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May 22, 2007

The Two Cultures of Mathematics Revisited

Posted by David Corfield

Where did we get to in our discussion of the two cultures of mathematics? To explore the possibility that interaction may be possible between what Gowers called ‘combinatorics’ and our Café subculture we were set the challenge of categorifying instances of the Cauchy–Schwarz inequality, which, unless I missed something, didn’t result in any noticeable success.

Now, an extreme wing of our subculture would take Urs’ remark

One knows one is getting to the heart of the matter when the definitions in terms of which one conceives the objects under consideration categorify effortlessly.

and replace the when by when and only when.

I was probing in this direction in my paper, Categorification as a Heuristic Device, when I put:

A happenstantial equation is one which cannot be categorified productively.

The term ‘happenstantial’ is supposed to contrast with ‘law-like’. I had written something on the idea of mathematical laws inspired by reading Poincaré say

Les faits mathématiques dignes d’être étudiés, ce sont ceux qui, par analogie avec d’autres faits, sont susceptibles de nous conduire à la connaissance d’une loi mathématique de la même façon que les faits expérimentaux nous conduisent à la connaissance d’une loi physique. Ce sont ceux qui nous révèlent des parentés insoupçonnées entre d’autres faits, connus depuis longtemps, mais qu’on croyait à tord étrangers les uns aux autres.

Now, if we can’t categorify the Cauchy–Schwarz inequality, this would seriously weaken the extreme view. But perhaps we can think of more subtle ways of bridging the gap between the cultures. Terence Tao reports on Shing-Tung Yau’s lectures in UCLA’s Distinguished Lecture Series in a field “adjacent” to his own areas of expertise. Plenty of Yau’s themes over the three lectures are adjacent to Café interests too.

Posted at May 22, 2007 9:49 AM UTC

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Re: The Two Cultures of Mathematics Revisited

Yau’s extraordinarily verbal lecture notes are here: 1, 2, 3.

Posted by: David Corfield on May 22, 2007 11:56 AM | Permalink | Reply to this

Re: The Two Cultures of Mathematics Revisited

David wrote:

To explore the possibility that interaction may be possible between what Gowers called “combinatorics” and our Café subculture we were set the challenge of categorifying instances of the Cauchy–Schwar inequality, which, unless I missed something, didn’t result in any noticeable success.

Hmm.

Here’s a preliminary attempt to categorify the Cauchy–Schwarz inequality.

Let Rep(G)Rep(G) be the category of finite-dimensional representations of a compact topological group GG — for example, a finite group.

For any representations A,BRep(G)A,B \in Rep(G), there’s an obvious inclusion of vector spaces:

hom(A,B)A *B hom(A,B) \hookrightarrow A^* \otimes B

Here:

  • hom(A,B)hom(A,B) is the space of intertwining operators from the representation AA to the representation BB. This is the external hom in Rep(G)Rep(G).
  • A *BA^* \otimes B is the space of all operators from the vector space AA to the vector space BB. This is the internal hom in Rep(G)Rep(G).

The inclusion says, ironically: the external hom is smaller than the internal hom. Some things look bigger from the inside!

If we take dimensions, we get

dim(hom(A,B))dim(A *B) dim(hom(A,B)) \le dim(A^* \otimes B)

This looks vaguely Cauchy–Schwarzian. Let’s see what it says!

Every representation is of the form

A= i a ie iA = \oplus_i \mathbb{C}^{a_i} \otimes e_i

where e ie_i ranges over a basis of irreducible representations of GG.

By Schur’s lemma we have this isomorphism of vector spaces:

hom(A,B) i( a i b i) hom(A,B) \cong \oplus_i (\mathbb{C}^{a_i} \otimes \mathbb{C}^{b_i})

but we also clearly have this isomorphism of vector spaces:

A *B( i a i)( j b j) A^* \otimes B \cong (\oplus_i \mathbb{C}^{a_i}) \otimes (\oplus_j \mathbb{C}^{b_j})

The inclusion

hom(A,B)A *B hom(A,B) \hookrightarrow A^* \otimes B

is just the obvious diagonal inclusion

i( a i b i)( i a i)( j b j) \oplus_i (\mathbb{C}^{a_i} \otimes \mathbb{C}^{b_i}) \hookrightarrow (\oplus_i \mathbb{C}^{a_i}) \otimes (\oplus_j \mathbb{C}^{b_j})

If we decategorify by taking dimensions, we thus get

ia ib i( ia i)( jb j) \sum_i a_i b_i \le (\sum_i a_i) (\sum_j b_j)

Alas, this ain’t the Cauchy–Schwarz inequality! Cauchy–Schwarz says:

( ia ib i) 2( ia i 2)( jb j 2) (\sum_i a_i b_i)^2 \le (\sum_i a_i^2) (\sum_j b_j^2)

Unlike Cauchy–Schwarz, the inequality I got doesn’t hold for all real numbers a ia_i and b jb_j. My proof only shows it holds for all natural numbers — which is also obvious directly.

But, all hope is not lost! Maybe we can get the Cauchy–Schwarz inequality using similar tricks. For example, by finding an inclusion

hom(A,B)hom(B,A)hom(A,A)hom(B,B)hom(A,B) \otimes hom(B,A) \hookrightarrow hom(A,A) \otimes hom(B,B)

Anyone want to take a stab at it?

I think, though, that we should call this:

ia ib i( ia i)( jb j) \sum_i a_i b_i \le (\sum_i a_i) (\sum_j b_j)

the TARDIS inequality.

Posted by: John Baez on May 23, 2007 1:57 AM | Permalink | Reply to this

Re: The Two Cultures of Mathematics Revisited

Looks similar to quantum probabilities: there are many evidences why quantum amplitude (with quadratic law) is most natural generalization of classic probability, but no distinct categorified evidence, as I can see. The problem of quadratic laws categorification, ah?

Posted by: osman on May 23, 2007 7:49 AM | Permalink | Reply to this

Re: The Two Cultures of Mathematics Revisited

A dumb question from someone who hasn’t taken the time to understand very much of this category stuff: all the “basic” instances of Cauchy-Schwarz start by looking at (f+g) 2(f+g)^2 (or f+g,f+g\langle f+g,f+g\rangle) and then splitting it into bits like f 2f^2, fgfg, etc. So why isn’t the natural place to start to be looking for some way (lets call it \vee just because this operator hasn’t been used elsewhere) of combining AA and BB such that hom(AB,AB)hom(A \vee B,A \vee B) can be broken into pieces? Ie, I don’t see why one would expect the space corresponding to the “squared cross terms” to be a included in the “product of square terms” rather than “complementary to it with respect to some combined structure”?

Posted by: dave tweed on May 23, 2007 9:47 AM | Permalink | Reply to this

Re: The Two Cultures of Mathematics Revisited

Minor correction: Cauchy-Schwarz actually begins considering (f+λg) 2(f+\lambda g)^2 for all λ\lambda. So the analogous combined space will be even more complicated than just ABA \vee B (for some \vee), but I think the basic question why one wouldn’t expect to start with some superstructure still arises?

Posted by: dave tweed on May 23, 2007 10:06 AM | Permalink | Reply to this

Re: The Two Cultures of Mathematics Revisited

In case it’s helpful to explicitly state for other neophytes here (gleaned from other comments), the “implicit” block to coming up with an categorical analogy to the |f+λg| 2|f+\lambda g|^2 proof of Cauchy-Schwarz is that you can’t use the notion of subtraction anywhere in the proof (because you want to work in contexts where subtraction may not be applicable); I’d only thought about not using it in the first step. This seems unlikely to be acheivable in an analogy to the quadratic form proof of C-S.

Posted by: dave tweed on May 23, 2007 7:33 PM | Permalink | Reply to this

Re: The Two Cultures of Mathematics Revisited

Why do I get the feeling that there won’t be a natural way of doing this? Back here when we were talking about Cauchy-Schwarz for sets, Terry Tao pointed out a reliance on Choice.

I’m hearing a faint tinkling of that phenomenon encountered when categorifying series to species. We saw that

cosh(2x)sinh 2(x)+cosh 2(x)cosh (2x) \equiv sinh^2(x) + cosh^2(x)

categorifies to

The species ‘2-coloured even numbered set’ is isomorphic to the species ‘either an ordered pair of odd numbered sets or an ordered pair of even numbered sets’.

However, trying to categorify

cosh 2(x)sinh 2(x)+1cosh^2(x) \equiv sinh^2(x) + 1,

one can’t arrive at an isomorphism. (Ah, found it in a trice (p. 65), using Mike Stay’s excellent index.)

I seem to remember an e-mail chat we had about this that to have an isomorphism we would need extra structure on the sets. Placing the structure ‘pair of even numbered sets’ on a pointed set, one could flick the pointed element into the other set to make ‘pair of odd numbered sets’.

I also remember a piece of wanton speculation to the effect that this difference reflected why the second equivalence is more interesting than the first, with its link to the hyperbola, etc.

