The n-Category Café

Sorry, it was silly of me to say…

No problem :-)

In fact it is still a long term goal of mine to understand this a little bit better - it’s just a little observation of mine that, if you need three years at least to be able to contribute to the topic, that pretty much excludes it as a topic for a PhD. Gerd Faltings once said in an interview that it took him 2 years of hard work to prove the Mordell conjecture, and that he was able to do it because he already had tenure (at the age of 27, which is still a record for mathematics in Germany, I think). For someone without tenure it is a considerable risk to invest all the time in one project that may not lead to any publications for years (or at all).

BTW: Has anybody done some interesting physics with the Connes-Kreimer theory yet? Something like: “Hey, now that we have a better grip at the combinatorics of Feynman graphs, we can calculate the next 5 digits of the anomalous magnetic dipole moment of the electron”?

So I’ve always wanted to write a paper on JB-algebras, to fool people into thinking these algebras are named after me.

Then there is no problem with JC-algebras, too, but we’ll have to do something about the JBW algebras.

BTW, I just noticed that the classic book Harald Hanche-Olsen and Erling Stormer: Jordan operator algebras is available for free because the publisher has returned all rights to the authors!

I’m still catching up with the Bernoulli-number discussion from 2008, but all those $\emptyset$s are unsettling me. Isn’t dealing with the vacuous species $e^X$ a bit problematic, since the vacuous structure can be put on the empty set? By contrast, the Bell numbers count the ways to partition n elements into non-empty boxes, so their generating function is given by the composition of the species “being a finite set” and “being a non-empty finite set”, $e^{e^X - 1}$.

If we tried to compose with the vacuous species instead of the uniform species, we’d naively get $e^{e^X}$; however, because we could interleave an arbitrary number of empty sets into our structure, the number of ways to build a “finite set of finite sets” would blow up in our faces. So, we have to move beyond ordinary combinatorial species into stuff types.

I’m basically copying this out of Section 4.2 of Jeffrey Morton’s “Categorified Algebra and Quantum Mechanics” (2006), which everybody else here has probably read and understood more thoroughly than I have.

Decategorified, the number concerned gives us: $1+2+3+4+\ldots$, or $\zeta(-1)$, which we know to be $-1/12$.

Recently, Terry Tao advised interpreting that $-1/12$ as the constant term in the asymptotic expansion of the regulated sum

$$\sum_{n = 1}^\infty n \eta(n/N) = -\frac{1}{12} + \int_0^\infty dx\,x \eta(x) N^2 + O\left(\frac{1}{N}\right).$$

More generally,

$$\sum_{n = 1}^\infty n^s \eta(n/N) = -\frac{B_{s+1}}{s+1} + \int_0^\infty dx\,x^s \eta(x) N^{s+1} + O\left(\frac{1}{N}\right),$$

where $B_s$ is the $s$th Bernoulli number, which arises because the expansion in the smoothing parameter $N$ is computed using the Euler–Maclaurin formula. (A physicist-y explanation of why the Bernoulli numbers appear in the Euler–Maclaurin formula starts with using the derivative operator $\partial_x$ as the generator of translations; the combination of exponentials in the sum considered by Euler–Maclaurin becomes the generating function for $B_n$.)

Regulating the infinite sum $1 + 2 + 3 + 4 + \ldots$ like this is a good physics thing to do. For example, in the Casimir effect, we might start by trying to sum up all the photon modes which are allowed to exist between two conductive plates, which — shock and horror! — leads to an infinite result, until we remember that the real world doesn’t provide us with ideal surfaces. Electrons only move so fast, so fields can only be cancelled so quickly. Instead of summing over all modes, we should introduce a cutoff which suppresses the contributions of those above the plasma frequency. The leading term in the Casimir force is universal, given by the $-1/12$, while the next-order corrections are material-specific.

Luckily for those of us who love abstruse formulas, it works out that when we have a sum which looks at first like $\sum_n n^s$ (but which we sneakily know we should regulate), we can get the constant term of the regulated answer by plugging in the value of $\zeta(-s)$ established via analytic continuation.

Blake wrote:

I’m still catching up with the Bernoulli-number discussion from 2008, but all those $\emptyset$s are unsettling me. Isn’t dealing with the vacuous species $e^X$ a bit problematic, since the vacuous structure can be put on the empty set?

Yes! More precisely: if $F$ is a species, then $F(e^X)$ is typically not a species — but it is a stuff type.

