have a much clearer formal understanding of the formal structure of this picture. Again this is not part of mirror symmetry, but of 4d field theory – the key part is cut and paste of the Riemann surface into local pieces, which is not part of the mirror symmetry on the Hitchin space story. The way I understand (some small part of) what Gaitsgory-Lurie are doing is pushing this BD/field theory picture to its logical conclusion. To describe the entire geometric Langlands conjecture this way we need to leave the comfortable world of abelian categories (in part since BD’s construction is about opers, which are the global version of Kostant’s slice of the adjoint quotient of a Lie algebra – ie the very nicest/simplest part of the story). One needs to learn how to integrate not just a vertex algebra but a chiral (or $E_2$) category, and then make enough progress in the local story to give the desired global results. This requires a lot of higher categorical sophistication!

[By the way I wasn’t claiming Gaitsgory-Lurie have announced the completion of this picture, I’m just very optimistic about what they’re doing :-) - I don’t want to be putting words in their mouths.]

Posted by: David Ben-Zvi on May 16, 2010 4:52 PM | Permalink | Reply to this

### Re: striking category theory

I think urs makes a good point here:

Category theory is what happens when the Yoneda lemma is applied, the adjoint functor theorem, the monadicity theorem, the Grothendieck construction-theorem and such things,

which complements Minhyong’s thought that

…group theory is not evidently 1-categorical just because groups form a 1-category.

And the evidence provided by David Ben-Zvi and Urs suggests then that the $(\infty, 1)$ theory is also being applied.

So perhaps the next question is when does it become clear that 2-category theory (or $(\infty, 2)$-category theory) is being applied. Perhaps when the categorifications of the Yoneda lemma, the adjoint functor theorem, the monadicity theorem, and the Grothendieck construction-theorem are being applied?

Posted by: David Corfield on May 17, 2010 10:43 AM | Permalink | Reply to this

### Re: striking category theory

David wrote:

So perhaps the next question is when does it become clear that 2-category theory (or (∞,2)-category theory) is being applied. Perhaps when the categorifications of the Yoneda lemma, the adjoint functor theorem, the monadicity theorem, and the Grothendieck construction-theorem are being applied?

I guess then I’ve never done 2-category theory, since I’ve never used those theorems.

Perhaps this is actually true! It could be a sign that I’m not a real 2-category theorist. But I would prefer to think that using these theorems is a sufficient but not necessary condition to be applying 2-category theory. For example, I’d like to think that using the Gray tensor product is another indication that you’re doing 2-category theory.

And while I’m sure the enemy will try to shear off 2-groups from 2-category theory as soon as they understand them well enough to do so, I personally consider 2-group theory to be part of 2-category theory.

Posted by: John Baez on May 17, 2010 8:59 PM | Permalink | Reply to this

### Re: striking category theory

I personally consider 2-group theory to be part of 2-category theory.

We all use sets all day. Are we all doing set theory?

Is distinguishing isomorphisms from identities doing category theory?

To conclude this comment with something that is not a question but a nice “small but striking” application of category theory, one where a little bit of abstract category theory reasoning yields a plethora of concrete results as special cases, I repeat my example from before: Tannaka duality. The $n$Lab entry Tannaka duality has been expanded a bit to make the point more forcefully, but only scratches the surface. For the full abstract story told, see the article

Daniel Schäppi, Tannaka duality for comonoids in cosmoi

a crucial conceptual point (at least for our discussion here) is that the Tannakian adjunction is the composite of the adjunction called the semantics-structure adjunction on p. 16 and the restriction adjunction on p. 18.

The first says that on very general abstract grounds, algebras (in the general sense: in a cosmos $V$) are recovered as the endomorphisms of the fiber functors from their category of all modules. (In the $n$Lab entry this is discussed as an instance of the Yoneda lemma. In Schäppi’s article this is encoded in the fact that the counit of the structure-semantic adjunction is an isomorphism, see the very last sentence of theorem 5.4.) The second one is about this statement when the fiber functor is restricted to just the compact modules.

This is category theory. It can be applied to the Tannakian reconstruction of representation categories of groups and 2-groups and this way help with group- and 2-group theory.

I’d say.

Posted by: Urs Schreiber on May 17, 2010 11:55 PM | Permalink | Reply to this

### Re: striking category theory

‘We all use sets all day. Are we all doing set theory?’