Posted by: David Corfield on May 23, 2007 12:24 PM | Permalink | Reply to this

Re: The Two Cultures of Mathematics Revisited

Analysis as a field is almost completely “nonsense”-free, and I think that this is another reason you get the sense that there’s not a natural way of doing this. Maybe, if one could articulate what the obstacle is a little more clearly, this is why there’s so little algebra in analysis.

Or maybe analysts are doing it wrong.

Posted by: Changbao on May 23, 2007 4:58 PM | Permalink | Reply to this

Re: The Two Cultures of Mathematics Revisited

How should we view Sato’s work in relation to this question of algebra in analysis? Pierre Schapira writes:

Looking back, forty years later, we realize that Sato’s approach to mathematics is not so different from that of Grothendieck, that Sato did have the incredible temerity to treat analysis as algebraic geometry, and that he was also able to build the algebraic and geometric tools adapted to his problems.

Would some analysts see the reach of Sato’s concepts as limited?

Posted by: David Corfield on May 24, 2007 8:58 AM | Permalink | Reply to this

Re: The Two Cultures of Mathematics Revisited

This identity is not true for species, but it is for K-species with K with enough structure (in this case, I seem to remember that you need Q-species where the usual species are N-species [rationals versus the rig of non-negative integers]). This can be found in a paper of Yeh, in LNM 1234.

Posted by: Jacques Carette on May 24, 2007 3:24 AM | Permalink | Reply to this

Re: The Two Cultures of Mathematics Revisited

David wrote:

Why do I get the feeling that there won’t be a natural way of doing this? Back here when we were talking about Cauchy-Schwarz for sets, Terry Tao pointed out a reliance on Choice.

Hmm! That’s usually a sign that one is doing something hugely non-canonical, which doesn’t deserve to be categorified.

The first time you guys had this discussion of categorifying Cauchy–Schwarz, I wasn’t paying much attention. Now I think I see what all the fuss is about.

In ordinary uncategorified math, to get an inequality, you usually need an inequality. Cauchy–Schwarz is really just a spinoff of the basic inequality

AA0A \cdot A \ge 0

where you see that we can only have

(A+λB)(A+λB)0(A + \lambda B) \cdot (A + \lambda B) \ge 0

for all λ\lambda if ABA \cdot B doesn’t get too big.

All of what I just wrote is perfectly easy to categorify; the problem — I think — is that to derive Cauchy–Schwarz you need to subtract.

While I’ve spent a lot of time trying to categorify subtraction and get a good theory of negative sets, it hasn’t fully gelled, so often I’m willing to follow Nancy Reagan’s advice and just say no when it comes to the use of subtraction in categorified settings: when tempted to use subtraction to make a result look prettier, just don’t do it!

If we followed this policy with regards Cauchy–Schwarz, we’d just try to categorify

(A+λB)(A+λB)0(A + \lambda B) \cdot (A + \lambda B) \ge 0

and this would be perfectly easy to do; for example, in many contexts we have a monomorphism

0(A+λB)(A+λB)0 \hookrightarrow (A + \lambda B) \otimes (A + \lambda B)

or

0hom((A+λB),(A+λB))0 \hookrightarrow hom((A + \lambda B), (A + \lambda B))

(an inner product being more like an internal hom than a tensor product).

I’m hearing a faint tinkling of that phenomenon encountered when categorifying series to species.

Yes, there are some nice identities that don’t categorify unless you bend the rules, and surely this is trying to tell us something.

I seem to remember an e-mail chat we had about this that to have an isomorphism we would need extra structure on the sets. Placing the structure ‘pair of even numbered sets’ on a pointed set, one could flick the pointed element into the other set to make ‘pair of odd numbered sets’.

I seem to remember that conversation. Right. You seem to have just proved cosh 2(x)=sinh 2(x)\cosh^2 (x) = \sinh^2(x), but of course this trick of equipping our set with a distinguished point doesn’t work for the empty set, which contributes the ‘+1+ 1’ to the identity cosh 2(x)=sinh 2(x)+1\cosh^2(x) = \sinh^2(x) + 1.

Equipping sets with extra structure, we increase our power to categorify identities, but diminish the interest of the results.

Posted by: John Baez on May 24, 2007 2:35 PM | Permalink | Reply to this

Re: The Two Cultures of Mathematics Revisited

Are there lessons to be learned about the difficulty of a good theory of negative sets?

By the way, with regard to your page on counting, I found a better translation of the Weyl quotation you have there:

Mathematics is not the rigid and petrifying schema, as the layman so much likes to view it; with it, we rather stand precisely at the point of intersection of restraint and freedom that makes up the essence of man itself.

Posted by: David Corfield on May 24, 2007 3:09 PM | Permalink | Reply to this

Re: The Two Cultures of Mathematics Revisited

Here’s a preliminary attempt to categorify the Cauchy-Schwarz inequality.

It seems to me that the John’s approach above has a somewhat different flavor than Terry Tao’s approach:

in one case the product is sought to be categorified by a categorical product – in the other case one attempts to identify it with a Hom-space!

I like John’s approach for the following reason: we know from examples arising in concrete applications that it is often indeed the case that the Hom-functor plays the role of the inner product on a vector space.

So, sticking with the culture of problem-solving, I now decide to be interested in the formal game of categorification here only in as far as I know it relates to concrete applications.

So: inner product should go to Hom.

Moreover, we know that complex conjugation should go to reversal of arrows, in this context.

More precisely, the existence of a structure that allows complex conjugation should translate to a category with duals, such that we have an operation () :(f:ab)(f :ba). (\cdot)^\dagger : (f : a \to b ) \mapsto (f^\dagger : b \to a) \,.

This should mean that we read a,b \langle a , b \rangle as Hom(a,b) Hom(a,b) and a,bb,a \langle a , b \rangle \langle b, a \rangle as Hom(a,b)Hom(b,a). Hom(a,b) \otimes Hom(b,a) \,.

I notice that we have a canonical morphism c:Hom(a,b)Hom(b,a)End(a)End(b) c : Hom(a,b) \otimes Hom(b,a) \to End(a) \otimes End(b) which acts as c:(f:ab)(g:ba)(gf)(fg). c : (f : a \to b)\otimes (g : b \to a) \mapsto (g \circ f) \otimes (f \circ g) \,.

This is in general not epi.

Of course it is also in general not mono, so this might not be quite the right answer yet, but I thought it might be worth mentioning nevertheless.

Posted by: urs on May 23, 2007 7:32 PM | Permalink | Reply to this

Re: The Two Cultures of Mathematics Revisited

I think I can strenghten that statement, exhibiting the entire natural transformation:

Consider a category CC with duals such as Hilb\mathrm{Hilb}, such that afbf a=aIda a \stackrel{f}{\to} b \stackrel{f^\dagger}{\to} a = a \stackrel{\mathrm{Id}}{\to} a for all morphisms ff.

In this case we have on top of the canonical Hom-functor Hom:C op×CSet \mathrm{Hom} : C^{\mathrm{op}} \times C \to \mathrm{Set} (I don’t want to be too specific about where Hom takes values) an endomorphism functor End:CSet \mathrm{End} : C \to \mathrm{Set} which acts as End:(afb)(End(a)Ad fEnd(b)). \mathrm{End} : (a \stackrel{f}{\to} b) \mapsto (\mathrm{End}(a) \stackrel{\mathrm{Ad}_f}{\to} \mathrm{End}(b)) \,.

Then, I claim, we have a canonical natural transformation

Hom(() ,)×σ *Hom(() ,)End()×End() \mathrm{Hom}((\cdot)^\dagger, \cdot) \times \sigma^*\mathrm{Hom}((\cdot)^\dagger, \cdot) \to \mathrm{End}(\cdot) \times \mathrm{End}(\cdot)

between functors C×CSetC \times C \to \mathrm{Set}, where σ\sigma is the standard braiding on Cat\mathrm{Cat} (i.e. just swaps the arguments).

Its component map is that from my previous comment.

Read

\;\;\; Hom is inner product

\;\;\; End is norm square

to see something like Cauchy-Schwarz.

Posted by: urs on May 23, 2007 8:21 PM | Permalink | Reply to this

Re: The Two Cultures of Mathematics Revisited

This gave me a sense of deja vu all over again :)

Simply reversing the arrows leads to some inner product that may ultimately not be what you want, but you can get what you want by introducing deformations once the mechanics are worked out.

Just a quick (and random) comment from the sidelines…

Posted by: Eric on May 23, 2007 10:08 PM | Permalink | Reply to this

Re: The Two Cultures of Mathematics Revisited

(1) I must be missing something, it seems to me that the map c isn’t linear. The definition given is c:(f:ab)(g:ba)(gf)(fg)c: (f: a \to b) \otimes (g : b \to a) \mapsto (g \circ f) \otimes (f \circ g) So c(f 1g+f 2g)=c((f 1+f 2)g)=((f 1+f 2)g)(g(f 1+f 2))c(f_1 \otimes g + f_2 \otimes g)=c((f_1 + f_2) \otimes g) = ( (f_1 + f_2) \circ g) \otimes (g \circ (f_1 + f_2) ). Since our category is abelian, this is (f 1g+f 2g)(gf 1+gf 2)(f_1 \circ g+f_2 \circ g) \otimes (g \circ f_1 + g \circ f_2). This is not c(f 1g)+c(f 2g)=(f 1g)(gf 1)+(f 2g)(gf 2)c(f_1 \otimes g)+c(f_2 \otimes g)=(f_1 \circ g) \otimes (g \circ f_1)+(f_2 \circ g) \otimes (g \circ f_2).

