Posts with this logo use 
March 31, 2008
Limits and Push-Forward
Posted by Urs Schreiber
The limit and colimit of a functor can be understood as the “push-forward of the functor to a point”: the image of the functor under the right or left adjoint functor of the pullback of functors from the terminal category .
Is there a useful generalization of this correspondence between limits and push-forward for the case of indexed limits?
March 30, 2008
This Week’s Finds in Mathematical Physics (Week 262)
Posted by John Baez
In week262 of This Week’s Finds, see the Southern Ring Nebula and the frosty dunes of Mars:
Then read about quantum technology in Singapore, atom chips, graphene transistors, nitrogen-vacancy pairs in diamonds, a new construction of , and a categorification of quantum .
March 29, 2008
Test Your Singlish
Posted by John Baez
Singlish is a creole language based on English, Malay, Hokkien, Teochew, Cantonese, Tamil and various other languages. I didn’t hear much Singlish during my recent visit to the Singapore, but I found a nice book about it in Kinokuniya, which is conceivably the world’s best bookstore chain.
There’s a lot of wit in some Singlish expressions, and I hope they catch on elsewhere in the English-speaking world. Try guessing what these mean:
- action (verb)
- arrow (verb)
- blur (adjective)
- catch no ball (verb)
- havoc (adjective)
- stylo mylo (adjective)
- Z-monster (noun)
(Of course you can resort to various online Singlish dictionaries, but that’s cheating.)
March 27, 2008
Categorified Quantum Groups
Posted by David Corfield
Once in a distant blog, John was quick to pour cold water on the suggestion I made that the aims of those categorifying might differ sufficiently to merit distinguishing types of ‘categorification’:
I don’t like this “Frenkelian” versus “Baezian” distinction. Baez was inspired to work on higher categories thanks to the work of Crane and Frenkel. Frenkel’s student Khovanov cites Baez’s work on 2-tangles in his first paper on categorified knot invariants. Frenkel’s student Khovanov has taken on Baez’s student Lauda as a postdoc at Columbia starting next fall. Will their work on categorifying quantum groups and using these to get 2-tangle invariants be “Frenkelian” or “Baezian”?
Some results of the collaboration are now out. Aaron has just posted A categorification of quantum sl(2) to the arXiv.
Did the Geometric Representation Theory Seminar reach the point of having categorified quantum groups? Not that I’m after differences of approach, of course.
What Has Happened So Far
Posted by Urs Schreiber
The -Category Café has recently passed beyond entries, trackbacks and comments. Maybe a good time to look back at what has happened so far.
Our subtitle says “A blog on math, physics and philosophy”. For me, there is one major question sitting at the intersection of these three subjects. It is
The fundamental question of quantum physics: What is a -model, really?
I have been exclusively talking about this question ever since we started the blog. I started referring to it as the question of the QFT of the charged -particle #. I still think this is the more descriptive term, but it was rightly indicated to me that it is not politically advisable for somebody in my position to make up new terminology.
Since it was also pointed out to me ## that it may at times be hard to remember the big picture, let me recall:
The proposed answer to the fundamental question of quantum physics: Pull-push of nonabelian differential cocycles.
We are in the setting of general cohomology theory, where generalized/homotopy/ana-morphisms between “spaces” (usually # presheaves with values in a homotopy category) are “cocycles” encoding higher fiber bundles. And also higher fiber bundles with connection, which are addressed as (nonabelian) differential cocycles #.
Given a (nonabelian, differential) cocycle on , and given another “space” , there is a canonical way to obtain a cocycle on : we pull-push through the correspondence
The pullback along (followed by the hom-adjunction) is transgression of the cocycle on to a cocycle on . The push-forward along is “taking sections” ## #.
Usually the push-forward along won’t exist. The chances that it exists increase when the original cocycle is pushed-forward along a representation
In the context of quantum physics, is the target space in which an “()-brane” (= -particle) with worldvolume # of shape propagates and is charged # # under a background field . The pull-push is quantization in the extended/localized # sense of Freed ##. is the Schrödinger picture # propagation. Applying an endomorphism functor sends it to the Heisenberg picture # of AQFT #. Since quantization sends differential cocycles to differential cocycles, we can iterate. This is second quantization #.
While following through this program, we ran into one big puzzle, concerning the proper nature of -curvature: it turned out that a differential cocycle “with values in ” is actually a certain constrained generalized morphism into # . Understanding that funny shift in dimension properly used up maybe 50 percent of my time here, and is probably the reason if the effort looked less than coherent at times.
