The n-Category Café

Here’s a question that came up as Aaron Lauda and I have been writing A prehistory of $n$-categorical physics. It’s about whether you can prove the existence of an exotic $\mathbb{R}^4$ with the help of Khovanov homology.

Here is a question on models for $(\infty,1)$-categories of $(\infty,1)$-functors.

In mathematics, “one-to-one” and “onto” are a pair of faithful workhorses. They don’t always get the attention they deserve – mainly because it seems hard to say anything new about them. But in fact, plenty of new issues arise as we start categorifying these concepts.

In a paper we wrote together, Mike Shulman and I said a lot about this stuff. On page 47, Mike left open some questions about “0-epic” and “1-epic” functors. Yesterday I talked with Jonas Frey at Université Paris 7. Jonas is a grad student of Paul-André Melliès; coming from a background in topos theory and logic, he’s now busy throwing higher categories and string diagrams into his arsenal of techniques. And, it turned out he has settled the questions Mike left open:

• Jonas Frey, On 2-categorical duals of (full and) faithful functors.

Let me say a bit more..

guest post by Minhyong Kim

Perhaps people here at the café with a bit of arithmetic inclination (which is possibly a large majority) might be interested in this workshop:

• Non-abelian fundamental groups in arithmetic geometry: introductory workshop, Newton Institute, Cambridge, July 27–31, 2009. Organized by John Coates (Cambridge), Minhyong Kim (UCL), Richard Taylor (Harvard), and Andrew Wiles (Princeton).

A while ago I had sent a question on a certain aspect of the notion of descent to Dominic Verity.

To my pleasant surprise, a few days later he sent me a detailed 13 page description and proof of the statement in question!

I was thinking of asking him to reproduce that document here, but then I hesitated for some reason. Now it so happens that Jim Stasheff emails me today, saying that Dominic Verity permits and that he himself suggests that I do a post.

Under these conditions of course I can’t resist and am very glad to do so. I made an $n$Lab entry of it

Verity on Descent for strict $\omega$–Groupoid valued Presheaves

Please see there for Dominic Verity’s document, some introductory comments and further links.

If you are one of the many esteemed contributors to the $n$-Café who go through the trouble from time to time to post valuable information in some of the discussions that we are having – or if you always thought about doing that but never got around to it – you might be interested in having your contribution, be it a small remark or a major exposition or anything in between, archived and hyperlinked in a more accessible and more robust form than a plain blog comment provides.

That’s one thing that the $n$Lab wiki is for! The $n$Lab accumulates hyperlinked information and expertise on all the topics we discuss around here.

It’s quick and easy to include your blog comment contributions into the $n$Lab network. Here’s a page that’ll help you get started:

$n$Lab: How to get started.

guest post by Thomas Fiore

Thomas Noll’s visit to Chicago in June was amazing! His two talks were filled to the brim with mathemusical insights and musical examples. Soon after the talk, I met him again at the International Conference for the Society of Mathematics and Computation in Music, where we heard many other exciting talks on mathematics and music. The field is growing quickly, I was delighted by the huge turnout and work presented.

I’ll report on just a couple of topics in Thomas’ talks in Chicago, he said way more than I can write here. Some topics were mentioned in the thread The Mathematics of Music at Chicago, begun by John back in May.

Physicists like to complain that economists don’t make enough testable predictions. There’s a geophysicist at UCLA named Didier Sornette who has tried to remedy this situation. Here is his latest paper on the arXiv:

• K. Bastiaensen, P. Cauwels, D. Sornette, R. Woodard, and W.-X. Zhou, The Chinese equity bubble: ready to burst, July 10, 2009.

My friend the combinatorist Bill Schmitt breezed through Paris recently, taking a train back from Hopf-in-Lux with Paul-André Melliès, and spending a day here before going home to DC. We had some time to talk, and during the course of it I realized I’d become less scared of certain topics involving monads.

Monads seem to bother a lot of people. There’s even a YouTube video called The Monads Hurt My Head! It’s a long aimless self-indulgent rant, and it’s not exactly focused on monads in the mathematical sense, but you may enjoy listening to it anyway if you start at this particular second: 7:30. Shortly thereafter, the woman speaking exclaims:

What the heck?! How do you even explain what a monad is?

I’m not completely convinced of $n$Lab acting as both reference wiki and as a place to work out ideas. Perhaps it can work by the method we’re using at the moment of flagging up tentative pages, but we have to be careful. The homotopy (as an operation) page is certainly tentative. It goes on to wonder whether we can dualize everything in sight at the cohomology page. Doing this threw up an interesting effect when it turned out that there was an already existing Cech homotopy as a candidate dual for Cech cohomology. As Tim Porter describes on the Cech homotopy page, there is work of long standing falling under that title, linked to (strong) shape theory. But is there a problem with its being linked to from a tentative page? Will anyone check the extent to which Cech homotopy is dual to Cech cohomology?

Elsewhere at sphere, we have Toby questioning the need for generalised homotopy theory. But we only know it’s Toby because he mentioned this at ‘Latest Changes’. It seems likely to me that an airing here would get wider attention, where views can be easily attributed to specific people.

Maybe the point is that discussing ideas we need to know identities. On ‘Latest Changes’ we read Urs saying that about twisted K-theory that he is

…feeling slightly uneasy about making this public, though, maybe later I get scared and remove that content again, or move it to my private web.

Expressing personal views on an anonymous wiki can be awkward. On the other hand, are people visiting each other’s private webs?