(2) There are very reasonable abelian categories, with finite dimensional Hom spaces, in which dim Hom(A,A) dim Hom(B,B) can be less than dim Hom(A,B) dim Hom(B,A). For example, consider the category of coherent sheaves on a disjoint union of two CP 2CP^2’s. Take A to be O(1) on the first CP 2CP^2 and O(0) on the second, and take B to be the reverse. Then Hom(A,A) consists of functions which are constant on each component, and hence Hom(A,A) is two dimensional. The same holds for Hom(B,B). But Hom(A,B) consists of functions which are zero on the first component but can be any section of O(1) on the second component, and hence Hom(A,B) is three dimensional. The same holds for Hom(B,A). 2*2 is not greater than 3*3.

I suspect that I can build a similar example on a connected variety which is proper but not projective. All I need is a line bundle L such that H 0(L)H^0(L) and H 0(L 1)H^0(L^{-1}) both have dimension bigger than 1. (Then I take A to be the structure sheaf and B to be L.) But I can’t think of such an example right now.

Posted by: David Speyer on May 23, 2007 11:46 PM | Permalink | Reply to this

Re: The Two Cultures of Mathematics Revisited

David Speyer commented my suggestion with

I must be missing something,

Well, as I said:

this might not be quite the right answer yet, but I thought it might be worth mentioning nevertheless.

One reason is

There are very reasonable abelian categories, with finite dimensional Hom spaces, in which dim Hom(A,A) dim Hom(B,B) can be less than dim Hom(A,B) dim Hom(B,A).

Yes, as I remarked, cc is in general not epi but also not mono.

So if you decategorify cc in “the obvious way” by passing to isomorphism classes, it doesn’t really yield Cauchy-Schwarz.

I am not sure, though, that this is the right way to look at it: if we were talking about an ordinary product, then yes. But it might be relevant that we are talking about an inner product (“scalar product”). It’s not obvious to me that this, as a concept, should be thought of, as seems to be the premise of other comments here, as nothing but an abbreviation for the component formula v,w= iv¯ iw i, \langle v,w \rangle = \sum_i \bar v_i w_i \,, and hence just as an abbreviation for a combination of ordinary products.

Posted by: urs on May 24, 2007 9:06 AM | Permalink | Reply to this

Re: The Two Cultures of Mathematics Revisited

Here is a variation on Urs’ approach, which has the advantage of being linear (so it is immune to David Speyer’s first objection), since one can write it down with the graphical calculus, which I don’t think you can do for Urs’ original formula (if you could , Urs’ formula would have been linear).

In other words, I’m continuing in the spirit of Urs’ approach - following John’s suggestion - and interpreting “categorifying the Cauchy-Schwarz inequality” as finding an injective morphism

(1)hom(A,B)hom(B,A)hom(A,A)hom(B,B). hom(A,B) \otimes hom(B,A) \hookrightarrow hom(A,A) \otimes hom(B,B).

Here I’m assuming I’m working in a 2-Hilbert space. This means we’re working in a linear category which is enriched in Hilb (i.e. there are inner products on the hom-sets), and which has an involution *:hom(A,B)hom(B,A)* : hom(A, B) \rightarrow hom(B,A).

As in John’s post, the best example of a 2-Hilbert space is the category Rep(G)Rep(G) of (possibly projective) unitary representations of a finite group, or groupoid, where the involution is the adjoint of a linear map.

As David essentially showed in his second objection, things like “coherent sheaves” aren’t 2-Hilbert spaces (though they’re closely related), so this interpretation of the Cauchy-Schwarz inequality won’t work there.

On the other hand, if one thinks of the Cauchy-Schwarz inequality as having to do with inner products in a vector space ( is that the right way to think of it?), and then one tries to categorify this concept, it is natural one should work in a “2-inner product space”, i.e. a 2-Hilbert space.

After this long introduction, here’s the graphical definition of the morphism. Remember these ribbon-like diagrams take place in the monoidal category Hilb. To be reminded how they work, look at this post of mine or John’s notes on classical vs quantum computation.

Here is the morphism (top-to-bottom):

pic

Recall that the “half-twist” maps are **-operations, while the upside-down Y’s are the adjoints of the composition maps.

One can show this map is isometric (and hence injective) by graphical manipulations. I’m running out of space, so let’s just use brute force, and see what the map actually does in terms of a chosen basis.

So let a i pa_i^p be a basis for AA, where ii runs through the simple objects (irreducible components), and pp runs through the ii‘th isotypic part of AA. Similarly let b i qb_i^q be a basis for BB.

We then get a basis (AB) i p,q(AB)_i^{p,q} for the hom-set hom(A,B)hom(A,B), where we are writing (AB) i p,q(AB)_i^{p,q} for the map ABA \rightarrow B which sends

(2)a i pb i q. a_i^p \mapsto b_i^q.

Similarly we get bases (AA) i p,p(AA)_i^{p,p'} and (BB) i q,q(BB)_i^{q,q'} for hom(A,A)hom(A,A) and hom(B,B)hom(B,B).

In terms of these bases, the map

(3)hom(A,B)hom(B,A)hom(A,A)hom(B,B) hom(A, B) \otimes hom(B,A) \hookrightarrow hom(A,A) \otimes hom(B,B)

sends

(4)(AB) i p,q(BA) j q,pδ ij(AA) i p,p(BB) i q,q, (AB)_i^{p,q} \otimes (BA)_j^{q', p'} \mapsto \delta_{ij} (AA)_i^{p,p'} \otimes (BB)_i^{q,q'},

which is clearly injective. Another way to think of it graphically is as a kind of “crossing” duality pic:

pic

Posted by: Bruce Bartlett on May 24, 2007 3:46 PM | Permalink | Reply to this

Re: The Two Cultures of Mathematics Revisited

Thanks Bruce, great! I was hoping you would look into this and see how to do it.

I am glad this blog exists and has contributors like you.

Posted by: urs on May 24, 2007 4:51 PM | Permalink | Reply to this

Re: The Two Cultures of Mathematics Revisited

Wow, that’s really pretty. So the point is, if we want dim Hom to act like a positive definite inner product, we need to already have some sort of positive definite structure (a Hilbert space, in this case) on Hom(V,W).

Thanks!

Posted by: David Speyer on May 24, 2007 5:35 PM | Permalink | Reply to this

Re: The Two Cultures of Mathematics Revisited

Dear Bruce, is the map (4) really injective? When i is not equal to j it looks like it’s mapping to zero, to me.

Posted by: Terry Tao on May 24, 2007 6:03 PM | Permalink | Reply to this

Re: The Two Cultures of Mathematics Revisited

Ah, you’re right, it isn’t injective! This oversight on my part comes from working things out “one isotypic part at a time”, and then adding together all the contributions at the end, without thinking carefully enough about how the final result pans out.

By the way, if our 2-Hilbert space is Rep(G)Rep(G) for a finite group GG, and we have given the hom-sets the “trace inner product”,

(1)(f,g)=Tr(f *g), (f,g) = Tr(f^* g),

then there should be a factor of dim(e i)dim(e_i) appearing (where e ie_i is the i’th irreducible representation) which I missed before,

(2)(AB) i p,q(BA) j p,qδ ijdim(e i)(AA) i p,p(BB) i q,q. (AB)_i^{p,q} \otimes (BA)_j^{p',q'} \mapsto \delta_{ij} \dim(e_i) \, (AA)_i^{p,p'} \otimes (BB)_i^{q,q'}.

I guess on the diagonal, i.e. where i=ji=j, the map is just a simple bijective rearrangement of the indices (multiplied by a scalar factor), while off the diagonal it’s zero. And that’s not exactly Cauchy-Schwarz pic.

I guess the map I presented in graphical notation can’t be of fundamental importance : it looks a little ugly.

Another thing that’s lacking in the approach I gave is a ‘TARDIS-like’ internal hom vs external hom dichotomy, which I found quite appealing.

While I’ve written this David Speyer has written down some interesting points.

Posted by: Bruce Bartlett on May 24, 2007 10:48 PM | Permalink | Reply to this

Re: The Two Cultures of Mathematics Revisited

Darn, Terry is right.

Moreover, there are representation theoretic reasons that we can’t do this in a natural way. Let C be the category of finite dimensional representations of the group Z/2. If we want to fit this into the 2-Hilbert space formalism, I think we can do this by having the representations come equipped with an invariant positive definite inner product, so that we get a Hilbert space structure on Hom(A,B) by f,g=Tr(gf *)\langle f,g \rangle=Tr(g f^*). We’ll write an object A of C as (A(+)A())(A(+) \oplus A(-)), where A(±)A(\pm) are the isotypic components of A. Then Aut(A)=GL(A(+))×GL(A())Aut(A)=GL(A(+)) \times GL(A(-)). If representations come equipped with invariant inner products, then the automorphism group of A is O(A(+))×O(A())O(A(+)) \times O(A(-)) (where O(V) is the orthogonal group on an inner product space V).

