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May 31, 2007
On the Bar Construction
Posted by Urs Schreiber
– guest post by Todd Trimble –
Here is a comment about the bar construction which I wanted to mention, since John told me that his course is ending soon and he might not get around to mentioning it himself.
I’d like to thank Urs for suggesting this be made a guest post, and all three hosts for the hard work they put into making the Café such a great success.
The Second Edge of the Cube
Posted by Urs Schreiber
What is a Cartan-Ehresmann connection, really?
If we know what a Cartan-Ehresmann connection with values in a Lie algebra really is, we should immediately know what an -Cartan-Ehresmann connection with values in any Lie -algebra is.
Kock on Higher Connections
Posted by Urs Schreiber
Building on his synthetic description of parallel transport, which I mentioned a while ago in Kock on 1-Transport, Anders Kock has now worked out a notion of higher order connections using synthetic differential geometry:
Anders Kock
Infinitesimal cubical structure, and higher connections
arXiv:0705.4406v1
This is rooted in the world of strict -fold groupoids: an -connection here is an -fold functor from cubical -paths in a space to an -fold groupoid :
Smoothness of this functor is described using synthetic differential reasoning (described in detail in his book).
In the strict cubical context, Anders Kock finds precisely the relation between parallel transport, curvature, and Bianchi identity which I describe in -Curvature:
For any given parallel -transport, the corresponding curvature is an -transport. The latter is necessarily flat, meaning that its curvature -transport is trivial. This flatness of the curvature -transport is the (higher order) Bianchi identity.
What is a Lie Derivative, really?
Posted by Urs Schreiber
For solving the problem that I am currently working on, it turned out I need to understand
What is a Lie derivative, really?
By which I mean
What is a Lie derivative, arrow-theoretically?
By which I mean
How can I think of a Lie derivative in an implementation-independent way, such that the concept may be a) internalized and, in particular, b) be categorified without effort (read: without running into problems that require thinking).
As David Corfield has put it in The Two Cultures of Mathematics Revisited:
“[…] for any worthwhile idea there is a story about it which gets to the heart of what it really is, and I’ll know when I’ve reached that point by the ease with which it categorifies.”
And this may be necessary for understanding what’s going on.
So here is my current take at the answer to “What is a Lie derivative, really?”. It’s maybe not quite the final answer yet, but the applications that I am looking at suggest that this is on the right track.
Let me know what you think!
May 28, 2007
This Week’s Finds in Mathematical Physics (Week 252)
Posted by John Baez
In week252 hear about the possibility of oceans on Neptune billions of years from now:
Learn the latest about hot Neptunes in other solar systems. See the electromagnetic snake at the center of the Galaxy. And, continue reading the Tale of Groupoidification! In this episode, with a nod to the work of Georg Frobenius and William Burnside, we begin to tackle the theme of "Hecke operators".
May 26, 2007
Link Homology and Categorification in Kyoto
Posted by John Baez
Aaron Lauda points out an interesting conference:
- Link Homology and Categorification, Kyoto University, May 10-25, 2007.
You can see notes for some of the talks!
May 25, 2007
Congratulations to David!
Posted by John Baez
Hurrah!
David Corfield just got a permanent job in the Philosophy Department of the University of Kent at Canterbury!
Derivation Lie 1-Algebras of Lie n-Algebras
Posted by Urs Schreiber
Here is a fact which should be important, but whose true meaning is not entirely clear to me at the moment:
To every Lie -algebra is canonically associated an ordinary Lie algebra .
This Lie 1-algebra is, in some sense, a Lie 1-algebra of derivations of the Lie -algebra. But it is not the derivation Lie -algebra .
What is , conceptually?
Why does it exist, as an ordinary Lie algebra?
What is its analog at the level of Lie -groups?
Below I give the detailed description and mention applications where this is relevant.
For more details see section 3.3.1 of
and section 5.3 of
(which has grown out of the original zoo).