I’m glad you’re studying this — it’s a lot of fun.

So, we have to move beyond ordinary combinatorial species into stuff types.

Right. Indeed, near the beginning of this very discussion, I emphasized that if $F$ and $G$ are species but $G(0) \ne 0$, the composite $F \circ G$ is typically not a species, but it is a stuff type.

So, let’s see. Naively,

$$\frac{1}{1 - e^X}$$

is the composite of the species

$$F(X) = \frac{1}{1 - X}$$

and the species

$$G(X) = e^X$$

So, we should naively expect it to be a stuff type. And naively, this is the stuff type ‘being a finite set written as the union of a possibly empty list of possibly empty subsets.’ Where ‘list’ means ‘linearly ordered set’.

But this is naive, since if we think of $F$ and $G$ as formal power series, the composite $F \circ G$ isn’t a well-defined formal power series! It blows up at $X = 0$.

Nonetheless, we could still hope to make sense of

$$\frac{X}{1 - e^X}$$

And there is where David’s remark comes in. I think you’re worried that he’s calling $\frac{X}{1 - e^X}$ a species instead of a stuff type. But really, there are much worse problems to worry about here! No?

I think you’re worried that he’s calling $\frac{X}{1−e^X}$ a species instead of a stuff type. But really, there are much worse problems to worry about here! No?

Indeed. I wasn’t really worried about the terminology, per se; it just seemed that the blowing-up to infinity seen in the Bernoulli-number case was a specific instance of a more general problem.

Blake wrote:

Indeed. I wasn’t really worried about the terminology, per se; it just seemed that the blowing-up to infinity seen in the Bernoulli-number case was a specific instance of a more general problem.

Right. But I didn’t express myself very clearly, since I was pretty confused. Let me try again, at the risk of being wrong.

In the case of

$$e^{e^X}$$

the coefficient of $X^n$ is proportional to the number of ways of chopping an $n$-element set into a (possibly empty) finite set of (possibly empty) finite sets. And this is infinite unless we remember the permutation symmetries acting on a finite set of empty sets, and use groupoid cardinality.

But in the case of

$$\frac{1}{1 - e^X}$$

the coefficient of $X^n$ is proportional to the number of ways of chopping an $n$-element set into a (possibly empty) ordered finite set of (possibly empty) finite sets. Here the ordering means there are no permutation symmetries. So, groupoid cardinality does not save us: the coefficient is really infinite.

And maybe that makes sense, since

$$\frac{1}{1 - e^X}$$

blows up at $X = 0$.

We still could hope for a sensible answer for

$$\frac{X}{1 - e^X}$$

But this seems to require zeta-function regularization!

So in short, I don’t think that the passage from species to stuff types is the main problem here.

How can we understand this zeta-function regularization combinatorially? I don’t know. But here’s something I’m reminded of:

There are some infinite groups like $SL(2,\mathbb{Z})$ where if you count the elements in a clever way, and use zeta-function regularization to extract a finite answer from the resulting sum, you get an answer that correctly matches the results of other curious calculations.

In particular, the reciprocal of the cardinality of $SL(2,\mathbb{Z})$ works out to be

$$1 + 2 + 3 + 4 + \cdots = \zeta(-1) = - \frac{1}{12}$$

But this is also the orbifold Euler characteristic of the upper half-plane mod the usual action of $SL(2,\mathbb{Z})$, as shown here! This thing is also called the ‘moduli stack of elliptic curves’.

Since the upper half-plane is contractible, we’d naively expect the orbifold Euler characteristic of this space mod $SL(2,\mathbb{Z})$ to be the reciprocal of the number of elements of $SL(2,\mathbb{Z})$. So, something is ‘working’ here — but in a very strange way.

I explained some of this back in week213. I have no idea how it’s related to what we’re talking about now, but I thought I’d throw it into the pot….

Looking again at Connes’ comments, there’s plenty to go on.

The passage from the tangent bundle to the normal bundle is represented on the level of formulas by the passage to the inverse. The mystery of the generating function of the Atiyah-Singer formula, containing the exponential function in the denominator is completely dispelled if we understand this passage to the inverse.

But there’s a more ‘satisfying answer with respect to Planck’s formula’:

It is a strict parallelism…between the local proof of the index theorem using the heat equation and Planck’s formula.