The proposed goal is to find an injection

Hom C(A,B)Hom C(B,A)Hom C(A,A)Hom C(B,B).Hom_C(A,B) \otimes Hom_C(B,A) \hookrightarrow Hom_C(A,A) \otimes Hom_C(B,B).

If this injection were natural, it would in particular be Aut(A)×Aut(B)Aut(A) \times Aut(B) equivariant. But the irreducible representation A(+)A() *B(+)B() *A(+) \otimes A(-)^* \otimes B(+) \otimes B(-)^* (where ** is dualization) appears on the left but not the right.

A similar argument with symmetric groups shows that there is no natural way to make the argument work with the finite set version either.

Posted by: David Speyer on May 24, 2007 9:40 PM | Permalink | Reply to this

Re: The Two Cultures of Mathematics Revisited

There’s a slightly nicer, more invariant way to write down the calculations John is performing above, as John is probably aware. It’s true that every representation is of the form

(1)A= i a ie i A = \oplus_i \mathbb{C}^{a_i} \otimes e_i

where e ie_i ranges over a basis of irreducible representations of GG, but this is a noncanonical isomorphism… it is only canonical to split AA up into its isotypic factors, but further splitting requires choices, as I’m sure we all agree.

A more invariant way to portray the noncanonical isomorphism

(2)hom(A,B) i a i b i hom (A, B) \cong \oplus_i \mathbb{C}^{a_i} \otimes \mathbb{C}^{b_i}

is to write

(3)hom(A,B) ihom(A,e i)hom(e i,B) hom (A,B) \cong \oplus_i hom(A, e_i) \otimes hom(e_i, B)

which is canonical, since it is the inverse of the canonical composition map,

(4)hom(A,e i)hom(e i,B)hom(A,B). hom (A,e_i) \otimes hom(e_i, B) \rightarrow hom(A,B).

I mentioned this in a previous post, where I also showed how one can write it in ribbon-like diagrams… which all of us at the n-cafe (hopefully!) learnt from John’s classical versus quantum computation notes :-).

Posted by: Bruce Bartlett on May 24, 2007 1:38 AM | Permalink | Reply to this

Re: The Two Cultures of Mathematics Revisited

I’ve just naively checked following code in Python:

ss = lambda x, y: (pow(x,x)*pow(y,y)) - (pow(x,y)*pow(y,x))
for i in xrange(1,1000):
....for j in xrange(i,1000):
........if 0 > ss(i,j): print i, j

In [1,1000] it’s always > 0!

Posted by: osman on May 25, 2007 12:37 PM | Permalink | Reply to this

Re: The Two Cultures of Mathematics Revisited

In other words, I’ve tried to calculate x xy yx yy xx^x y^y-x^y y^x for integer xx and yy in interval from 1 to 1000. And in this interval always 0x xy yx yy x0 \leq x^x y^y-x^y y^x

Posted by: osman on May 25, 2007 4:12 PM | Permalink | Reply to this

Re: The Two Cultures of Mathematics Revisited

Oh, I’m so stupid! If y=x+ny=x+n then x xy yy xx y=y nx n0x^x y^y - y^x x^y = y^n - x^n \geq 0! :(

Posted by: osman on May 25, 2007 9:33 PM | Permalink | Reply to this

Re: The Two Cultures of Mathematics Revisited

Nobody answers to me, so I feel I say something irrelevant or something well-known. Anyway I must complete the proof whithout errors. So, excuse me for one more post.

I want to prove:

in category of finite sets Hom(A,B)×Hom(B,A)Hom(A,B) \times Hom(B,A) is isomorphic to some CHom(A,A)×Hom(B,B)C \subseteq Hom(A,A) \times Hom(B,B).

In this case it’s enough to prove that for any natural xx and yy we have x xy yx yy x0x^x y^y - x^y y^x \geq 0. This expression is symmetric for xx and yy, so we can choose y=x+ny = x + n for some natural nn. So we have:

x xy yx yy x=x xy x(y nx n)(y nx n)0x^x y^y - x^y y^x = x^x y^x (y^n - x^n) \geq (y^n - x^n) \geq 0.

Posted by: osman on May 26, 2007 10:33 AM | Permalink | Reply to this

Re: The Two Cultures of Mathematics Revisited

Sorry that no-one replied to your comments; they were difficult to understand at first.

I think it’s clear that there exists an injection

(1)Hom(A,B)×Hom(B,A)Hom(A,A)×Hom(B,B) Hom(A,B) \times Hom(B,A) \hookrightarrow Hom(A,A) \times Hom(B,B)

in the category of finite sets (or using \otimes in a linear category), merely by a counting argument, as you’ve shown. I think the point of this discussion was to try and find a natural injection, as originally suggested by John.

Urs tried one way, and I tried another way, but both methods basically failed, for one reason or another.

Then David Speyer essentially proved a “no-go” theorem, which suggested that any attempt to find a natural injection

(2)Hom(A,B)×Hom(B,A)Hom(A,A)×Hom(B,B) Hom(A,B) \times Hom(B,A) \hookrightarrow Hom(A,A) \times Hom(B,B)

would fail (either in finite sets or in some category like Rep(G)Rep(G)), because it would necessarily violate symmetry.

Thus it seems we reached a dead-end in this direction, and I think people have moved on now to try more exotic methods. That’s the way I interpret this discussion anyway.

Posted by: Bruce Bartlett on May 26, 2007 5:26 PM | Permalink | Reply to this

Re: The Two Cultures of Mathematics Revisited

Thank you Bruce! At leats I understood how to use itex here. :) I’m sorry for my first obscure posts.

It’s a little difficult for me to watch you conversations, because I still don’t know enough mathematics. But now, thanks to your explanations, I know the main result of this thread.

Posted by: osman on May 27, 2007 11:58 AM | Permalink | Reply to this

Re: The Two Cultures of Mathematics Revisited

Hi again! I’m happy to see you take up this puzzle again - welcome to the “problem solving” culture of mathematics :-).

I have a few comments. Firstly, I recommend reading Gowers’ thoughts on the Cauchy-Schwarz inequality. He mostly discusses the v+λw 2\| v + \lambda w\|^2 approach to proving the Cauchy-Schwarz inequality, and only briefly mentions the Lagrange identity approach at the end. Of the two I feel the Lagrange identity approach has a slightly better chance of categorifying, since it is after all an identity. (A third approach is algorithmic, manipulating v and w by incremental changes to narrow the gap in the inequality until equality is attained, but this probably has the least chance of being categorified - on the sphere of mathematics, I would consider the mathematics of algorithms is being almost the antipode of the mathematics of categories (try defining the category Algorithm and you’ll see what I mean).)

Secondly, one thing about the “problem solving” culture as opposed to the “theory building” is that we tend to make progress by working upward from special cases, rather than downward from more general formulations. In the last post on this topic, Robin was able to categorify the arithmetic mean-geometric inequality

A×B+B×AA×A+B×B A \times B + B \times A \to A \times A + B \times B

for finite sets A, B by introducing reasonable axioms on Set. I think the next step is to obtain a relative version of this injection, namely

A× CB+B× CAA× CA+B× CB A \times_C B + B \times_C A \to A \times_C A + B \times_C B

where we are given maps ACA \to C and BCB \to C, thus A and B are basically bundles over C. If we think non-categorically, the latter injection is a trivial consequence of the former, as we can work on each fibre separately and sum up; but I believe you can’t get away with that categorically, and you probably need to introduce relativised versions of Robin’s axioms (or maybe create an abstract axiom of relativisation).

The key difficulty here seems to me that whereas on the non-relative setting, we either had A larger than B or B larger than A, in the relative setting, the relative size of the A-fibre and the B-fibre depends on the choice of fibre, and so you have to make that part of the argument (i.e. the splitting into two cases) fibre-dependent, and thus embedded inside the category instead of being external to it.

I believe that if the above relative AM-GM inequality (which is basically trying to capture the fact that “an arbitrary sum of squares of integers is positive”) is categorified then it will be straightforward to get Cauchy-Schwarz in the form

(A× CB)×(B× CA)(A× CA)×(B× CB) (A \times_C B) \times (B \times_C A) \to (A \times_C A) \times (B \times_C B)

by categorifying Lagrange’s identity appropriately, since the error term in that identity is also a sum of squares. Indeed, from relative AM-GM we have

(A×B)× C×C(B×A)+(B×A)× C×C(A×B)(A×B)× C×C(A×B)+(B×A)× C×C(B×A) (A \times B) \times_{C \times C} (B \times A) + (B \times A) \times_{C \times C} (A \times B) \to (A \times B) \times_{C \times C} (A \times B) + (B \times A) \times_{C \times C} (B \times A)

which rearranges to form a doubled version of Cauchy-Schwarz. (One then has to “divide by 2”, i.e. deduce ABA \to B from A+AB+BA + A \to B + B, but that seems to be an easier problem.)