Going Hi-Tech
Posted by David Corfield
Perhaps we should be going more hi-tech at the Café. If you enjoy a bit of fancy PowerPoint, try out Chris Schommer-Pries’ slides for a talk ‘What is an N-category?’.
Chris is a student of Peter Teichner at Berkeley.
May 24, 2007
Cohomology and Computation (Week 24)
Posted by John Baez
In the 1950s, Cartan, Eilenberg, Mac Lane and others systematically studied the cohomology of many different algebraic gadgets — groups, associative algebras, Lie algebras, and so on. It was later realized that underlying all these different cohomology theories there’s a marvelous unifying idea: the bar construction. In this week’s seminar on Cohomology and Computation, we began trying to understand what makes the bar construction tick:
-
Week 24 (May 17) - The bar construction. Why do adjoint functors give simplicial objects? First, is the free monoidal category on a monoid object — or "the walking monoid", for short. Second, adjoint functors give certain monoids, called "monads".
Supplementary reading:
- Todd Trimble, On the bar construction.
- Todd Trimble, Bar constructions.
May 22, 2007
The Two Cultures of Mathematics Revisited
Posted by David Corfield
Where did we get to in our discussion of the two cultures of mathematics? To explore the possibility that interaction may be possible between what Gowers called ‘combinatorics’ and our Café subculture we were set the challenge of categorifying instances of the Cauchy–Schwarz inequality, which, unless I missed something, didn’t result in any noticeable success.
Now, an extreme wing of our subculture would take Urs’ remark
One knows one is getting to the heart of the matter when the definitions in terms of which one conceives the objects under consideration categorify effortlessly.
and replace the when by when and only when.
May 21, 2007
Questions on n-Categories and Topology
Posted by John Baez
Here are some questions on n-categories and topology from Bruce Westbury. I’ll post a reply later — but why don’t some of you take a crack at them first?
May 19, 2007
Chern Lie (2n+1)-Algebras
Posted by Urs Schreiber
I mentioned Lie -algebras coming from invariant symmetric polynomials of degree on some Lie algebra in Zoo of Lie -Algebras.
Since it is a really simple and short computation, I want to give here all the details needed to understand it.
May 18, 2007
Calculations Inside Semisimple Categories
Posted by Urs Schreiber
– guest post by Bruce Bartlett –
Hi guys,
I’ve got a question regarding performing calculations inside semisimple categories.
I posted a version of it back in March. I’ve made quite a lot of progress, but I’m still missing some vital ingredient which I can’t put my fingers on. John hinted (rather sneakily!) in one of his TWF comments that he had made some progress on it too. Since everyone has been working so diligently on it, I thought it’s time to reconvene the homework group and compare notes.
Recall that the problem is about defining adjoints (or ‘daggers’) of natural transformations. There are two ways to define them : a left-handed way and a right-handed way, and we’d like to check they are the same.
Don’t worry about the specific problem. Just think : we have some calculation to perform inside a semisimple category. What are we going to do?
Linear Algebra Done Right
Posted by David Corfield
– guest post by Tom Leinster –
One of the best things about the n-Category Café is the willingness of its clientele to re-examine notions that are usually thought of as too basic to bother with — things that you might think were beneath your dignity.
Another great thing about the n-Café is the generosity of its proprietors in letting me post here! Thanks to them.
I’m not going to talk about fancy linear algebra: just the good old theory of finite-dimensional vector spaces, like you probably met when you were an undergraduate. Or rather, not quite like what you met…
The inspiration is Sheldon Axler’s textbook Linear Algebra Done Right. Unfortunately you can’t download it, but you can download an earlier article making the same main points, Down with determinants!.
Penrose on Angular Momentum: An Approach to Combinatorial Space-Time
Posted by John Baez
Georg Beyerle has created an electronic version of a classic paper on spin networks that was previously available only in an out-of-print book:
- Roger Penrose, Angular momentum: an approach to combinatorial space-time.
Roger Penrose has given me permission to put it on my website. Take a copy! If you spot typos, please let me know.