Hey, now you sound like me!

More on David’s great post after I wade my way through the volcanic ash to arrive in Korea tomorrow.

Posted by: Minhyong Kim on May 18, 2010 10:41 AM | Permalink | Reply to this

### Re: striking category theory

So perhaps the next question is when does it become clear that 2-category theory (or (∞,2)-category theory) is being applied.

The single article out there on $(\infty,2)$-categories develops the theory and then applies it to the study of Goodwillie calculus.

Posted by: Urs Schreiber on May 18, 2010 1:36 PM | Permalink | Reply to this

### Re: striking category theory

I don’t think I’d consider that an honest application as of yet: the formalism of (infty,2)-categories is only used at the end to restate the earlier results, not to prove or improve upon them. However, I believe that the formalism of (infty,2)-categories will be extremely useful for getting deeper into the Goodwillie calculus (studying higher derivatives, understanding the chain rule, and so forth).

As for the general question under discussion, I think you can ask it even a level below ordinary category theory. Say, with the theory of abelian groups. If you are interested in proving theorems about real numbers, you might use addition (or multiplication) all over the place. So some might say you are “using” group theory. But you could proceed perfectly well without developing a theory: there’s no need to introduce the general notion of an abelian group if you’re only ever going to discuss one example.
The “theory” emerges when you have seen several examples of abelian groups and realize that they have a lot of similarities: at this point it becomes useful to axiomatize the notion of abelian group so that you don’t have to go through the same reasoning again and again. A little further down the line, you might start to view abelian groups as objects of interest in their own right, and contemplate natural things that you can do with them (like forming tensor products). Axiomatizing the structures you see there will lead you to category theory and the whole story can play out again, one level of abstraction higher.

I would say that the geometric Langlands example (at least the picture envisioned by Dennis Gaitsgory, insofar as I understand it) plays out more or less two levels of abstraction higher than the discussion in the previous paragraph. The “ambient playground” is a particular 2-category (or, more precisely, (infty,2)-category): namely, the (infty,2)-category of (infty,1)-categories which carry an action a loop group LG. Actually, there
are two a priori different (infty,2)-categories which are relevant: representations of LG and representations of LG^, where G^ is the Langlands dual group of G. The most categorically sophisticated (though probably not the hardest) part of the conjectural picture is that these two
(infty,2)-categories are equivalent to each other.
If you believe this, the rest of the picture is concerned with saying how particular objects of these (infty,2)-categories match with each other under the equivalence.

I would say that this doesn’t really need a theory
of (infty,2)-categories for its formulation (just as one
doesn’t need to develop a theory of groups to understand a specific example of a group isomorphism). However, it does need a good theory
of (infty,1)-categories, since these are the basic objects which inhabit the playground (two levels down: you will have difficulty understanding an isomorphism of groups if you don’t what the elements of those groups are).

Posted by: Jacob Lurie on May 19, 2010 8:44 PM | Permalink | Reply to this

### Re: striking category theory

The “ambient playground” [of geometric Langlands] is a particular 2-category (or, more precisely, $(\infty,2)$-category): namely, the $(\infty,2)$-category of $(\infty,1)$-categories which carry an action a loop group $L G$. Actually, there are two a priori different $(\infty,2)$-categories which are relevant: representations of $L G$ and representations of $L \hat G$, where $\hat G$ is the Langlands dual group of $G$. The most categorically sophisticated (though probably not the hardest) part of the conjectural picture is that these two $(\infty,2)$-categories are equivalent to each other.

Is there any place where more details on this can be found?

I understand that you have in preparation something titled

Toric varieties, elliptic cohomology at infinity, and loop group representations

which I suppose is related, but is there anything already available? I’d be quite interested in learning more details about this $(\infty,2)$-categorical picture of Geometric Langlands.

Posted by: Urs Schreiber on May 20, 2010 11:17 AM | Permalink | Reply to this

### Re: striking category theory

Not that I know of. Dennis wrote some informal notes on the picture that may be floating around somewhere.
I’d expect more details to emerge if any of the conjectures get turned into theorems (or, at the very least, get formulated in a precise way).

The (future) paper of mine that you’re referring to isn’t related (it will be about loop group representations in the usual sense: loop groups acting on vector spaces, not categories).