Posted by: Terry Tao on May 23, 2007 5:08 PM | Permalink | Reply to this

Re: The Two Cultures of Mathematics Revisited

“… trying to capture the fact that an arbitrary sum of squares of integers is positive is categorified…”

BUT there can be extra structure. For example, we should not neglect that every nonnegative integer can be represented as the sum of 4 squares of nonnegative integers, when we categorize. Right?

Posted by: Jonathan Vos Post on May 23, 2007 5:17 PM | Permalink | Reply to this

Re: The Two Cultures of Mathematics Revisited

when we categorize

Once upon a time I used to make that mistake, too. The right word is categorify.

Posted by: urs on May 23, 2007 6:41 PM | Permalink | Reply to this

right word is categorify; Re: The Two Cultures of Mathematics Revisited

I apolify.

Posted by: Jonathan Vos Post on May 24, 2007 12:59 AM | Permalink | Reply to this

Re: The Two Cultures of Mathematics Revisited

Terry Tao wrote:

I recommend reading Gowers’ thoughts on the Cauchy-Schwarz inequality.

I enjoyed them too. Here is the link.

Posted by: Tom Leinster on May 23, 2007 6:42 PM | Permalink | Reply to this

Re: The Two Cultures of Mathematics Revisited

…try defining the category Algorithm…

Noson Yanofsky gave it a go here.

We define an algorithm to be the set of programs that implement or express that algorithm. The set of all programs is partitioned into equivalence classes. Two programs are equivalent if they are “essentially” the same program. The set of all equivalence classes is the category of all algorithms.

Perhaps John has seen other attempts in his reading for his ‘Cohomology and Computation’ course.

Posted by: David Corfield on May 23, 2007 7:07 PM | Permalink | Reply to this

Re: The Two Cultures of Mathematics Revisited

Could anything be done in a “K-theoretic” way? So in the unrelativized case, for any pair of sets, AA and BB, there are DD and EE such that A+DA + D and B+EB + E are isomorphic.

A relativized version would similarly postulate DD and EE over CC.

If, as noted here,

FINITE SETS ARE LIKE ‘VECTOR SPACES OVER THE FIELD WITHOUT ELEMENTS’,

there is a hint of something literally K-theoretic in the air, sums of vector bundles anyway.

Posted by: David Corfield on May 24, 2007 11:11 AM | Permalink | Reply to this

Re: The Two Cultures of Mathematics Revisited

Terry writes:

Secondly, one thing about the “problem solving” culture as opposed to the “theory building” is that we tend to make progress by working upward from special cases, rather than downward from more general formulations.

It’s all very complicated.

Theory-builders may act as if general theories come full-blown into their heads, just waiting to be applied to special cases. But often the most exciting part is finding those general theories in the first place. For this, one has to dream up patterns that would make sense of obscure clues. This often progresses from special to general.

I think of my style as ‘finding paths of least resistance’, or ‘following the tao of mathematics’. I’m more of a hiker than a rock-climber. I prefer to know lots of fun trails, and occasionally bump into a new one while I’m strolling about, than spend my day trying to climb up a sheer cliff. Instead of trying to do anything hard, I prefer to keep lots of facts, patterns and questions in mind, and make obvious guesses and deductions when I notice patterns turning up.

Given this rather lazy approach to mathematics, the only way to discover anything new is by knowing a little about lots of things, and knowing some good patterns that haven’t been fully exploited. ‘Categorification’ is my favorite pattern, because it’s extremely broad, and nowhere near being tapped out. But, it’s just one of many.

We can imagine two extremes. At one extreme we have someone who tries very hard to open a specific door, perhaps melting the lock with an acetylene torch if necessary, or even blowing the door open with dynamite. At the other extreme, we have someone who carries a huge ring of keys and tries them all on any door they happen to walk by.

The Wikipedia article on lock picking makes for interesting metaphoric reading:

Lock picking is the ideal way of opening a lock without the correct key, while not damaging the lock, allowing it to be rekeyed for later use, which is especially important with antique locks that would be impossible to replace if destructive entry methods are used.

Usually it is possible to bypass a lock without picking it. Most common locks can be quickly and easily opened using a drill, bolt cutters, or a hydraulic jack. The hasp, door, or fixture they are attached to can be cut or broken.

A lock that offers high resistance to picking does not necessarily make unauthorized access more difficult, but it will make surreptitious unauthorized access more difficult. They are often used in combination with alarms to provide layered security.

Some people enjoy picking locks recreationally, because they find it brings high satisfaction and hack value.

Posted by: John Baez on May 24, 2007 8:32 PM | Permalink | Reply to this

Re: The Two Cultures of Mathematics Revisited

“Some people enjoy picking locks recreationally…”

[Insert canonical Feynman Los Alamos safecracking anecdote here]

I had the distinct impression that Feynman followed the Tao of safecracking and lockpicking in his applications of mathematics, where he was wildly creative and sneaky, and willing to let others add rigor later, after he got the right answer elegantly.

In this, he was a consummate member of the ‘problem solving’ culture – but left apparatus of interest to the ‘theory building’ culture.

Like Terrence Tao, he went out of his way to show how he did it, and did not hide behind formal paper-writing “behold, consider this bizarre equation; see, it works; never mind how I found it” variety.

Posted by: Jonathan Vos Post on May 25, 2007 1:19 AM | Permalink | Reply to this

Re: The Two Cultures of Mathematics Revisited

By, the way, I’ll mention a take on this that I had last night. It might be better to take the K-theoretic approach and consider our inner product to be E(A,B):=(1) iExt i(A,B)E(A,B) := \sum (-1)^i Ext^i(A,B), rather than just Ext 0(A,B)=Hom 0(A,B)Ext^0(A,B)=Hom^0(A,B). Now, E(A,B)E(A,B) won’t be symmetric. There are three reasonable ways out: define Cauchy-Schwartz to be the claim E(A,B)E(B,A)E(A,A)E(B,B)E(A,B) E(B,A) \leq E(A,A) E(B,B) whether or not EE is symmetric; define Q(A,B)=E(A,B)+E(B,A)Q(A,B)=E(A,B)+E(B,A) and investigate the claim for QQ; use Tor instead of Ext.

No matter what approach we take, our problem breaks into two halves. First, figure out whether the non-categorical, plain numeric Cauchy-Schwartz is true. (Or, more likely, what hypotheses we need to add to make it true.) Second, find the categorical statement explaining the numerical result.

For the second option, QQ is not always positive definite on the KK-group of our category; it is not even true that Q(A,A)0Q(A,A) \geq 0 for actual objects of our category. In the category of (finite dimensional) representations of (acyclic) quivers, there is a beautiful result that QQ is positive definite on the KK-group precisely for the Dynkin quivers. So it might be worth seeing whether there are some more general good properties of abelian categories (with various finiteness conditions) where QQ is positive definite.

Option 33 is reminiscent of Serre’s multiplicity conjectures. If I get a chance, I’ll check with some commutative algebraists and see whether anyone has thought about it. A question for the category theorists – why is Hom a better analogue for inner products than \otimes?

Option 1 I have no thoughts about at all.

Posted by: David Speyer on May 24, 2007 9:59 PM | Permalink | Reply to this

Re: The Two Cultures of Mathematics Revisited

why is Hom a better analogue for inner products than \otimes?

Try page 3 of Higher Dimensional Algebra II. The similarity between adjoint operators and adjoint functors is perhaps the best evidence.

Posted by: David Corfield on May 25, 2007 8:43 AM | Permalink | Reply to this

Re: The Two Cultures of Mathematics Revisited

The similarity between adjoint operators and adjoint functors is perhaps the best evidence.

Yes, there is various, maybe surprising, evidence that really nature wants us to think of inner products as Homs.

As far as I am aware, this observation originates, as David points out, in John’s HDA II and found various confirmations afterwards.

So in a way it is an empirical fact.

One might wonder what this should be telling us, assuming that it is not just a coincidence.

Once I thought about this a bit, when thinking about quantum mechanics, quantum field theory, and how one might be obtainable from the other by categorification.

I am not sure I found the right answer, but I thought that the following might be remarkable:

for categorifying quantum mechanics to get quantum field theory, it is helpful to formulate everything one needs “arrow-theoretically” (like in the Rosetta stone section 1.2).

Quantum mechanically, the inner product v,w \langle v, w \rangleis a “correlator”: an object “in state ww comes in”, some reaction happens, and then an object “in state vv comes out”, and the inner product is some measure for this to happen.

This physics intuition is potentially helpful for finding the right arrow theory — since, as John puts it, higher categories are ultimately about states, processes, processes of processes, etc.

Now, it it is not a surprise that a good way to think of the inner product w,v\langle w, v \rangle is hence as the composition

- of a vector vVv \in V, regarded as a morphismvV \mathbb{C} \stackrel{v}{\to} V

- a dual vector w *=w,w^* = \langle w, \cdot \rangle, regarded as a morphismVw *, V \stackrel{w^*}{\to} \mathbb{C} \,,hencevVw *. \mathbb{C} \stackrel{v}{\to} V \stackrel{w^*}{\to} \mathbb{C} \,.

This describes the process where vv enters, nothing happens and we just check what part of ww comes out.

