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June 6, 2007

Whose 2-Vector Spaces?

Posted by David Corfield

Back here, Urs said:

…while it is quite interesting that the “K-theory” of Baez-Crans 2-vector bundles is K×KK \times K and that of Kapranov-Voevodsky is BDR-2K, the original hope was that there is a kind of 2-vector bundle such that its K-theory is something more closely resembling elliptic cohomology, somehow.

This goal has not been achived yet, as far as I am aware.

And I argued that this is maybe no wonder: while the notions of 2-vector spaces used so far in these studies all have their raison d’être, they are all comparatively restricted, as compared with the most general notion of 2-vector space one would imagine.

A fuller explanation is given in this post.

Now, a paper out today, Two-vector bundles define a form of elliptic cohomology, pushes further the BDR-2K project. Does the “a form” still suggest the need to follow Urs’ advice and look towards Bim(Vect)?

Now who were those experts Urs persuaded to work on the first n-category Café millennium prize?

Posted at June 6, 2007 8:49 AM UTC

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Re: Whose 2-vector spaces?

Now, a paper out today, Two-vector bundles define a form of elliptic cohomology, pushes further the BDR-2K project.

There are two (old) claims: that the “classifying space” (the (group completion, really, of the) realization of the nerve of the topological 2-category

a) of Baez-Crans 2-vector spaces

b) of Kapranov-Voevodsky 2-vector spaces

“is” (homotopy theorist’s “is”)

a) two copies of the ordinary K-theory spectrum

b) the algebraic K-theory spectrum (that of a ring) of the ordinary K-theory spectrum, with the latter regarded as a ring (since it is a ring spectrum) (in short: “K-theory of K-theory”).

For quite a while, it had not been fully clear to me to what degree these claims have been supplied with complete proofs.

It was only recently that the proofs had been completed. I mentioned this in What Does the Classifying Space of a 2-Category Classify?

(Birgit Richter played a crucial role in that. Her office happens to be right next to mine.)

What do these nerves of categories have to do with 2-vector bundles?

The idea is this: as we have discussed at length here at the blog (and are still discussing), ordinary bundles are given by isomorphism classes of functors (the “transition functions”) g:Y [2]ΣG, g : Y^{[2]} \to \Sigma G \,, where Y [2]YXY^{[2]} \stackrel{\to}{\to} Y \to X is the groupoid coming from a good cover YY of XX by open sets, and where σG\sigma G is the structure group, regarded as a category.

By applying the nerve realization functor ||:Cat TopTop |\cdot| : \mathrm{Cat}_{\mathrm{Top}} \to \mathrm{Top} from topological categories to topological spaces, we find that such functors are the same as homotopy classes of maps |g|:|Y [2]||ΣG|. |g| : |Y^{[2]}| \to |\Sigma G| \,. But |Y [2]|X|Y^{[2]}| \simeq X. Moreover, |ΣG|:=BG|\Sigma G| := B G, the classifying space of GG-bundles (or, equivalently, “of the category ΣG\Sigma G”).

So |g|:XBG. |g| : X \to B G \,.

This is the familiar classification of principal 1-bundles.

Now, these authors take the point of view that suich transition data is all there is to nn-bundles. Hence what they do is computing the nerves of higher categories which are to be thought of as categorifications of ΣG\Sigma G.

The original hope was that 2-vector bundles would have a “categorified K-theory” which is directly related to elliptic cohomology. This idea dates back to

Baas, Dundas & Rognes, 2-Vector bundles and forms of elliptic cohomology.

I have tried to give a rather detailed discussion of this literature and lots of trelated things in Seminar on 2-Vector bundles and Elliptic Cohomology, I, II, III.

Early on it was rather clear that by using Baez-Crans 2-vector spaces and KV 2-vector spaces does not really yield the classifying spaces one was hoping to see.

As you mention, I believe there is a pretty plausible reason for that: both these notions of 2-vector spaces exhaust only a tiny fraction of the 2-category of all 2-vector spaces.

Probably one doesn’t need the full thing, but I am pretty sure that one does need at least all 2-vector spaces which admit a 2-basis: this are those equivalent to module categories of algebras.

One reason for this is the existence of the “canonical 2-rep” of any strict 2-category, and hence in particular the existence of String-2-bundles (which are to be thought of a 2-Spin-bundles). This I describe in Connections on String-2-Bundles.

Now who were those experts Urs persuaded to work on the first n-category Café millennium prize?

I don’t want to talk about names here - yet. Let’s see how things proceed!

Posted by: urs on June 6, 2007 12:02 PM | Permalink | Reply to this

Re: Whose 2-vector spaces?

But K-theory in its roginal manifestation
corresponded to (equivalence classes of) STABLE vector bundles. Only by a miracle
(Bott periodicity) and for restricted base spaces did it correspond to a classifying space of a group.

So are you thingking of STABLE 2-bundles?

Posted by: jim stasheff on June 8, 2007 9:45 PM | Permalink | Reply to this
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 2:20 PM

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