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August 31, 2010

Categorification in Portugal

Posted by John Baez

Jeffrey Morton has gotten married!

And he’s moved to Portugal, where he’s taken a job at the Instituto Superior Técnico in Lisbon. This is a center for work in categorification, thanks in large part to Roger Picken, who with Joao Faria Martins has recently been studying connections on 2-bundles and 3-bundles.

Check out his latest blog entry:

You’ll see pictures of Lisbon, as well as descriptions of talks from the XIXth Oporto meeting, given by Rafael Diaz, Yazuyoshi Yonezawa, Aleksandar Mikovic, Aaron Lauda, Sabin Cautis, Catharina Stroppel, Ben Webster, and Dylan Thurston.

Posted at 8:23 AM UTC | Permalink | Post a Comment

August 30, 2010

Enriching Over a Category of Subsets

Posted by Simon Willerton

Several of us here at the Café are fans of Lawvere’s paper on enriched categories:

What I would like do here is expand on a part of the paper that I haven’t seen mentioned elsewhere, although I haven’t looked very hard, about enriching over a category of subsets of some fixed set. I will show how this leads, for instance, to the following generalized metric on the set of continuous functions [0,1][0,1]\to \mathbb{R}, where μ\mu means the Lebesgue measure:

d (f,g)μ{x[0,1]f(x)>g(x)}.d^\le(f,g)\coloneqq\mu\{x\in[0,1]\mid f(x)\gt g(x)\}.

It also leads to the following symmetrized version which is a metric in the classical sense:

d =(f,g)μ{x[0,1]f(x)g(x)}.d^=(f,g)\coloneqq\mu\{x\in[0,1]\mid f(x)\neq g(x)\}.

I would be interested to hear if anyone has any other examples of categories enriched over subsets of some set.

Posted at 5:21 PM UTC | Permalink | Followups (28)

August 28, 2010

The Difference between Measure Zero and Empty Interior

Posted by Tom Leinster

This is a post in the category of “Small things I have learned”.

Close your eyes and picture a set of measure zero, where by “set” I mean subset of n\mathbb{R}^n. Now open them again so that you can carry on reading…

Close them again and picture a set with empty interior.

For me, the two mental images are about the same. Should they be?

Posted at 3:52 AM UTC | Permalink | Followups (28)

August 27, 2010

This Week’s Finds (Week 301)

Posted by John Baez

I won’t keep pestering people with This Week’s Finds here now that it’s moved to another blog and switched topics, but:

I invite everyone here to check out the first issue of the new This Week’s Finds. You’ll see a basic overview of the global warming problem, other ecological problems, and their common underlying cause. We’ll get into lots more detail later.

Posted at 6:41 AM UTC | Permalink

August 26, 2010

The Geometry of Monoidal Fibrations?

Posted by Mike Shulman

In my paper framed bicategories and monoidal fibrations I wrote down a definition of a “monoidal fibration,” which you can think of as a family of monoidal categories indexed by a (usually cartesian) monoidal category. (I certainly wasn’t the first person to write down such a thing.) A canonical example is that if VV is any monoidal category, then the categories V XV^X, for XX a set, are each monoidal and are indexed by the cartesian monoidal category Set\mathrm{Set}.

What I want to know now is – is there a string diagram calculus for these things? Never mind proving anything about it; I can’t even visualize what it ought to look like!

Posted at 4:52 AM UTC | Permalink | Followups (19)

August 25, 2010

Free High School Science Texts Needs You!

Posted by Tom Leinster

Over the years at the Café we’ve talked a lot about the exploitative practices of certain commercial publishers, and the virtues of free, openly-licensed texts. Here’s your chance to contribute to the open textbook movement.

Free High School Science Texts is

a project that aims to provide free science and mathematics textbooks for Grades 10 to 12 science learners in South Africa.

The texts are openly licensed, so can be adapted and used anywhere in the world.

It was started by physicist Mark Horner (then a grad student in Cape Town), who is supported by the Shuttleworth Foundation — Mark Shuttleworth being the South African entrepreneur who donated millions of dollars of his fortune to Ubuntu and other free software projects.

For reasons I don’t understand, Free High School Science Texts suddenly needs a lot of proof-reading done fast. I just received a copy of an email from Horner beginning

Hi guys

I’m in a completely crazy position—the textbooks will be distributed by national government next week if we can do another round of proof-reading of them all by Wednesday [1 September].

If you have a postgraduate degree in math, physics, chemistry, or a related field, they want your help. Here’s what to do:

  1. Have a look at Horner’s calls for help: yesterday and today.
  2. Send an email to the address mentioned in the second link, so that you can be assigned a section to read.

That’s it!

Posted at 9:57 PM UTC | Permalink | Followups (12)

August 21, 2010

What is the Langlands Programme?

Posted by Tom Leinster

You probably know that the 2010 Fields Medals have been announced.

One of the four medallists, Ngô Bảo Châu, works on the Langlands programme. Now, I know the Langlands programme is famous. In particular, Laurent Lafforgue won a Fields medal for his work on the Langlands programme in 2002, so there was a flurry of publicity surrounding it then.

The thing is, I’ve never succeeded in understanding the slightest thing about it. It’s as if it’s got a hard, shiny shell—it resists all attempts at explanation, even when the person listening is a trained, interested, mathematician. But I’d like to believe that isn’t so.