More generally, in between vv will go through some process or other. This will be modelled by an edomorphismU:VV. U : V \to V \,.This gives us the processvVUVw *, \mathbb{C} \stackrel{v}{\to} V \stackrel{U}{\to} V \stackrel{w^*}{\to} \mathbb{C} \,,which physicists, following Dirac, indeed usually write in a sort of arrow notation asw|U|v. \langle w | U | v \rangle \,.

But of course one is interested in this entity not just for one given endomorphism UU, but for all of them. So the information of interest encoded by the “states” vv and ww is really, in physics, the mapw||v:End(V). \langle w | \cdot | v \rangle : \mathrm{End}(V) \to \mathbb{C} \,.

But that wants to be read as an application of the Hom-functor!

Hom(V v ,V w * ):(VUV)V U V v w * \mathrm{Hom} \left( \array{V \\ \uparrow^v \\ \mathbb{C}}, \array{V \\ \downarrow^{w^*} \\ \mathbb{C}} \right) : (V \stackrel{U}{\to} V) \;\; \mapsto \;\; \array{ V &\stackrel{U}{\to}& V \\ \uparrow^v && \downarrow^{w^*} \\ \mathbb{C} && \mathbb{C} }

Looking at correlators and inner products in quantum mechanics this way, we are lead to a more or less obvious categorification. One can then check if the 2-quantum mechanics obtained this way does capture aspects of 2-dimensional quantum field theory. I think it does, as I have tried to indicate in
D-Branes from tin cans: Homs of homs.

Once again, I am not claiming that this is the final answer to our problems here, but it looks suggestive enough to appear remarkable to me.

I know that Bruce has more — and possibly deeper — things to say about this. But probably not in public.

Posted by: urs on May 25, 2007 10:10 AM | Permalink | Reply to this

Re: The Two Cultures of Mathematics Revisited

this observation originates, as David points out, in John’s HDA II and found various confirmations afterwards.

One such confirmation I wish I knew more about (and maybe will be treated here at some point) is Schur functors, as defined by Kapranov and.. someone else I can’t recall offhand.

Anyhow, the fascinating thing about these is that if hom is an inner product then a Schur functor is its asymmetry! This grabs me personally because the loop value in the Kauffman bracket is the trace of the asymmetry of a bilinear form, so a really nice categorification of the bracket should feature the “trace” of a Schur functor prominently.

Posted by: John Armstrong on May 25, 2007 3:43 PM | Permalink | Reply to this

Re: The Two Cultures of Mathematics Revisited

if hom is an inner product then a Schur functor is its asymmetry!

Could you explain that?

Posted by: urs on May 25, 2007 3:50 PM | Permalink | Reply to this

Re: The Two Cultures of Mathematics Revisited

Grr… still not awake from the drive back to New Haven last night, I screwed up some things. :( I shouldn’t have said “inner product”, but more generally “bilinear form”. More importantly, it’s “Serre functor”, not “Schur” (/me facepalm).

Just in case, I’ll explain what I mean now that I have the terminology right. First the decategorified picture:

A nondegenerate bilinear form B(x,y)B(x,y) gives us two (generally different) isomorphisms between VV and V *V^*. The asymmetry measures how different these two are. More specifically, there is a unique linear transformation GG on VV so that B(x,y)=B(y,Gx)B(x,y)=B(y,Gx). This GG is the asymmetry of the form.

A Serre functor on a \mathbb{C}-linear category is an additive (covariant) functor SS satisfying hom(X,Y)hom(Y,S(X)) *\hom(X,Y)\cong\hom(Y,S(X))^*, where the dual on the right is there to make the variance work out right. The analogy with asymmetries should be clear.

Posted by: John Armstrong on May 25, 2007 4:36 PM | Permalink | Reply to this

Re: The Two Cultures of Mathematics Revisited

Thanks, great!

Since you said

the loop value in the Kauffman bracket is the trace of the asymmetry of a bilinear form, so a really nice categorification of the bracket should feature the “trace” of a [Serre] functor prominently.

You might also have to remind me what the Kauffman bracket is and what its categorification is (expected to be?).

So, are there any places where these Serre functors actually show up in a way that we could point at them and say: “hey, see, if we now interpret the hom as an inner product, then suddenly all this is saying is just xxx”?

Posted by: urs on May 25, 2007 4:46 PM | Permalink | Reply to this

Re: The Two Cultures of Mathematics Revisited

Okay, the Kauffman bracket is a regular isotopy invariant of links. The categorification we know is Khovanov homology, which does have an extension to tangles.

What I did was start from extensions to tangles and found the connection to bilinear forms. Every nondegenerate bilinear form over a commutative ring R gives a functor extending a bracket evaluation in R, and two such functors are naturally isomorphic if and only if the bilinear forms are equivalent. And equivalence of bilinear forms (over fields at least) always comes down to similarity of the asymmetries (plus other conditions depending on characteristic and algebraic closure), so that asymmetry transformation shows up again. It’s really everywhere you look once you understand how the bracket extends to tangles.

Posted by: John Armstrong on May 25, 2007 8:22 PM | Permalink | Reply to this

Re: The Two Cultures of Mathematics Revisited

Andrei Caldararu and Simon Willerton (who happens to be my supervisor!) have done some really interesting things with these Serre functors. If you work with the 2-category of varieties and integral kernels (objects are complex manifolds and Hom(X,Y)Hom(X,Y) is the derived catgory of coherent sheaves on X×YX \times Y), you can write down everything nicely - including the Serre functors - using string diagrams.

It’s pretty cool : Serre functors force one to think pretty carefully about the notion of “duality”.

Anyhow, one nice result which comes out is that you can write down a nice string-diagram construction of the generalized Mukai pairing, i.e. a nondegenerate pairing on the Hochschild homology of an algebraic variety (or something like that),

(1)(,):HH *(X)HH *(X). ( \, , \, ) : HH_* (X) \otimes HH_* (X) \rightarrow \mathbb{C}.

When XX is Calabi-Yau (so that homology and cohomology are isomorphic) and we’re working in the “type II-B twisted model” (the experts will know what that means, even though I’ve never really understood it) then this inner product on homology and the ring structure on cohomology mesh together to give a Frobenius algebra, as one would expect from a TQFT.

Anyhow, it seems that Serre functors offer a nice point of contact between abstract categorical considerations about “duality” and mainstream algebraic geometry.

Posted by: Bruce Bartlett on May 25, 2007 5:24 PM | Permalink | Reply to this

Re: The Two Cultures of Mathematics Revisited

David Speyer writes:

why is HomHom a better analogue for inner products than \otimes?

As David points out, the analogy between adjoint functors

Hom(A *x,y)Hom(x,Ay)Hom(A^* x,y) \cong Hom(x,A y)

and adjoint operators

A *x,y=x,Ay\langle A^*x, y \rangle = \langle x, A y \rangle

is a huge clue. Even if someone hadn’t been smart enough to use the exact same word ‘adjoint’ to describe both of them, it’s pretty evident that the equation between numbers

A *x,y=x,Ay\langle A^*x, y \rangle = \langle x, A y \rangle

is a decategorified version of the isomorphism between sets

Hom(A *x,y)Hom(x,Ay)Hom(A^* x,y) \cong Hom(x,A y)

This is what led me to explicitly seek a categorification of the whole theory of Hilbert spaces, which you can find in HDA2: 2-Hilbert spaces. Bruce Bartlett has been digging deeper in this direction and coming up with marvelous stuff.

But, the real reason why HomHom is a categorified version of the inner product becomes clear when you think about quantum mechanics, especially path integrals. In a path integral, we’re taking a set of ways to get from here to there — which is clearly a Hom-set — and integrate over it to get a number, the amplitude to get from here to there.

(Integration over a set to get a number is a classic form of decategorification. The simplest example is counting. Given an object of the category of finite sets, we ‘count’ it — meaning take its isomorphism class — and get an element of the set \mathbb{N}. Indeed, the main reason for being interested in \mathbb{N} is that it’s the decategorification of the category of finite sets — that is, the set of isomorphism classes of objects.)

Lately I’ve been going further in this direction, showing more explicitly how path integrals are a way of turning categories into Hilbert spaces, with integration over Hom-sets giving the inner product. You can read about this in my course notes; I hope my student Alex Hoffnung will work out a bunch of details in his thesis.

Posted by: John Baez on May 25, 2007 6:46 PM | Permalink | Reply to this

Re: The Two Cultures of Mathematics Revisited

It might be better to take the K-theoretic approach and consider our inner product to be E(A,B):=(1) iExt i(A,B), E(A,B) := \sum (-1)^i \mathrm{Ext}^i(A,B) \,, rather than just Ext 0(A,B)=Hom(A,B)\mathrm{Ext}^0(A,B) = \mathrm{Hom}(A,B).

[…]

there is a beautiful result that QQ is positive definite on the K-group precisely for the Dynkin quivers.

Sounds very interesting, indeed. Can you see whether or not this might help fix the injectivity in Bruce’s approach?