This parallelism shows how this surprising function appears in exactly the same way as in Planck’s formula. This function is simply a sum of exponentials, the geometric series $1/(1 - exp(-\beta))$. It appears in the same way when one calculates the trace of the heat operator using supersymmetry, as Witten has shown. (p. 61)

I wish I understood those remarks of Connes better, so I could extract some sort of combinatorial essence.

Connes’ remarks made a little more sense to me after I read P. Windey’s “Supersymmetric quantum mechanics and the Atiyah–Singer index theorem” [Acta Physica Polonica, Vol. B15, No. 5 (1984)].

Well, since Blake has been thinking about the index theorem, maybe it’ll pay for me to say exactly what I’m confused about! Connes writes:

The passage from the tangent bundle to the normal bundle is represented on the level of formulas by the passage to the inverse.

I understand this when it comes to Chern classes. Every vector bundle $E$ over a space $M$ gives rise to an element $c(E)$ in the cohomology of $M$, called the total Chern class of $E$. Taking the direct sum of bundles corresponds to multiplying their total Chern classes:

$$c(E_1 \oplus E_2) = c(E_1) \cup c(E_2)$$

where $\cup$ stands for the product in the cohomology ring. The total Chern class of a trivial vector bundle is just 1.

Any manifold $M$ has a tangent bundle $T M$, and we can find a vector bundle over $M$, called the ‘stable normal bundle’ $N M$, such that $T M \oplus N M$ is a trivial vector bundle.

So, from what I’ve said, the total Chern class of the stable normal bundle is the inverse of the total Chern class of the tangent bundle:

$$c(T M) \cup c(N M) = 1$$

At first this sounds what Connes could be talking about. But then:

The mystery of the generating function of the Atiyah-Singer formula, containing the exponential function in the denominator is completely dispelled if we understand this passage to the inverse.

The formula for the topological index involves not the total Chern class, but rather the Chern character and the Todd class.

He must be talking about the Todd class, since the definition of that involves Bernoulli numbers and the generating function

$$\frac{x}{1 - e^{-x}}$$

Hmm, I see the Todd class is multiplicative too!

$$td(E_1 \oplus E_2) = td(E_1) \cup td(E_2)$$

So, my remarks about the total Chern class for the stable normal bundle being the inverse of the total Chern class for the tangent bundle would also apply to the Todd class.

However, the ‘mystery’ that Connes wants to ‘completely dispel’ is the appearance of the exponential function in the denominator of

$$\frac{x}{1 - e^{-x}}$$

I don’t understand how this is explained by the stuff about normal bundles.

All this stuff about ‘normal bundle as inverse of tangent bundle’ and multiplicativity for characteristic classes is making me — yet again! — want to understand the splitting principle more deeply. I understand it well enough in a textbook way, but there really should be a nice category-theoretic way of precisely stating the essential idea, namely: while it’s not true that every vector bundle is a direct sum of 1-dimensional vector bundles, for the purposes of defining characteristic classes you can pretend that it is true.

There’s something very general going on here. The category of vector bundles $Vect(X)$ on some space $X$ is a symmetric monoidal category with direct sums. This lets us define the concept of a line object: namely, an object $E$ that has an inverse with respect to tensor product, where the canonical ‘symmetry’ map $E \otimes E \to E \otimes E$ is the identity.

It’s not true that every object in $Vect(X)$ is a direct sum of line objects, but there’s a full embedding of $Vect(X)$ in a larger symmetric monoidal category with direct sums where this is true. And this embedding gets along with the symmetric monoidal structure. I think that’s roughly the idea of the splitting principle.

There should be some very broad class of symmetric monoidal categories with direct sums for which this idea works! Does someone know a theorem to this effect?

And when this idea works, another idea comes along for the ride:

What kind of a category is symmetric monoidal, with direct sums, where every object is a direct sum of line objects? Well, the classic example is the category of representations of an abelian group!

So I’m getting the feeling that any category where the splitting principle works can be embedded in the category of representations of some abelian group. Maybe there’s not always a specific ‘best choice’, but it would be really cool if there were a best choice.

This idea is somehow implicit in the concept of Adams operations and their link to the Galois group of the maximal abelian extension of the rationals!

It seems that whenever we apply the splitting principle to embed our symmetric monoidal category with direct sums into the category of representations of an abelian group, it’s natural to do it in a way so that this abelian group contains $Gal(\mathbb{Q}^{ab}/\mathbb{Q})$.

