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October 18, 2006

Topology in Trondheim and Kro, Baas & Bökstedt on 2-Vector Bundles

Posted by Urs Schreiber

The Nordic Conference in Topology takes place in Trondheim, 24-25 November 2006.

A few 2-categorical talks are announced, in particular concerning 2-bundles.

B. Dundas gives a talk titled 2-vector bundles and KK-theory of KK-theory, which, I am being told, is concerned with new progress on the old approach by Baas, Dundas & Rognes to conceive “forms of elliptic cohomology” in terms of 2-vector bundles.

In the same vein, Tore A. Kro talks about 2-categorical KK-theories, which is apparently based on

N. Baas, M. Bökstedt & T. A. Kro
2-categorical K-theories
Royal Swedish Academy of Sciences, Institut Mittag-Leffler
preprint, spring 2006, nr. 43

Abstract: For 2-categories 2C2C we define a notion of associated charted 2C2C-bundles and 2K2K-theories. For the 2-category of 2-vector spaces in the sense of M. Kapranov and V. Voevodsky this gives the 2K2K-theory of Baas, Dundas and Rognes. When the 2-category is a nice 2-groupoid we prove that its 2-nerve is a classifying space for the associated charted 2C2C-bundles. In the process of proving this result we develop a lot of powerful machinery which may be useful in further studies of 2-categorical topology. As a corollary we get a new proof of the classification of principal bundles. As another corollary it follows that if we take the 2-category to be the 2-vector spaces introduced by J. Baez, then the associated 2K2K-theory is just two copies of ordinary KK-theory.

The last result, that <Baez-Crans 2-vector>-bundles give rise to a classification by two copies of ordinary KK-theory (instead of one copy of “2-KK-theory”, as one might expect) has surfaced before, but, as far as I am aware, only now appears in (pre-)print.

So, Baez-Crans 2-vector spaces, while useful for many purposes, are not what one wants to use as the typical fiber in an interesting 2-<vector bundle>.

Hence, instead, Baas, Dundas, Rognes & Kro use Kapranov-Voevodsky 2-vector spaces.

But - evidently - these are still not quite what we want as fibers of really interesting 2-vector bundles. Because ultimately one wants a geometric realization (in terms of classes of 2-bundles) not of KK-theory of KK-theory, but of elliptic cohomology proper.

In case anyone cares, as I have remarked before (also here), I think there are several indications that we want to pass to a 2-category of 2-vector spaces which is somewhat larger than that of KV 2-vector spaces.

Let KK be some field. I believe Baez-Crans 2-vector spaces are module categories for the monoidal category Disc(K)\mathrm{Disc}(K), the discrete category of the set KK.

(1)BC2Vect Disc(K)Mod. BC2\mathrm{Vect} \simeq {}_{\mathrm{Disc}(K)}\mathrm{Mod} \,.

On the other hand, KV 2-vector spaces are a very special case of categories that are Vect K\mathrm{Vect}_K-modules

(2)KV2Vect K Vect KMod. KV2\mathrm{Vect}_K \subset {}_{\mathrm{Vect}_K}\mathrm{Mod} \,.

A “nn-dimensional” KV 2-vector space is, in fact, a category of modules for the algebra K nK^{\oplus n}.

Being careful about distinguishing the two different levels of “modules” that play a role here, it is noteworthy that every category of right modules for an ordinary KK-algebra is itself a module for a left action by Vect K\mathrm{Vect}_K.

In fact, there is an embedding

(3)Bim(Vect K) Vect KMod \mathrm{Bim}(\mathrm{Vect}_K) \subset {}_{\mathrm{Vect}_K}\mathrm{Mod}

which sends KK-algebras AA to their categories of AA modules

(4)AMod A, A \mapsto \mathrm{Mod}_A \,,

which sends AA-BB bimodules to functors

(5) AN B((N A?):Mod AMod B), {}_A N{}_B \mapsto ((N \otimes_A ?) : \mathrm{Mod}_A \to \mathrm{Mod}_B) \,,

and so on.

The way KV2VectKV2\mathrm{Vect} sits inside Vect KMod{}_{\mathrm{Vect}_K}\mathrm{Mod} is just a special case of that

(6)KV2VectBim VectMod. KV2\mathrm{Vect} \subset \mathrm{Bim} \subset {}_{\mathrm{Vect}}\mathrm{Mod} \,.

But KV2VectKV2\mathrm{Vect} is too small. It doesn’t have sufficiently many multiplicative inverses. That’s what makes constructing KV2VectKV2\mathrm{Vect}-bundles a little unwieldy: the transitions on double intersections are not in general (not even weakly) invertible.

Bim(Vect K)\mathrm{Bim}(\mathrm{Vect}_K) is larger. It has plenty of inverses (all weakly invertible bimodules). Is it large enough?

At least, there are several indications that Bim(Vect K)\mathrm{Bim}(\mathrm{Vect}_K) 2-vector bundles are the 2-vector bundles that appear in physics.

A U(1)U(1)-gerbe with curving is a 2-functor to bimodules, taking values in bimodules of algebras of compact operators. Since these are Morita equivalent to the ground field, that’s a “1-dimensional” 2-vector bundle. A line 2-bundle.

Quite similarly, a String(n)\mathrm{String}(n)-bundle with connection is a 2-functor with values in bimodules for the representation algebra of a loop group.

It is quite straightforward to write down local descriptions in terms of transitions for these bimodule 2-vector bundles. This I can do.

What I cannot do right now without investing a forbidding amount of time it to work out the classification of the bimodule 2-vector bundles.

But I am hoping that somebody will.

Posted at October 18, 2006 4:29 PM UTC

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Read the post What does the Classifying Space of a 2-Category classify?
Weblog: The n-Category Café
Excerpt: Notes by Baas, Boekstedt and Kro on classifying spaces for 2-vector bundles.
Tracked: December 4, 2006 3:36 PM
Read the post Whose 2-Vector Spaces?
Weblog: The n-Category Café
Excerpt: 2-vector spaces for elliptic cohomology.
Tracked: June 6, 2007 5:00 PM
Read the post Extended Quantum Field Theory and Cohomology, I
Weblog: The n-Category Café
Excerpt: On understanding extended quantum field theory and generalized cohomology.
Tracked: June 8, 2007 2:17 PM
Read the post 2-Vectors in Trondheim
Weblog: The n-Category Café
Excerpt: On line 2-bundles.
Tracked: November 5, 2007 9:46 PM
Read the post Concordance
Weblog: The n-Category Café
Excerpt: On the notion of concordance of 2-bundles and, more generally, on a notion of omega-anafunctor and a possible closed structure on the category of omega-categories with omega-anafunctors between them.
Tracked: November 23, 2007 6:08 PM

Re: Topology in Trondheim and Kro, Baas & Bökstedt on 2-Vector Bundles

Does anyone know if Dundas’ talk in Spain recently would update Urs’ version here?

Posted by: jim stasheff on September 20, 2008 1:58 PM | Permalink | Reply to this

Re: Topology in Trondheim and Kro, Baas & Bökstedt on 2-Vector Bundles

Out today, Stable bundles over rig categories by Nils A. Baas, Bjorn Ian Dundas, Birgit Richter, John Rognes

The point of this paper is to prove the conjecture that virtual 2-vector bundles are classified by K(ku)K(ku), the algebraic KK-theory of topological KK-theory. Hence, by the work of Ausoni and the fourth author, virtual 2-vector bundles give us a geometric cohomology theory of the same telescopic complexity as elliptic cohomology.

Posted by: David Corfield on September 10, 2009 1:43 PM | Permalink | Reply to this

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