Posted by: urs on May 25, 2007 3:32 PM | Permalink | Reply to this

Re: The Two Cultures of Mathematics Revisited

I’m not too familiar with categorification, and maybe this has been answered before. I think classical Cauchy-Schwarz is fundamentally about projective modules over a ring with an archimedean norm. Might it be more natural to look at something more analytic like measure spaces instead of sets or vector spaces as the ring without subtraction, and hunt for some plausible-sounding “module” categories? Aside from obvious choices (e.g. G-equivariant measure spaces), I don’t have any good ideas for the form such a category would take.

Posted by: Scott Carnahan on May 25, 2007 2:41 AM | Permalink | Reply to this

Re: The Two Cultures of Mathematics Revisited

I think classical Cauchy-Schwarz is fundamentally about projective modules over a ring

Ah, that’s very interesting indeed. Could you elaborate on that point of view? When I am sure about what you have in mind, I can maybe indicate why this sounds like it would be very helpful for categorification.

So, given two such (left) modules VV and WW, and denoting by V¯\bar V, etc. the induced right module (using some extra structure lying around), is Cauchy-Scharz some statement relating V¯ RWW¯ RV \bar V \otimes_R W \otimes \bar W \otimes_R V with V¯ RVW¯ RW \bar V \otimes_R V \otimes \bar W \otimes_R W ?

I am inclined to still stick to that statement which David Corfield started this thread with, and which he called “extreme”:

I expect that there is a good way to think of what the Cauchy-Schwarz inequality is really about (as opposed to sums ia ib i\sum_i a_i \cdot b_i) and that the categorification of the definition stated this way will be so obvious as to not require any serious thinking at all.

(Okay, I should admit that by my private definintion of “really”, that’s true by definition. But you get the point.)

Posted by: urs on May 25, 2007 9:19 AM | Permalink | Reply to this

Re: The Two Cultures of Mathematics Revisited

I called it “extreme” imagining someone who held that view sitting in judgment of others. That person might express the thought as follows:

“I’m only interested in ideas, themes and constructions which lend themselves to being categorified, because for any worthwhile idea there is a story about it which gets to the heart of what it really is, and I’ll know when I’ve reached that point by the ease with which it categorifies.”

It would be one thing to fit your own research to this way of thinking. But if you believed completely in that “really”, it would significantly effect the kind of mathematics you would encourage others to do. Perhaps wondering about arithmetic progressions amongst the primes wouldn’t feature.

I once heard that Michael Atiyah steered Oxford graduate students away from point set topology in the 70s. It’s clear from his informal writings why he would do this. As you know, I’m fascinated by this kind of value judgment. I believe it is of much greater philosophical interest than the “what kind of thing is a number?” type question which dominates Anglo-American philosophy.

Posted by: David Corfield on May 25, 2007 10:13 AM | Permalink | Reply to this

Re: The Two Cultures of Mathematics Revisited

“I’m only interested in ideas, themes and constructions which lend themselves to being categorified, because for any worthwhile idea there is a story about it which gets to the heart of what it really is, and I’ll know when I’ve reached that point by the ease with which it categorifies.”

I like that way to put it. Only that I would not say “only interested in”. That would be too extreme! :-)

(But it’s true: personally I am not all that interested in primes in arithmetic progress…)

Posted by: urs on May 25, 2007 10:23 AM | Permalink | Reply to this

Re: The Two Cultures of Mathematics Revisited

This takes us back to my first post on this subject. As I said there, at the level of results there’s no apparent reason to be more or less dismissive of results about progressions in primes than about Fermat’s Last Theorem. What’s important in each case is the underlying theory: work around the Taniyama-Shimura-Weil conjecture and Tao and Green’s work involving, among other things, ‘quadratic Fourier analysis’. Now, the interesting point is whether someone could dismiss something like quadratic Fourier analysis on the grounds that it had no categorification potential, if that indeed is the case.

Reading Gowers’ article, I remember wondering if it really could be the case that the general principles he perceives operating behind what he calls ‘combinatorics’ have nothing to do with those operating in the world of Grothendieck and Langlands. Is it possible that there is no way to bridge the gap between the cultures? Or might some genius in 2050 find a way to fit them together?

How strongly do we believe in the unity of mathematics?

Posted by: David Corfield on May 25, 2007 11:15 AM | Permalink | Reply to this

Re: The Two Cultures of Mathematics Revisited

I’ll at this point say why I sometimes stick my oar in at the cafe with my viewpoint: David’s devil advocate view point of “I’m only interested in ideas, themes and constructions which lend themselves to being categorified…” is a good restriction of one’s focus if one is occasionally put face to face with other viewpoints. One might well decide not to be interested in the things important from the other perspective — upon brief consideration when one encounters them — but perhaps the biggest problem is that if a group forms which is too homogeneous, there’s no way for the occasional “non-group-canonical” view to be expressed, which is useful in bringing new ideas and tools to bear on the problems under consideration. As an example, although I haven’t remotely had the time to really get to know category theory well, I’m interested in the “reverse process”: from some categorified facts, can they be specialised on a specific structure (I think that’s what “decategorified” means) to produce interesting and useful new facts that weren’t known before.

To express my viewpoint by analogy, I’m at least as interested in chainsaws as stories :-) .

Posted by: dave tweed on May 25, 2007 11:17 AM | Permalink | Reply to this

Re: The Two Cultures of Mathematics Revisited

what “decategorified” means

In its strict meaning, decategorification means forming the set of isomorphism classes of a category.

If the category has extra structure, this set will have extra structure.

For instance if the category is abelian and monoidal, its decatgorification is a ring. So an abelian monoidal catgegory is a categorification of a ring.

Here we are trying to play this game for vector spaces or sets:

the isomorphism classes in the category of finite-dimensional vector space or in the category of finite sets are indexed by natural numbers (giving the dimension of the vector spaces in the class or the cardinality of the sets in that class).

So the idea here is to take true statements about natural numbers and try to figure out if there are true statements about finite sets or finite-dimensional vector spaces such as after remembering of these only what they do on isomorphism classes, we reobtain those statements about numbers.

(But maybe we need to be a little more open minded here.)

Posted by: urs on May 25, 2007 11:43 AM | Permalink | Reply to this

Re: The Two Cultures of Mathematics Revisited

Thanks for the clarification urs. What I was sort of thinking about was the sort of thing that happens in this data structure/computer science paper, where slightly different notions of embedding one tree in another all categorify to equivalent categories, and then results proved about this “kind of” category. Concrete-ish algorithms for embeddings are obtained by going from the categories back to the original types of trees. I’m not sure if there’s a different term for this process.

Posted by: dave tweed on May 25, 2007 12:56 PM | Permalink | Reply to this

Re: The Two Cultures of Mathematics Revisited

David Tweed wrote:

I’m interested in the “reverse process”: from some categorified facts, can they be specialised on a specific structure (I think that’s what “decategorified” means) to produce interesting and useful new facts that weren’t known before.

As Urs noted, that’s not what ‘decategorified’ means. Here’s my little story about categorification and decategorification, from week99:

If one systematically studies categorification one discovers an amazing fact: many deep-sounding results in mathematics are just categorifications of stuff we all learned in high school. There is a good reason for this, I believe. All along, mathematicians have been unwittingly “decategorifying” mathematics by pretending that categories are just sets. We “decategorify” a category by forgetting about the morphisms and pretending that isomorphic objects are equal. We are left with a mere set: the set of isomorphism classes of objects.

I gave an example in “week73”. There is a category FinSet whose objects are finite sets and whose morphisms are functions. If we decategorify this, we get the set of natural numbers! Why? Well, two finite sets are isomorphic if they have the same number of elements. “Counting” is thus the primordial example of decategorification.

I like to think of it in terms of the following fairy tale. Long ago, if you were a shepherd and wanted to see if two finite sets of sheep were isomorphic, the most obvious way would be to look for an isomorphism. In other words, you would try to match each sheep in herd A with a sheep in herd B. But one day, along came a shepherd who invented decategorification. This person realized you could take each set and “count” it, setting up an isomorphism between it and some set of “numbers”, which were nonsense words like “one, two, three, four,…” specially designed for this purpose. By comparing the resulting numbers, you could see if two herds were isomorphic without explicitly establishing an isomorphism!

According to this fairy tale, decategorification started out as the ultimate stroke of mathematical genius. Only later did it become a matter of dumb habit, which we are now struggling to overcome through the process of “categorification”.

Here’s a nice example of categorification at work. Take the generating function for your favorite counting problem, say the generating function for nn-leaved binary trees:

C n1x n \sum C_{n-1} x^n

where C nC_n are the Catalan numbers. Then, evaluate it at some number within the radius of convergence, say x=1/10x = 1/10 in this case. We get some funny number — in this case,

C n110 n=13/52 \sum C_{n-1} 10^{-n} = {1 - \sqrt{3/5} \over 2}

What’s the combinatorial meaning of this number? In other words: what is this number the number of?

This is the kind of question that has a very nice answer if one pursues categorification.

So, ironically, categorification often amounts to a process of making mathematics less abstract. Instead of just playing with numbers 1, 2, 3,…, we deal with the finite sets — of sheep, say — actually counted by these numbers. And, instead of just playing with numbers like 13/52{1 - \sqrt{3/5} \over 2} obtained by formally plugging numbers into some generating function, we study the things actually counted by these numbers!

