The n-Category Café

I am further thinking about some issues which we discussed at the recent entry Ben-Zvi on geometric function theory (see $n$Lab: geometric function theory for some context).

From where I am coming the site over which we are looking at our generalized spaces in form of $\infty$-stacks is not an algebraic site and the $(\infty,1)$-stack QC of quasicoherent sheaves which Ben-Zvi-Francis-Nadler use so fruitfully as a model for nice geometric functions on generalized spaces is not manifestly available.

There is another construction naturally desiring to take its place, though, and I am wondering how the two perspectives would connect.

Here’s a new improved version of a paper for Bob Coecke’s book New Structures for Physics:

• John Baez and Mike Stay, Physics, topology, logic and computation: a Rosetta Stone — 4th draft version or arXiv version.

We thought it was done almost a year ago, but there was a serious mistake at the very end of the section on computation. Now we’ve completely rewritten that section — so, we’d love to hear what you think about it. Typos? Fine, we’ll fix them. Serious mistakes? I hope not, but if so, please tell us about them now!

Reading again Proofs and Refutations for my first Utrecht talk, I came across this part of the dialogue:

Omega: But I want to discover the secret of Eulerianness!

Zeta: I understand your resistance. You have fallen in love with the problem of finding out where God drew the boundary dividing Eulerian from non-Eulerian polyhedra. But there is no reason to believe that the term ‘Eulerian’ occurred in God’s blueprint of the universe at all. What if Eulerianness is merely an accidental property of some polyhedra? In this case it would be uninteresting or even impossible to find out the zig-zags of the demarcation line between Eulerian and non-Eulerian polyhedra. Such an admission however would leave rationalism unsullied, for Eulerianness is then not part of the rational design of the universe. So let us forget about it. One of the main points about critical rationalism is that one is always prepared to abandon one’s original problem in the course of the solution and replace it by another one. (pp. 67-68)

Almost a year ago, Mike Stay and I proudly announced the completion of our Rosetta Stone paper, which explains how symmetric monoidal closed categories show up in physics, topology, logic and computation. But then Theo and Todd helped us spot a serious mistake in our description of a programming language that was supposed to be suitable for work inside any symmetric monoidal closed category — and hence, good for both classical and quantum computation.

We’re almost done fixing that problem now, and Mike is starting to write his thesis on related issues: classical and quantum computation and how they relate to ‘linear’ and also ‘categorified’ versions of the lambda-calculus.

But as a small side-effect, we’ve stumbled upon a deviant definition of ‘monoidal closed category’. I haven’t proved it’s equivalent to the usual definition. I bet it’s not. But I don’t know counterexamples. I wonder if anyone here has thought about such entities before.

Since several experts on monoidal closed categories read this blog, I’m optimistic.

This sounds like a really cool way to learn some high-powered applications of category theory:

• School on Homotopy Theory and Algebraic Geometry, September 7th-13th 2009, Mathematical Research Institute of the University of Sevilla, Spain. Organized by Luis Narváez Macarro, Beatriz Rodríguez González and Michel Vaquié.

Three puzzles based on a conversation with the writer Tim Powers:

1. Which founder of Caltech’s Jet Propulsion Laboratory was also a practicing Satanist, and invoked the god Pan whenever a rocket was launched?

2. Aleister Crowley sent this person a warning. What was the warning about, and was it heeded?

3. How did this person die?

Can you help Lieven out with some questions?

Have you seen a first-year group theory course starting off with groupoids? Do you know an elegant way to prove a classical group-result using groupoids?

Last Friday saw me in Utrecht to deliver a couple of talks for the From Plato to Predicativity seminar. I spent a very pleasant morning in a typical Dutch café chatting with Klaas Landsman about mathematics, physics and philosophy.

At the seminar in the afternoon I spoke first on Lakatos and then on Lautman, uniting them under the banner of ‘dialectical realists’. One obvious difference between the two is how contemporary were Lautman’s case studies. In the 1930s he’s talking about class field theory, where Lakatos’s main work from the late 1950s and early 1960s concerned early to mid-nineteenth century mathematics. His aesthetic antenna was so finely tuned to detect a certain kind of structure similarity that, had his life and work not been curtailed by the 1939-45 war, I wonder whether Lautman might have prompted his Bourbaki friends to take up category theory more rapidly.

On February 3rd, John Conyers of the US House of Representatives re-introduced a bill to repeal the National Institute of Health’s public access policy, which says that research funded by this agency must be made freely available on a database called PubMed Central.

But that’s not all! This bill, ironically called the Fair Copyright in Research Works Act, would also ban all other federal agencies from adopting open-access policies!

Igor Baković is an energetic young mathematician from Croatia… I bet we’ll be hearing a lot from him as time goes on. He’s very excited about topos theory, nonabelian cohomology, 2-bundles and the like. I met him in Göttingen last week, and he said he and Branislav Jurčo were almost done with a paper on this subject. Now it’s out!

• Igor Baković and Branislav Jurčo, The classifying topos of a topological bicategory.

David Ben-Zvi notes that we can now see these talks by Jacob Lurie:

• Jacob Lurie, TQFT and the cobordism hypothesis, videos of 4 lectures at the Geometry Research Group, Mathematics Department, University of Texas Austin.

Lecture notes should be forthcoming. For a brief summary of the cobordism hypothesis and its generalizations, visit the nLab. Experts should improve this page!