Can someone help me out? This stuff should be well-understood by someone out there…

I can’t see that it’s got anything directly to do with homotopical species.

The abstract runs

In 1967 Quillen introduced model categories “to cover in a uniform way” “a large number of arguments [in the different homotopy theories encountered] that were formally similar to well-known ones in algebraic topology”. We show the same formalism “covers in a uniform way” a number of arguments in (naive) set theory. We argue that the formalism is curious as it suggests to look at a homotopy-invariant variant of Generalised Continuum Hypothesis which has less independence of ZFC, and first appeared in PCF theory independently but with a similar motivation. Techically, we show how a naive, diagramme chasing homotopy theory approach to set theory leads to a construction of a model category (in the sense of Quillen) modelling some invariants in set theory. These invariants, the covering numbers of PCF theory, appear, in homotopy theory, as values of (minor variations of) the derived functor of cardinality. Homotopy theory suggests to look at a homotopy-invariant version of Generalised Continuum Hypothesis (hGCH) replacing cardinality by its homotopy invariant approximation, the derived functor, and we observe that ZFC proves strong bounds towards hGCH for many cardinals, either by PCF theory or trivially.

and a final hope

homotopy theory provides a non-trivial analogy between Poincare’s continuous and Cantor’s infinite.

Gromov’s Structures, Learning and Ergosystems is quoted frequently. Is there some distinctly Russian drive to speculate about the larger place of mathematics? My friend Alexandre Borovik has written ‘Mathematics under the Microscope: Notes on Cognitive Aspects of Mathematical Practice’ and ‘Shadows of the Truth’. Yuri Manin has several such papers.

Interestingly, both Manin and Gromov point to n-categories as the mathematics of the future, Manin in Georg Cantor and his heritage and Gromov in the above-mentioned paper

Probably, the full concept of “ergo-brain” can be expressed in a language similar to that of n-categories which make an even more refined (than categories/functors) tool for dealing with “arbitrariness and ambiguity” (p. 53)

With a little luck Mike finds five minutes and can tell us more… ;-)

It would take more than that. It’s basically impossible for me to read, written in a very opaque and roundabout style (plus unfinished), and most of it is scanned at low resolution. The most I can make out is that he’s trying to put a model structure on some large poset of sets where the relation $X\le Y$ is something like “for all $x\in X$ there is a $y\in Y$ with $x\subseteq y$.” Perhaps weak equivalence models something like equipotence, possibly up to finiteness or countability? One problem is that this category has no terminal object, so it can’t actually support a model structure, although perhaps it satisfies some of axioms.

I did not read much (lack of time and eye-strain with that pdf),
but I am puzzled by the description of the trivial cofibrations
in 1.0.1 (3) on page 4: they do not seem to be closed under transfinite
composition.

I have been lurking at the n-cafe for a while, and was now rather quite pleased to find my paper mentioned here…I wish to add a link to a 15-page summary of the paper discussed above which is shorter and therefore might be easier to read.

I can’t see that it’s got anything directly to do with homotopical species.

Neither do I (although I do not know much about homotopical species).

but I am puzzled by the description of the trivial cofibrations
in 1.0.1 (3) on page 4: they do not seem to be closed under transfinite
composition.

Thank for your remark and I am sorry for the confusion but I am not sure I understand your remark. The definition of the trivial cofibrations may be found on page 25 (Def.2(wc)) and its set-theoretic characterisation at page 25, claim 3(wc): A –> B is a trivial cofibration iff every element of B is a subset of an element of A up to finitely many elements. And thus it seems to be closed any transfinite induction.

The notation is unfortunate but it’s standard in the CS world: F-algebra or F-algebra. Lists and trees can both form F-algebras.

Oh, so an $F$-algebra is just an algebra for an endofunctor! Why didn’t you say so? I always complain about my calculus students who call the method of doing integrals by substitution “$u$-substitution”, because it amounts to taking the name of a variable too seriously. But I see they’re not alone.

Okay… well, I don’t see how we can ‘glue together’ algebras of an arbitrary endofunctor in the manner I described. However, there should be some conditions on an endofunctor that let us do this, turning the free vector space on the set of isomorphism classes of algebras of that endofunctor into a pre-Lie algebra. This sounds like a nice puzzle: find those conditions!

Any monad experts out there who can see an elegant answer?