Posted by: Jacob Lurie on May 21, 2010 1:49 PM | Permalink | Reply to this

### Re: Math Overflow

Let me explain my very brief response, which I think maybe you misunderstood — in which case, I only have myself to blame.

A group is, almost by definition, the system of automorphisms of an object. (To make a more precise statement, a group is a small one-object category in which all morphisms are isomorphisms.) Keeping track of the isomorphisms means noticing when two automorphisms are different. In other words, it means noticing that an object can have non-identity automorphisms. In other words, it means noticing that not every group is trivial. In other words, it means noticing that there is such a subject as group theory.

I guess the existence of group theory counts as a “tangible benefit” — not that I’d want to call it “tangible” in front of a non-mathematician.

Posted by: Tom Leinster on May 12, 2010 11:31 PM | Permalink | Reply to this

### Re: Math Overflow

It’s quite easy to argue that group theory is a tangible benefit. Even my mother was somewhat convinced by Marcus du Sautoy’s book. But it’s not so easy to make a compelling case that it’s good to think of it as a one-object category, say to a skeptical but open-minded fellow mathematician. If you have a good argument of that sort, I’d love to hear it. From what I’ve seen of your past writings, I suspect you can come up with one. I apologize for putting it in slightly crude terms, but I’m referring to some argument that’s not just preaching to the choir.

By the way, in case it’s not obvious, there’s no need to convince me personally. For example, in this old paper on the Lefschetz trace formula, there was no way to define sensible local terms without thinking of the group action in terms of groupoids and two-categories.

Posted by: Minhyong Kim on May 13, 2010 10:11 AM | Permalink | Reply to this

### Re: Math Overflow

Well, you could remove the parenthetical reference to categories. It’s not necessary. Don’t we usually tell our students that a group is the system of symmetries of a thing? And a symmetry is a particular type of isomorphism (what grownups call an automorphism). So in this situation, keeping track of isomorphisms means keeping track of symmetries.

Posted by: Tom Leinster on May 13, 2010 1:43 PM | Permalink | Reply to this

### Re: Math Overflow

I agree. But if that were all, the need to be careful about distinguishing equality and isomorphism would almost never arise. So what I would like is to *retain* the reference to categories, but still be convincing.

Posted by: Minhyong Kim on May 13, 2010 2:29 PM | Permalink | Reply to this

### Re: Math Overflow

And in regard to Tom’s comment, I think I understand what’s meant. But I believe we would be hard pressed to convince someone even mildly skeptical that usual group theory has to be interpreted in terms of ‘distinguishing between isomorphism and equality.’

That is, we might visualize a group as a set of some sort equipped with a composition. Now, elements will be isomorphisms of something when the group is acting somewhere. But then, in any of the commonplace situations, the danger of someone regarding $y=gx$ as an equality between $x$ and $y$ will almost never come up. In short, it would be nice to have accessible examples where the subtle lessons people like to emphasize on this site come into play very actively and clearly.

Posted by: Minhyong Kim on May 12, 2010 11:25 PM | Permalink | Reply to this

### Re: Math Overflow

But I believe we would be hard pressed to convince someone even mildly skeptical that usual group theory has to be interpreted in terms of ‘distinguishing between isomorphism and equality.’

But if we don’t distinguish between isomorphism and equality, then, say, a nontrivial permutation of $n$ letters is not distinguished from the trivial permutation. Hence no group theory.

Posted by: Urs Schreiber on May 12, 2010 11:35 PM | Permalink | Reply to this

### Re: Math Overflow

Of course. But most people will be able to make that particular distinction and prove good theorems even in complete ignorance of a category-theoretic framework, which I believe you regard as essential.

Posted by: Minhyong Kim on May 12, 2010 11:43 PM | Permalink | Reply to this

### Re: Math Overflow

But most people will be able to make that particular distinction and prove good theorems even in complete ignorance of a category-theoretic framework, which I believe you regard as essential.

Many people even compute complicated integrals correctly, in completey ignorance of measure theory!

Okay, that’ll be it from me for today. I need to call it quits now. Up to isomorphism (in German I call “quits” something else :-).

Posted by: Urs Schreiber on May 12, 2010 11:55 PM | Permalink | Reply to this

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