Making recourse to the “rationalized” approximation of -connections # the pattern was finally understood, and now there are very nice relations emerging # between this question and major programs of my co-bloggers: higher topos theory and geometric representation theory/groupoidification.
There is one main class of examples which motivates all this effort: quantization of # (higher) Chern-Simons bundles with connection to Chern-Simons QFT ## and its holographic # #boundary theory. Indeed, the realization # that the known modular category theoretic formulation of 2-dimensional CFT # # was in fact secretly a differential cocycle was what originally lead to the proposed answer above. This is being worked out with Jens Fjeldstad #.
The hardest part of figuring out the pull-push of a given cocycle is in top dimension. This is no surprise, since there it must reproduce the “path integral”. But first consistency checks in simple toy examples suggest that it does work # # allright.
But with the big picture finally stabilizing, many details need to be worked out further.
March 22, 2008
Nonabelian Differential Cohomology in Street’s Descent Theory
Posted by Urs Schreiber
As a followup to our recent discussion #:
Nonabelian differential cohomology
in Street’s descent theory
(pdf, 20 pages)
Abstract: The general notion of cohomology, as formalized -categorically by Ross Street, makes sense for coefficient objects which are -category valued presheaves. For the special case that the coefficient object is just an -category, the corresponding cocycles characterize higher fiber bundles. This is usually addressed as nonabelian cohomology. If instead the coefficient object is refined to presheaves of -functors from -paths to the given -category, then one obtains the cocycles discussed in [BS, SWI, SWII, SWIII] which characterize higher bundles with connection and hence live in what deserves to be addressed as nonabelian differential cohomology. We concentrate here on -categorical models (strict globular -categories) and discuss nonabelian differential cohomology with values in -groups obtained from integrating L(ie)- algebras.
March 21, 2008
Groupoidfest in Riverside
Posted by John Baez
The next Groupoidfest is here in Riverside!
- Groupoidfest, November 22-23, 2008, Mathematics Department, University of California, Riverside, organized by Aviv Censor.
I hope some of you can come! If you want to, contact Aviv as described on the conference website.
Crossed Menagerie
Posted by Urs Schreiber
Tim Porter kindly made the following notes available online:
Tim Porter
Crossed Menagerie:
an introduction to crossed gadgetry and cohomology in algebra and topology
(pdf with the first 7 chapters (237 pages))
March 20, 2008
This Week’s Finds in Mathematical Physics (Week 261)
Posted by John Baez
In week261 of This Week’s Finds, learn about the Engraved Hourglass Nebula:
Then read an ode to the number 3, which explains how all these entities are connected:
- the trefoil knot
- cubic polynomials
- the group of permutations of 3 things
- the three-strand braid group
- modular forms and cusp forms
March 17, 2008
The World of L
Posted by David Corfield
March 13, 2008
Slides: On Nonabelian Differential Cohomology
Posted by Urs Schreiber
On nonabelian differential cohomology
(52 pdf slides)
March 12, 2008
Chern-Simons Actions for (Super)-Gravities
Posted by Urs Schreiber
Just as
electromagnetism is a theory of line 1-bundles with connection coupled to electric 1-particles and magnetic 1-particles,
we have that
supergravity # in eleven dimensions is a theory of line 3- and line 6-bundles with connection coupled to electric 3-particles and magnetic 6-particles.
(There is a beautiful discussion of essentially this statement by D. Freed, which I talked about here, and here. Freed doesn’t say “-bundle with connection”, but instead says “differential cocycle”. But it’s the same kind of thing.)
Wonders never cease, and hence there are indications that there is more to 11-dimensional supergravity than meets the eye. The question is: what? What is 11-dimensional supergravity really about?
One idea is: it is really about 1-particles on the “-group manifold”. This we talked about before.
Another idea is: it is really about the higher Chern-Simons theory # of an invariant degree 6-polynomial on a super Lie algebra not unlike super- #.
This speculation was put forward in
Petr Hořava
M-Theory as a Holographic # Field Theory
(arXiv)
The jargon in the title is such as to make certain physicists excited. A completely different, but possibly just as exciting jargon would be: it is speculated here that, very fundamentally, physics is about those representations of extended cobordism categories which are naturally induced from Chern-Simons -bundles with connection.
I was reminded of that by the appearance of the very nicely written basic review
Jorge Zanelli
Lecture notes on Chern-Simons (super-)gravities
(arXiv)
which was updated a few days ago. (Thanks to It’s equal but It’s different for noticing.)
This reviews the action functionals for theories of gravity one obtains by picking a -dimensional manifold , a structure group like together with a degree invariant polynomial