It dawned on me that pre-Lie algebras make perfect sense in the context of recursive data types. In particular, containers. A container is just a functor, F, on the category of types and functions. Examples are F(X)=X², the container of pairs of the same type, or F(X)=1+XF(X), the list container.

We can make a hole in a container F by differentiating the functor using the Leibniz rule to give F’. Pointwise multiplication of two functors gives us the product of containers, ie. a container consisting of one instance of one container and one instance of the other. Given two functors, F and G, define F*G = F G’, where the product on the right is pointwise. This has the combinatorial interpretation of a G container with one element excised and replaced by an F container. Ie. it’s just like sticking in an F as a child of a G. It’s pretty easy to show that * gives a pre-Lie structure. The only catch is that we can’t subtract types. So to make sense of it we need to rearrange the definition of a pre-Lie algebra so that the subtractions become additions.

Dan wrote:

Given two functors, $F$ and $G$, define $F \ast G = F G’$, where the product on the right is pointwise. This has the combinatorial interpretation of a $G$ container with one element excised and replaced by an $F$ container. I.e. it’s just like sticking in an $F$ as a child of a $G$. It’s pretty easy to show that $\ast$ gives a pre-Lie structure.

Hey, that’s nice!

I guess this makes sense whenever $F$ and $G$ are ‘polynomial functors’ in the sense of Joachim Kock, or even the more general ‘analytic functors’ of André Joyal.

If we like, we can think of each such functor $F$ as giving a ‘first-order differential operator’ $F \partial_x$ which acts on other functors of this type. Each of these differential operators can also be used to differentiate other differential operators of the same type. So, we can define your $\ast$ operation by:

$$(F \partial_x) \ast (G \partial_x) = (F \partial_x G) \partial_x$$

All this is a categorification of Cayley’s realization that polynomial coefficient vector fields form a pre-Lie algebra, with the pre-Lie operation $\ast$ given by

$$(F \partial_x) \ast (G \partial_x) = (F \partial_x G) \partial_x$$

You know, genus-independence isn’t so important — a unitary 2d TQFT corresponds to an orthogonal basis in the genus-sensitive case. This still gives you a decomposition into 1d subspaces, it’s just that each 1d subspace now comes equipped with a real-valued weighting. Genus-independence makes this basis orthonormal.

Good point, Jamie. But now I think I’ve been a bit sloppy about ‘decompositions into orthogonal subpspaces’ versus ‘orthonormal bases’. The map from {orthonormal bases} to {decompositions into orthogonal subpspaces} is many-to-one, with fiber $\U(1)^n$ if we’re working with an $n$-dimensional Hilbert space.

In week299, I had said that a genus-independent unitary 2d TQFT gives a decomposition of the Hilbert space $Z(S^1)$ into orthogonal subspaces. But it gives more: it gives an orthonormal basis, right? $Z(S^1)$ is a special commutative Frobenius algebra, so it gets a basis of minimal idempotents… which turns out to orthonormal. So I should fix what I said.

Similarly, there’s a difference between ‘decompositions into orthogonal subspaces where each subspace has a positive-real-valued weighting’ and ‘orthogonal bases’ — but you seem above to be saying they’re the same.

You know all this stuff, of course. In fact, I guess you’ve busy categorifying all this stuff…

I just want to find the cleanest relation between some sort of 2d TQFT and some sort of ‘classical structure’. I had defined a classical structure to be a decomposition into orthogonal subspaces, but now I realize that a 2d TQFT really gives more. Getting the details exactly right may help me understand the philosophical big picture (if there is one).

Joachim, from a physical point of view the utility of operads (and generalisations) is clearer in twistor techniques, such as BCFW recursion, than it is in local QFT. See this paper by Kreimer and Suijlekom. Cheers, Marni.

Neat! I do remember seeing that double categories come up, and being happy about it. Presumably this point of view on operads/multicategories is related to the double-categorical approach to generalized multicategories.

Thanks, Tim! Lol, yes Dan. But this is often necessary in real physics papers, where the readership can be very diverse.

Gee, I sure wasted a lot of time reading Hartshorne when all the algebraic geometry I need is in that appendix.

I didn’t say that I got things to work with Safari.
I just said that I get a comment that a new Quicktime version has to be installed.

I have a mac firefox version that may be the problem.

If I would switch to Safari then I would need to transfer all my bookmarks, if this is possible at all.