Currently i am travelling and talking about something at least closely related.

I am not thinking in the algebraic context that Toën and Vezzosi do, but that is more a question of implementation than of pinciple.

I have been pointing out

in

Nonabelian differential cohomology in Street’s descent theory

and in the developing

$\Sigma$-models and nonabelian differential cohomology

that $\omega$-categorical sheaves, hence sheaves with values in $\omega\mathrm{Cat}$, hence $\omega$-categories internal to sheaves (all on your site of choice) albeit much “stricter” than other models for “$\infty$-stacks” are not only convenient and useful for describing higher bundles (principal or associated, with or without connection) hence higher cohomology, but also sufficient.

The fact that makes this work (in this context) is the central result in arXiv:0705.0452 for $n=1$ and its generalization to $n=2$ and higher $n$ refining the discussion of math.DG/0511710 which I have talked about a lot here, and which is about to be published, which says that an $n$-bundle with structure $n$-group $G$ on X, what other call a $G$-torsor for the constant $G$-sheaf, in terms of an $n$-categorical total space fibration
$\array{
P
\\
\downarrow
\\
X
}$
are equivalent to fiber-assigning functors
$\array{
G\mathrm{Tor}
\\
\uparrow
\\
X
}$
which are all those such functors which admit a local trivialization that induces descent in the sense of Ross Street with coefficients in the $\omega$-category valued sheaf
$\omega\mathrm{Cat}(-,\mathbf{B}G)
\,.$

While $n$-categorical total space fibrations
$P \to X$
form an $n$-stack, since their pullback is defined “only” by a universal property and hence respects composition of pullback functions only weakly, the fiber-assigning functors
$X \to G\mathrm{Tor}$
pull back simply by precomposition
$Y \to X \to G\mathrm{Tor}$
and hence always form a *rectified* $n$-stack, in fact a sheaf.

The same game can be played with $n$-vector spaces replacing $G\mathrm{Tor}$ by $\mathrm{n}\mathrm{Vect}$.

For instance some of the aspects about K-theory and 2-vector bundles that Toën and Vezzosi mention in the introduction are described in my slides

On nonabelian differential cohomology

(say from slide 21 on) and in the extensive

On String and Chern-Simons $n$-transport

where the section on 2-vector bundles starts on slide 324.

Toën and Vezzosi cite Stolz-Teichner’s comparatively old proposal for realizing elliptic cohomology but do not cite the combined result of math.QA/0504123 and arXiv:0801.3843/ arXiv:math/0612549 which shows that Stolz-Teichner’s String bundles indeed *are* 2-bundles (bundles with categorical fibers, as Toën and Vezzosi are looking for).

I have talked at various times about the corresponding associated 2-vector bundles coming from the canonical 2-representation of the String 2-group, for instance in

The canonical 2-representation.

These representations are on a sub-2-category of $2\mathrm{Vect} := Vect-\mathrm{Mod}$ considerably larger than the Kapranov-Voevodsky 2-vector spaces, and I have argued many times that the fact that Baas-Dundas-Rognes didn’t find the expected elliptic cohomology from KV-2vector bundles but something else (K-theory of K-theory) is not all that surprising, given that they are working with such a small subcategory of all 2-vector spaces.

From the Witten genus argument one expects elliptic cohomology to be related to String group representations like K-theory is related to Spin group representations, so I expect that 2-vector bundles associated by the canonical 2-rep of the String 2-group to principal String 2-bundles should yield ellitpic cohomology, essentially following the original argument by Stolz-Teichner, but refining it properly 2-categorically.
I am not the topologist who could compute the corresponding spectrum. But I keep mentioning it in the hope that somebody looks into it who does.

Concerning the search for Chern-characters on 2-vector bundles: we had discussed characteristic classes of 2-bundles here in the context of arXiv:0801.1238, arXiv:0801.3843 and arXiv:0801.3480.

In my notes on nonabelian differential cohomology I start discussing how the $L_\infty$-connections and their characteristic classes of arXiv:0801.3480 may be integrated to nonabelian differential cocycles, thus yielding characteristic classes for these.