So: what ‘generalized set’ has 13/52{1 - \sqrt{3/5} \over 2} elements? I’ll leave this as a puzzle for now. Anyone who wants a little help figuring out the answer can read these notes, especially the notes from week 3.

Posted by: John Baez on May 26, 2007 12:50 AM | Permalink | Reply to this

Re: The Two Cultures of Mathematics Revisited

I thought “decategorify” might be slightly different. I came to all this category theory stuff through monads in programming, where various kinds of “process” – possibly-ill-defined computations, processes with exceptions, I/O processes, collections, etc – are such that the patterns of “doing operations on things” turn out to share properties, constraints, etc. The impression I got was that “non-categorical mathematics” studies what happens under “fixed-notion” operations (like addition) essentially independently of the type of object (“abstract out the data”), whilst category theory studies what happens “patterns” hold in the process of application of operations essentially independently of what those operations are (“abstract out the operations”), 2-category theory is about patterns in the patterns (“abstract out the patterns”). However, I thought that, eg, proofs, properties and procedures established at the category level could be used, after some form of translation, back at the “ordinary mathematics” level. For instance, if I prove some “property” - or come up with some process/algorithm - that depends only on the properties of the monad and not what precise monad it is, then I can look to see what it says when looked at as being about “possibly ill-defined” computations and what it says looked at as being about about collections and …. Presumably those things might look different, and might be utilised for different ends, but they come from a common “pattern about operations”.

So, a concrete question (where I’m not using any precise technical meanings that might accidentally exist for any of the following terms): a given categorification of a mathematical gadget/entity/whatever, say, C-S, has exactly one “corresponding `ordinary mathematical’ gadget/entity/whatever” and that’s precisely the original gadget/entity/whatever you originally categorified?

Posted by: dave tweed on May 27, 2007 3:45 PM | Permalink | Reply to this

Re: The Two Cultures of Mathematics Revisited

Sorry if I was unclear - I didn’t mean anything very deep. I just thought that both sides of the inequality naturally take values in the reals, so it might be overly restrictive to use objects that admit an integer-valued norm like dimension. The first thing that came to mind was measure spaces as an analytification of sets.

The other thing I managed not to say was that classical Cauchy-Schwarz seems to require some kind of parallelogram law on our norm to define the inner product, placing us squarely in archimedean land. Is there a way to categorify that without subtraction?

Posted by: Scott Carnahan on May 25, 2007 8:59 PM | Permalink | Reply to this

Re: The Two Cultures of Mathematics Revisited

David Speyer wrote:

why is HomHom a better analogue for inner products than \otimes?

As David Corfield points out, the analogy between adjoint functors

Hom(A *x,y)Hom(x,Ay)Hom(A^* x,y) \cong Hom(x,A y)

and adjoint operators

A *x,y=x,Ay\langle A^*x, y \rangle = \langle x, A y \rangle

is a huge clue. Even if someone hadn’t been smart enough to use the exact same word ‘adjoint’ to describe both of them, it’s pretty evident that the equation between numbers

A *x,y=x,Ay\langle A^*x, y \rangle = \langle x, A y \rangle

is a decategorified version of the isomorphism between sets

Hom(A *x,y)Hom(x,Ay)Hom(A^* x,y) \cong Hom(x,A y)

This is what led me to explicitly seek a categorification of the whole theory of Hilbert spaces, which you can find in HDA2: 2-Hilbert spaces. Bruce Bartlett has been digging deeper in this direction and coming up with marvelous stuff.

But, the real reason why HomHom is a categorified version of the inner product becomes clear when you think about quantum mechanics, especially path integrals. In a path integral, we’re taking a set of ways to get from here to there — which is clearly a Hom-set — and integrate over it to get a number, the amplitude to get from here to there.

(Integration over a set to get a number is a classic form of decategorification. The simplest example is counting. Given an object of the category of finite sets, we ‘count’ it — meaning take its isomorphism class — and get an element of the set \mathbb{N}. Indeed, the main reason for being interested in \mathbb{N} is that it’s the decategorification of the category of finite sets — that is, the set of isomorphism classes of objects.)

Lately I’ve been going further in this direction, showing more explicitly how path integrals are a way of turning categories into Hilbert spaces, with integration over Hom-sets giving the inner product. You can read about this in my course notes; I hope my student Alex Hoffnung will work out a bunch of details in his thesis.

Urs has been pursuing this clue in a slightly different way, but we share a common goal: to understand the quantum mechanics of point particles as a decategorified version of something deeper.

It’s interesting that in all the years I’ve thought about this, I never tried to categorify the Cauchy–Schwarz inequality! I think it’s a great example of my instinctive ability to follow the tao and avoid tough problems.

Posted by: John Baez on May 25, 2007 7:32 PM | Permalink | Reply to this

Re: The Two Cultures of Mathematics Revisited

I think it’s a great example of my instinctive ability to follow the tao and avoid tough problems.

The burning question of this thread is whether this is avoidance or evasion.

Posted by: David Corfield on May 27, 2007 12:16 PM | Permalink | Reply to this

Triality; Re: The Two Cultures of Mathematics Revisited

You seem to be implying:

“Nothing is certain, except death, taxes, and n-categories!”

Posted by: Jonathan Vos Post on May 30, 2007 6:36 AM | Permalink | Reply to this
Read the post What is a Lie derivative, really?
Weblog: The n-Category Café
Excerpt: On the arrow-theory behind Lie derivatives.
Tracked: May 31, 2007 11:30 AM
Read the post Category Theory in Machine Learning
Weblog: The n-Category Café
Excerpt: Does category theory have a future in machine learning?
Tracked: September 5, 2007 4:04 PM
Read the post Obstructions for n-Bundle Lifts
Weblog: The n-Category Café
Excerpt: On obstructions to lifting the structure n-group of n-bundles.
Tracked: September 12, 2007 11:25 PM
Read the post Detecting Higher Order Necklaces
Weblog: The n-Category Café
Excerpt: Nils Baas on higher order structures,Enrico Vitale on weak cokernels and a speculation on weak Lie n-algebras triggered by discussion with Pavol Severa.
Tracked: September 28, 2007 12:44 AM
Read the post Categorified Clifford Algebra and weak Lie n-Algebras
Weblog: The n-Category Café
Excerpt: On weak Lie n-algebras, differential graded Clifford algebra and Roytenberg's work on weak Lie 2-algebras.
Tracked: October 9, 2007 4:57 PM
Read the post Bridge Building
Weblog: The n-Category Café
Excerpt: Might the cohomology of dynamical systems provide a meeting ground for researchers on the 'combinatorics' side of mathematics, and those on the 'theory-building' side?
Tracked: December 23, 2008 1:20 PM
Read the post The Space of Robustness
Weblog: The n-Category Café
Excerpt: A discussion of a blog post by Tim Gowers on the structure of mathematics.
Tracked: January 14, 2009 1:15 PM

Re: The Two Cultures of Mathematics Revisited

Noam Elkies on the culture divide in analytic number theory:

The Harvard math curriculum leans heavily towards the systematic, theory-building style; analytic number theory as usually practiced falls in the problem-solving camp. This is probably why, despite its illustrious history (Euclid, Euler, Riemann, Selberg, … ) and present-day vitality, analytic number theory has rarely been taught here.

Now we shall see that there is more to analytic number theory than a bag of unrelated ad-hoc tricks, but it is true that partisans of contravariant functors, adèlic tangent sheaves, and étale cohomology will not find them in the present course. Still, even ardent structuralists can benefit from this course.

An ambitious theory-builder should regard the absence thus far of a Grand Unified
Theory of analytic number theory not as an insult but as a challenge. Both
machinery- and problem-motivated mathematicians should note that some of
the more exciting recent work in number theory depends critically on symbiosis
between the two styles of mathematics.

Posted by: David Corfield on March 25, 2009 5:22 PM | Permalink | Reply to this

Teaching the Teachers; Re: The Two Cultures of Mathematics Revisited

I read Noam Elkies papers frequently, but had not seen this fine quotation before. Thank you!

That there are multiple Cultures of Mathematics has even trickled down to American Colleges of Education, which are aware that urban public schools are under-serving Math students.

My professor who assigns grad students to Directed Teaching (75 days of unpaid full-time student teaching) commented on one of my research reports and associated DVD of my teaching one class:

“Your work is critically important. We need to keep kids engaging with 3-D math through modeling.”

“We also need to help them connect these experiences with reality, with other math concepts, and with algorithmic math.”

“I need you to show this piece to your master teacher but to always remember that students must understand and then connect to what they know and what they need to know.”

In this same Charter College of Education, at Cal State Los Angeles, the head of Mathematics for Secondary Schools is Dr. Fred Uy, who agrees with me on where the California Framework for Mathematics goes horribly astray: the committee members who wrote it (with whom he’s spoken in Sacramento) not only have never taught in actual middle schools nor high school, but they consider that “irrelevant” to their pedagogical theory!

Posted by: Jonathan Vos Post on March 25, 2009 6:00 PM | Permalink | Reply to this

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