Gavin Wraith proposed this puzzle for the $n$-Café:

Problem: Suppose we are given two finite sets, say one of men, the other of women, and a relation — e.g. a Grand Ball at which men and women dance in couples. After the ball the men like to discuss the women and the women the men. Let us call a group “discussive” if there is a person of the opposite sex with whom each member of the group has danced at the ball. Let $M_n$ be the number of discussive groups of $n$ men, and $W_n$ the number of discussive groups of $n$ women. Show that the alternating sums

$$M_1 - M_2 + M_3 - \cdots$$

and

$$W_1 - W_2 + W_3 - \cdots$$

are equal.

James Propp asks:

Can you steer me toward the YouTube video of someone using pebble-moves to prove that $x^6 - 1$ is a multiple of $x^2 - x + 1$, by starting with a single pebble and managing to move it six spaces to the left (or maybe it was to the right?) with moves that replace a pebble at $n$ with pebble at each of $n-1$ and $n+1$, and other moves that do the reverse?

The argument goes back to Andreas Blass’ paper Seven trees in one — he used it to give a nice bijection between the set of binary planar trees and the set of 7-tuples of such trees. The argument was generalized by Robbie Gates and later Marcelo Fiore and Tom Leinster. I discussed it in week202 of This Week’s Finds. But I don’t remember a video proof!

guest post by Orit Davidovich and Alex Hoffnung

Hi again! The following is a second set of notes which largely follows the second of David Ben-Zvi’s talks at a workshop on topological field theories, held at Northwestern University in May 2009. This post follows our previous post found here. We’ll again give a brief introduction, and then send you over to a PDF file for the full set of notes.

I’ve been having a fun conversation with Gavin Wraith, who allowed me to post parts of it here.

It leads up to a puzzle about this famous formula:

$$1^3 + 2^3 + \cdots + n^3 = (1 + 2 + \cdots + n)^2$$

I hope you know this one. Take the first $n$ natural numbers: the sum of their cubes is the square of their sum!

guest post by Orit Davidovich and Alex Hoffnung

Hi! What follows are some notes on the first of David Ben-Zvi’s talks at a workshop on topological field theories, held at Northwestern University in May 2009. We’ll start by sketching the basic ideas, and then we’ll send you over to a PDF file for more details and pretty pictures.

My previous attempt to finish this paper did not succeed:

• John Baez and Aaron Lauda, A prehistory of $n$-categorical physics — 2nd draft version or arXiv version.

I got blindsided by final exams, my arrival in Paris, my work with Paul-André Melliès here, and the need to work on other papers.

But I’m back at it again… and today, tour of the Catacombs instilled me with a new sense of urgency.

Over at $n$Lab we’re itching for some discussion as to whether there can be something which is to homotopy as Nonabelian (unstable) cohomology is to cohomology. Can we free things up so we don’t just map spheres into spaces? At the entry homotopy (as an operation) you can read the suggestion of a ‘homotopy with co-coefficients in $B$’, rather than a sphere.

I suppose Moore spaces as domain would be a start as suggested here for spaces of type $(A, 2)$ and suspensions.

Fernando Muro has kindly pointed out that there will be a school on topological quantum field theory in Almería, Spain:

• Advanced course on topological quantum field theories, October 19th to 23rd, 2009, University of Almería, Spain, organized by David Llena, Fernando Muro, Frank Neumann, José L. Rodríguez Blancas (coordinador), Miguel Ángel Sánchez Granero, and Antonio Viruel.

This course aims to engage PhD students and postdoctoral researchers in new developments while letting them meet international experts in the field. There will be a limited number of grants covering travel and lodging expenses for young participants — so register now. The course will be followed by the XVI Spanish Topology Meeting, which runs from October 23rd to the 24th.

$n$-Café regulars will know about Representative Conyer’s bill that would repeal the National Institute of Health’s public access policy and forbid other US funding agencies from mandating open access to research papers written with the help of federal grant money. Conyers’ argument in favor of this bill was hilariously misinformed. He wrote: “Journal publishers organize and pay for peer review with the proceeds they receive from the sale of subscriptions to their journals.”

But laughing at the folly of the world is not really much fun. Now some good news, for a change! A bill has been introduced that would do quite the opposite. It would ensure free, timely, online access to the published results of research funded by the National Science Foundation and ten other US federal agencies!

We all know how Elsevier has been running fake medical journals for the drug company Merck, devoted to saying good things about Merck products. But this isn’t all they’re up to.

For example, on a recent thread here at the $n$-Café, Ben pointed out an interesting BBC news report. Apparently Elsevier offered Amazon gift certificates to academics who would write 5-star reviews of their textbook Clinical Psychology!

Caught red-handed, Elsevier blamed this action on an unnamed ‘rogue employee’. They did not say whether any disciplinary action would be taken.

[guest post by Martin Laubinger in the context of Smootheology: the study of generalized smooth spaces]

I have posted a preprint which contains the central new result I obtained in my dissertation.

The result may not directly relate to higher category theory. Still, I would appreciate feedback and problems for further investigation:

Martin Laubinger
A Lie algebra for Frölicher groups
(arXiv)

Here is a short description: The category of Frölicher spaces is a cartesian closed category which contains the category of smooth finite-dimensional manifolds as a full subcategory. The same is true for the closely related category of diffeological spaces. However, it is easier to define tangent spaces to Frölicher spaces than to diffeological spaces. Many groups have natural Frölicher structures, including all Lie groups, but also certain groups of mappings such as $C^\infty(M,G)$ or $\Diff(M)$, which can not be given a manifold structure in general. A basic question is whether the tangent space (in the Frölicher sense) at the identity of these groups can be equipped with a Lie bracket. I have been able to construct such a Lie bracket, but there is an additional condition which has to be verified. This condition is very natural, but I have not found a general proof. In my thesis, I did not have an example for a group which satisfies the extra condition, but in the meantime I verified the condition for the additive group $\mathbb{R}^J$ (product of the reals) if J is not too big. This is explained in detail in the preprint.