## Re: Categorical Sheaves

Currently i am travelling and talking about something at least closely related.

I am not thinking in the algebraic context that Toën and Vezzosi do, but that is more a question of implementation than of pinciple.

I have been pointing out

in

Nonabelian differential cohomology in Street’s descent theory

and in the developing

$\Sigma$-models and nonabelian differential cohomology

that $\omega$-categorical sheaves, hence sheaves with values in $\omega\mathrm{Cat}$, hence $\omega$-categories internal to sheaves (all on your site of choice) albeit much “stricter” than other models for “$\infty$-stacks” are not only convenient and useful for describing higher bundles (principal or associated, with or without connection) hence higher cohomology, but also sufficient.

The fact that makes this work (in this context) is the central result in arXiv:0705.0452 for $n=1$ and its generalization to $n=2$ and higher $n$ refining the discussion of math.DG/0511710 which I have talked about a lot here, and which is about to be published, which says that an $n$-bundle with structure $n$-group $G$ on X, what other call a $G$-torsor for the constant $G$-sheaf, in terms of an $n$-categorical total space fibration $\array{ P \\ \downarrow \\ X }$ are equivalent to fiber-assigning functors $\array{ G\mathrm{Tor} \\ \uparrow \\ X }$ which are all those such functors which admit a local trivialization that induces descent in the sense of Ross Street with coefficients in the $\omega$-category valued sheaf $\omega\mathrm{Cat}(-,\mathbf{B}G) \,.$

While $n$-categorical total space fibrations $P \to X$ form an $n$-stack, since their pullback is defined “only” by a universal property and hence respects composition of pullback functions only weakly, the fiber-assigning functors $X \to G\mathrm{Tor}$ pull back simply by precomposition $Y \to X \to G\mathrm{Tor}$ and hence always form a

rectified$n$-stack, in fact a sheaf.The same game can be played with $n$-vector spaces replacing $G\mathrm{Tor}$ by $\mathrm{n}\mathrm{Vect}$.

For instance some of the aspects about K-theory and 2-vector bundles that Toën and Vezzosi mention in the introduction are described in my slides

On nonabelian differential cohomology

(say from slide 21 on) and in the extensive

On String and Chern-Simons $n$-transport

where the section on 2-vector bundles starts on slide 324.

Toën and Vezzosi cite Stolz-Teichner’s comparatively old proposal for realizing elliptic cohomology but do not cite the combined result of math.QA/0504123 and arXiv:0801.3843/ arXiv:math/0612549 which shows that Stolz-Teichner’s String bundles indeed

are2-bundles (bundles with categorical fibers, as Toën and Vezzosi are looking for).I have talked at various times about the corresponding associated 2-vector bundles coming from the canonical 2-representation of the String 2-group, for instance in

The canonical 2-representation.

These representations are on a sub-2-category of $2\mathrm{Vect} := Vect-\mathrm{Mod}$ considerably larger than the Kapranov-Voevodsky 2-vector spaces, and I have argued many times that the fact that Baas-Dundas-Rognes didn’t find the expected elliptic cohomology from KV-2vector bundles but something else (K-theory of K-theory) is not all that surprising, given that they are working with such a small subcategory of all 2-vector spaces.

From the Witten genus argument one expects elliptic cohomology to be related to String group representations like K-theory is related to Spin group representations, so I expect that 2-vector bundles associated by the canonical 2-rep of the String 2-group to principal String 2-bundles should yield ellitpic cohomology, essentially following the original argument by Stolz-Teichner, but refining it properly 2-categorically. I am not the topologist who could compute the corresponding spectrum. But I keep mentioning it in the hope that somebody looks into it who does.

Concerning the search for Chern-characters on 2-vector bundles: we had discussed characteristic classes of 2-bundles here in the context of arXiv:0801.1238, arXiv:0801.3843 and arXiv:0801.3480.

In my notes on nonabelian differential cohomology I start discussing how the $L_\infty$-connections and their characteristic classes of arXiv:0801.3480 may be integrated to nonabelian differential cocycles, thus yielding characteristic classes for these.