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If we work over a field $k$, and take the recursive definition of $n$-vector spaces (as, e.g. in Topological Quantum Field Theories from Compact Lie Groups, arXiv:0905.0731) then a $2$-vector space is a $k$-algebra $A$, to be thought as a placeholder for its categoty of modules; morphisms between $A$ and $B$ are $B$-$A$-bimodules, ad 2-morphisms are morphisms of bimodules.

Now consider a 2-vector bundle over some space $X$. How does one see that its global sections are a 2-vector space?

The answer somehow depends on the notion of section one adopt, but in any case the relation between the various definitions should be investigated. Basically there are two notions coming to my mind:

i) natural transformations from the trivial 2-bundle to the given bundle. This is a very neat object, but it is not clear (to me) that this is a 2-vector space: which is the underlying algebra?

ii) the limit in 2-Vect of the functor from the nerve of a good open cover of $X$ to 2Vect defining the 2-bundle. This is manifestly a 2-vector space, but it is not clear (to me) that this limit exists.

Clearly the dream statement here would be that i) has a natural structure of 2-vector space, and that this 2-vector space represents the limit ii), but I'm unable to prove this.

(or the dual version of the above, under suitably finiteness assumptions)

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Can you say more precisely what a 2-vector bundle is? Just a bundle of $k$-algebras? –  Konrad Waldorf Jun 7 '11 at 12:29
    
My (maybe too naive) picture of a 2-vector bundle over a space $X$ is in terms of local data on a good open cover, which encode a simplicial morphism $\mathcal{N}(\mathcal{U})\to 2Vect$. That is, over each $U_i$ I have a bundle of $k$-algebras $A_i$; over the double intersections $U_{ij}$ a bundle of $A_i$-$A-j$-bimodules $B_{ij}$ (these are "linear" functors between the categories of $A_i$- and $A_j$-modules), and on triple intersections $U_{ijk}$ I have endomorphisms of the $A_i$-$A_i$-bimodule $B_{ij}\otimes_{A_j}B_{jk}\otimes_{A_k}B_{ki}$, with all the natural compatibility conditions. –  domenico fiorenza Jun 7 '11 at 14:13

2 Answers 2

I think it's somewhat misleading to just say a 2-vector space is an algebra (just like specifying objects of a category without morphisms carries little information up to equivalence): the point is it's the Morita theory of algebras, with morphisms given by bimodules. This Morita theory embeds fully into the two-category of linear categories and linear functors (with some exactness etc depending on the details of your context). In other words, a 2-vector space is just a linear category, with the property that it has a generator, but without fixing the generator. The endomorphisms of this generator (the corresponding algebra) are a somewhat misleading "invariant" of the 2-vector space.

In any case your dream statement certainly holds in the derived version of this story, and is certainly the right intuition, though I can't vouch for the precise statement in the nonderived context..

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Hi David, thanks. Yes, I meant "an algebra" as a placeholder for its category of modules, with bimodules as morphisms; this pretended o be implicit in the "recursive definition", but as you and Konrad remarked this was absolutely not clear from the way I wrote my question, so now I'm editing it accordingly. Could you add a few details on the derived version of this story? Thanks. –  domenico fiorenza Jun 7 '11 at 16:19

In the special case $k=\mathbb C$ (or $\mathbb R$), you can take "2-bundle" to mean continuous field of C*-algebras (possibily with the extra condition that it be locally trivial, or locally trivial-up-to-Morita-equivalence). If you don't know what acontinuous field of C*-algebras, you can safely think of it as a bundle of C*-algebras.

The Čech data that you mention in your comment indeed provide examples of continuous fields of C*-algebras. So, in that sense, continuous fields of C*-algebras really behave like 2-categorical bundles (they also behave like 1-categorical bundles, but that's irrelevant).

Now, you can simply take the global sections to be the C*-algebra of global sections. In other words, take the 1-categorical global sections.

Starting with the simplest kind of non-trivial Čech data: all the algebra bundles are trivial (with fiber $\mathbb C$); all the bimodules are trivial (with fiber $\mathbb C$); the coherences are given by a two-cocycle with values in $S^1$, then the corresponding algebra of global sections is a continuous trace C*-algebra: a non-unital C*-algebra that is locally Morita equivalent to an abelian C*-algebra.

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