John Huerta is a student of mine who’s really interested in particle physics. Pretty soon he’ll plunge into his thesis work on exceptional algebraic structures and their role in physics — especially super-Yang–Mills theory, superstring theory and supergravity, but maybe also grand unified theories. But first he needs to pass his oral. Here are the slides for his talk:

• John Huerta, The Algebra of Grand Unified Theories. Also available in printer-friendly form.

With any luck, I’ll be in Paris this summer and thus able to learn more about the ‘practical applications’ of categorification to knot theory:

• Summer School on Link Homology, June 27 – July 3, 2009, at the Institut Henri Poincaré, 11 rue Pierre et Marie Curie, Paris. Organized by Christian Blanchet, Vincent Colin, Emmanuel Ferrand, Nadine Fournaiseua, Bernhard Keller and Gregor Masbaum.

Yesterday I had reported (here) some aspects of Ieke Moerdijk’s talk at Higher structures II in Göttingen about dendroidal sets, which are to simplicial sets as operads are to categories. Hence there should be a notion of $\infty$-operad (or $(\infty,1)$-operads, really) which is to $(\infty,1)$-categories as dendroidal sets are to simplicial sets.

In today’s talk Ieke Moerdijk looked into more details of the homotopical description of $(\infty,1)$-operads within all dendroidal sets. Here are some aspects reproduced from the notes that I have taken during the talk.

This morning I was in Hamburg chatting with David Ben-Zvi about his work (see his recent guest post for a pedagogical introduction) then I jumped on the train and arrived just in time in Göttingen at the workshop Higher Structures II (see this post for the announcement) to hear Ieke Moerdijk’s talk on $\infty$-Operads and dendroidal sets.

Now after conference dinner it’s already late, but I thought I’d try to produce at least parts of my notes of Ieke Moerdijk’s talk. I am doing this with an eye towards our discussion about hyperstructures over at the $n$Lab (which started at this blog entry) which is clearly somehow related (inconclusive as it is at this stage) to the idea underlying dendroidal sets.

From one point of view the simple underlying idea is that

- simplicial sets and hence in particular for instance Kan complexes and $(\infty,1)$-categories are presheaves on the simplex category $\Delta$, which can be regarded as the full subcategory of $Cat$ on those posets which are “linear” (totally ordered) $\cdots \leftarrow \leftarrow \leftarrow \cdots$

- a dendroidal set is a presheaf on a subcategory of $Cat$ (not a full one, though) on slightly more general posets, namely those which are “tree shaped” $$\array{ &&& \swarrow & \leftarrow & \cdots \\ \cdots &\leftarrow &\leftarrow &\leftarrow& \cdots \\ & \nwarrow \\ && \leftarrow & \cdots }$$

- while there are some situations which are naturally modeled by presheaves on more general posets, for instance those of the form $$\left( \array{ && \top \\ & \nearrow && \nwarrow \\ a &&&& b } \right)^{\times n}$$ as considered in Marco Grandis’s work on Cospans in Algebraic Topology for describing extended cobordisms

- or even entirely general ones, possibly, as considered tentatively at $n$Lab. hyperstructure.

One way to think about this is that every (finite) poset can be regarded as defining one of the geometric shapes for higher structures. For instance the posets $G'\downarrow [n]$ arising as over-categories of the globe category, which locally look like the poset that Tom Leinster recently mentioned here, are the poset incarnation of the $n$-globe: $c_i$ and $d_i$ represent its two sub-$i$-globes and the morphisms indicate which subglobe sits inside which higher sub-globe (the poset depicted above is actually that of the boundary of the $(n+1)$-globe, with the top $(n+1)$-cell missing). These globe-posets play a crucial role in Michael Batanin’s work (see his comment here).

As Ronnie Brown kindly points out at hyperstructure, the idea of defining higher structures without commiting oneself to a single or to one of the standard shapes (globes, simplices, cubes) is an old one (I am being told that this goes back to Grothendieck’s dérivateurs, but am lacking currently further information on that) which has for instance been studied by D. Jones in A general theory of polyhedral sets and their corresponding T-complexes.

All this may or may not be directly related to the main point of dendroidal sets and $\infty$-operads, but I felt like mentioning it in any case. More concretely, dendroidal sets are supposed to be precisely the notion that completes the analogy

$$\array{ category & (\infty,1)-category & simplicial set & weak Kan complex \\ operad & \infty-operad & dendroidal set & Kan dendroidal set } \,.$$

The following is a reproduction of some of the notes that I took in Ieke Moerdijk’s talk today. A closely related survey talk I had recently reproduced here.

guest post by Jeffrey Morton

In this guest post, I thought I would step back and comment about big picture of the motivation behind what I’ve been talking about on my own blog. I recently gave a talk at the University of Ottawa, which tries to give some of the mathematical/physical context. It describes both “degroupoidification” and “2-linearization” as maps from spans of groupoids into (a) vector spaces, and (b) 2-vector spaces. I will soon write a post setting out the new thing in case (b) that I was hung up on for a while until I learned some more representation theory. However, in this venue I can step even further back than that.

Tim Gowers is engaged in a new venture in open source mathematics. As one might expect from a leading representative of the ‘problem-solving’ culture, Gowers has proposed a blog-based group problem solving challenge.

He motivates his choice of problem thus:

Does the problem split naturally into subtasks? That is, is it parallelizable? I’m actually not completely sure that that’s what I’m aiming for. A massively parallelizable project would be something more like the classification of finite simple groups, where one or two people directed the project and parcelled out lots of different tasks to lots of different people, who go off and work individually. But I’m interested in the question of whether it is possible for lots of people to solve one single problem rather than lots of people to solve one problem each.