I’ve heard numerous attempted explanations in the past, but as soon as I run into a term like “automorphic form” or “reductive group” I’m lost. (I can go and look those up, and I have, but I still haven’t learned anything about the Langlands programme itself.) I’ve tried reading the two explanations on the ICM page, but one’s pitched too high and the other too low. I’ve tried the Wikipedia page, but it jumps suddenly from sentences such as

It is a way of organizing number theoretic data in terms of analytic objects

to sentences such as

The Artin reciprocity law applies to a Galois extension of algebraic number fields whose Galois group is abelian, assigns L-functions to the one-dimensional representations of this Galois group; and states that these L-functions are identical to certain Dirichlet L-series or more general series (that is, certain analogues of the Riemann zeta function) constructed from Hecke characters.

I’d love it if someone could explain just one thing about the Langlands programme in terms I’ll understand. If you can do that, I’ll have learned more about it from you than I ever have from anyone else.

Posted at 12:51 AM UTC | Permalink | Followups (102)

August 19, 2010

Minicourse on Nonabelian Differential Cohomology

Posted by Urs Schreiber

During September I’ll be at the ESI-institute in Vienna to give one third of a Minicourse on higher gauge theory .

My plan is at

Meanwhile with Domenico Fiorenza and Zoran Škoda we are discussing various aspects of the \infty-Chern-Weil theory that is part of this here on the nForum.

Posted at 12:15 AM UTC | Permalink | Followups (2)

August 11, 2010

This Week’s Finds in Mathematical Physics (Week 300)

Posted by John Baez

In the last issue of This Week’s Finds in Mathematical Physics, learn how to categorify the Riemann zeta function and get a groupoid whose cardinality is

ζ(2)=π 26 \zeta(2) = \frac{\pi^2}{6}

Posted at 1:55 PM UTC | Permalink

August 7, 2010

Means

Posted by Tom Leinster

I’ve been interested for a while in abstract notions of size, and this has got me thinking about abstract notions of measure—hence, inescapably, of integration. Now integration can be thought of as an averaging process: for instance, the mean value of a function f:[0,1]Rf: [0, 1] \to \mathbf{R} is simply 0 1f\int_0^1 f. So means provide another way to think about measure and integral (and probability, indeed).

For most types of integration there are restrictions on the functions involved: you can only integrate an integrable function. But I’m going to describe three slightly unusual settings in which the challenge is to find a notion of integration, or mean, in which every function is allowed. Here they are:

  1. The mean of binary digits
  2. Amenable groups
  3. Arrow’s Theorem on voting systems.

Then there’s a postscript on voting and enriched categories.

Posted at 4:00 AM UTC | Permalink | Followups (21)

August 6, 2010

Thermodynamics and Wick Rotation

Posted by John Baez

My grad student Mike Stay is a busy guy with many interests. He has a full-time job at Google working on Caja, which is a ‘source-to-source translator for securing Javascript-based web content’. He’s working with me on monoidal bicategories and their applications to computer science. And on the side, he’s been pondering some questions about thermodynamics and the idea of ‘Wick rotation’.

I asked him to post those latter questions here. Can you help him out?

Posted at 3:43 AM UTC | Permalink | Followups (54)

August 4, 2010

LaTeX Overflow

Posted by Mike Shulman

The new buzzing thing on the math blogosphere is a StackExchange site (that means a question+answer forum like MathOverflow) intended for TeX and LaTeX questions. It’s just getting started, but given the success of MO, it looks like it has the potential to become a great resource. Now if only it had an associated wiki to record the accumulated knowledge….

Posted at 4:34 AM UTC | Permalink | Followups (2)

August 1, 2010

Copenhagen Meeting on Differential Characters

Posted by Urs Schreiber

Am in Denmark on vacation. And for attending a day or two of the

Copenhagen meeting on differential characters.

Differential characters are one model for what Deligne cohomology and abelian nn-gerbes with connection are other models for: the differential cohomology refinement of the ordinary Eilenberg-MacLane cohomology theory K(,)K(-,\mathbb{Z}).

Myself, I’ll talk about this:

\infty-topos theoretic differential characters and the \infty-Chern-Weil homomorphism on principal \infty-bundles.

Posted at 5:52 PM UTC | Permalink | Post a Comment

Pullback-homomorphisms

Posted by Tom Leinster

Matías Menni and I have been having some interesting conversations about notions of Möbius inversion in categories, prompted by his talk at Category Theory 2010, his recent paper with Bill Lawvere, and my paper a while ago on Euler characteristic of categories.

This post is about an offshoot of our conversation. It solely concerns some very standard notions of category theory. Take a monad TT on some category. A TT-algebra is, of course, an object AA together with a map TAAT A \to A satisfying some axioms, and a map f:ABf: A \to B of TT-algebras is a commutative square TA Tf TB A f B. \begin{matrix} T A &\stackrel{T f}{\to} &T B \\ \downarrow & &\downarrow \\ A &\stackrel{f}{\to} &B. \end{matrix} Call ff a pullback-homomorphism if this square is a pullback.

How should we think about pullback-homomorphisms? What properties do they have? Are they a useful notion? (And is there a better name?)

Surely, surely, someone has looked into this before. Does anyone know anything about it?

Posted at 12:16 PM UTC | Permalink | Followups (29)