The n-Category Café

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

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

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

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

Here:

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

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

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

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

Every representation is of the form

$$A = \oplus_i \mathbb{C}^{a_i} \otimes e_i$$

where $e_i$ ranges over a basis of irreducible representations of $G$.

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

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

but we also clearly have this isomorphism of vector spaces:

$$A^* \otimes B \cong (\oplus_i \mathbb{C}^{a_i}) \otimes (\oplus_j \mathbb{C}^{b_j})$$

The inclusion

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

is just the obvious diagonal inclusion

$$\oplus_i (\mathbb{C}^{a_i} \otimes \mathbb{C}^{b_i}) \hookrightarrow (\oplus_i \mathbb{C}^{a_i}) \otimes (\oplus_j \mathbb{C}^{b_j})$$

If we decategorify by taking dimensions, we thus get

$$\sum_i a_i b_i \le (\sum_i a_i) (\sum_j b_j)$$

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

$$(\sum_i a_i b_i)^2 \le (\sum_i a_i^2) (\sum_j b_j^2)$$

Unlike Cauchy–Schwarz, the inequality I got doesn’t hold for all real numbers $a_i$ and $b_j$. 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

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

Anyone want to take a stab at it?

I think, though, that we should call this:

$$\sum_i a_i b_i \le (\sum_i a_i) (\sum_j b_j)$$

the TARDIS inequality.

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 $\| v + \lambda w\|^2$ 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

$$A \times B + B \times A \to A \times A + B \times B$$

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

$$A \times_C B + B \times_C A \to A \times_C A + B \times_C B$$

where we are given maps $A \to C$ and $B \to C$, 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

$$(A \times_C B) \times (B \times_C A) \to (A \times_C A) \times (B \times_C B)$$

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

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

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

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 $E(A,B) := \sum (-1)^i Ext^i(A,B)$, rather than just $Ext^0(A,B)=Hom^0(A,B)$. Now, $E(A,B)$ won’t be symmetric. There are three reasonable ways out: define Cauchy-Schwartz to be the claim $E(A,B) E(B,A) \leq E(A,A) E(B,B)$ whether or not $E$ is symmetric; define $Q(A,B)=E(A,B)+E(B,A)$ and investigate the claim for $Q$; 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, $Q$ is not always positive definite on the $K$-group of our category; it is not even true that $Q(A,A) \geq 0$ for actual objects of our category. In the category of (finite dimensional) representations of (acyclic) quivers, there is a beautiful result that $Q$ is positive definite on the $K$-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 $Q$ is positive definite.

Option $3$ 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.

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.

David Speyer wrote:

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

As David Corfield points out, the analogy between adjoint functors

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

and adjoint operators

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

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

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

is a decategorified version of the isomorphism between sets

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

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 $Hom$ 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.

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.

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!