## April 9, 2008

### Categorical Sheaves

#### Posted by David Corfield

In Toën and Vezzosi’s A note on Chern character, loop spaces and derived algebraic geometry, the authors explain how from considerations of elliptic cohomology

…there should exist an interesting notion of categorical sheaves, which are sheaves of categories rather than sheaves of vector spaces, useful for a geometric description of objects underlying elliptic cohomology.

But having listed a set of desiderata, argue that such a theory won’t be forthcoming. They resort instead to derived categorical sheaf theory.

As we will see the notion of 2-vector spaces appear naturally in this setting as the dualizable objects, exactly in the same way that the dualizable modules are the projective modules of finite rank. After arguing that this notion of 2-vector space is too rigid a notion to allow for push-forwards, we will consider dg-categories instead and show that they can be used in order to categorify homological algebra in a similar way as linear categories categorify linear algebra.

Posted at April 9, 2008 9:49 AM UTC

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### 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 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

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.

Posted by: Urs Schreiber on April 9, 2008 11:38 AM | Permalink | Reply to this

### Re: Categorical Sheaves

So is this right? You have all the push-forwards you want, whereas for T&V

One major problem with this notion of 2-vector bundles is the lack of push-forwards in general. (p. 7)

Posted by: David Corfield on April 9, 2008 11:59 AM | Permalink | Reply to this

### Re: Categorical Sheaves

You have all the push-forwards you want

This I didn’t say. The recent discussion on Limits and push-forward was supposed to educate me further on the question of push-forward of fiber-assigning functors (or transport functors). I wish I understood this better.

I was hoping to be able to reduce the question to some nice abstract machinery, making full use of the fact that all my $n$-bundles with connection are $n$-functors ($\omega$-functors).

Push-forward of these is Kan-extension. In this comment, following lots of help from Robin Houston, I recall how the formula for the Kan extension of a functor, applied to a very simple toy case, does precisely encode the idea of push-forward of vector bundles, in that the coend formula sums up all the contributions of fibers that are being sent to the same point downstairs.

So if we consider an ordinary vector bundle as a fiber-assigning functor $F : X \to Vect \,,$ where $Vect$ is the category of not-neccesarily finite vector spaces, then the Kan-extension of that functor along some map $X \to Y$ produces the fiber assigning functor on $Y$ which represents the pushed-forward vector bundle. (One needs to be careful with the extra structure around, though, like smoothness for instance).

I am trying to use the fact that Kan extensions work in the arbitrary enriched context and that I think I am able to realize $n$-vector spaces entirely in terms of strict globular $n$-categories to define the push-forward of my fiber-assigning (parallel transport) $V$-functor $X \to T$ for $V = \omega\mathrm{Cat}$ to be the (left) Kan extension along the corresponding morphism. I was thinking that general existence theorems on the Kan extension when the domain $V$-category is small (as it is for the cases of interest) would guarantee that this exists, and I’d just need to work it out.

But I am not there yet.

Posted by: Urs Schreiber on April 9, 2008 1:44 PM | Permalink | Reply to this

### Re: Categorical Sheaves

I should say what this $\omega$-categorical formulation of $\omega$-vector spaces is that I am referring to:

given an $L_\infty$-algebra $g$ and a cochain complex $v$ (all assumed to be finite dimensional here) I am defining a representation of $g$ on $v$ to be an extension of the DGCA $CE(g)$ by the DGCA $\wedge^\bullet v^*$: $\wedge^\bullet v^* \leftarrow CE(g,v) \leftarrow CE(g) \,.$ This is supposed to be the dual differential version of the action groupoid $V \to V//G \to \mathbf{B} G \,.$ Ordinary reps of Lie algebras are of this form and every $L_\infty$-algebra has an adjoint rep on itself with this definition. The BRST-complex is also a special case, as we discussed.

So that makes me think that this definition is “good” and “right”. But this then means that I can deduce what the good and right $\omega$-vector spaces are which we need to $\omega$-vector bundles, since they should come from reps of $L_\infty$-algebras.

We integrate an $L_\infty$-algebra $g$ by applying first the functor $S : DGCAs \to SmoothSpaces$ which is part of the adjunction induced by the ambimorphic sheaf of forms, and then form path $\omega$-groupoids, $\Pi_n$ or $\Pi_\omega$.

We can apply this to the entire sequence defining the $L_\infty$-rep $\Pi_\omega \circ S (\wedge^\bullet v^* \leftarrow CE(g,v)\leftarrow CE(g)) \,.$

The right item will integrate to $\mathbf{B}(-)$ of the “simply connected” Lie $\omega$-group integrating $g$, so this wants to become $V \to V//G \to \mathbf{B}G$ with the left $\omega$-groupoid $V = \Pi_\omega(S(\wedge^\bullet v^*))$ being the $\omega$-vector space that $G$ is represented on.

Such a $V$ is a bit of a mixture of pure chain complexes (“Baez-Crans $\infty$-vector spaces”) and something richer. Its space of $k$-morphisms is the collection of $v$-valued differential forms on $D^k$ modulo this and that and being closed or satisfying some closure relations depending on the differential structure of $v$.

Then an associated $\omega$-vector bundle would be a transport functor $\Pi_\omega(x) \to \omega\mathrm{Cat}$ with local $V$-structure following our general prescription, possibly with some quotients applied on the right $\omega$-groupoid if we don’t want to talk about the “simply connected” structure group $G$ obtained from the integration process, but some quotient by it, usually by $\mathbf{B}^k \mathbb{Z}$s.

I was thinking that by just using abstract nonsense on $\omega Cat$-enriched functors I get a guarantee of push-forwards of such $\omega$-vector bundles. But I still need to think much more about this.

Posted by: Urs Schreiber on April 9, 2008 2:11 PM | Permalink | Reply to this

### Re: Categorical Sheaves

Posted by: Bruce Bartlett on April 9, 2008 4:01 PM | Permalink | Reply to this

### Re: Categorical Sheaves

Just one general comment, for the record, without claiming anything:

One main aspect of dg-enrichment is of course that it is enrichment in a model category.

$\omega Cat$-enrichment is also enrichment in a model category: Yves Lafont, Francois Metayer, Krzysztof Worytkiewicz A folk model structure on omega-cat

(Thanks to Mike Shulman for this reference.)

Posted by: Urs Schreiber on April 9, 2008 7:37 PM | Permalink | Reply to this

### Re: Categorical Sheaves

With regard to the T and V article which inspired this thread and especially the remark about categorified homological algebra’, how much of the implied contortions are due to wanting to do this for algebraic geometry?

Posted by: jim stasheff on April 11, 2008 1:00 PM | Permalink | Reply to this

### Re: Categorical Sheaves

I cannot help feeling that somewhere or other the Kan extensions being used need souping up to be homotopy Kan extensions’ especially in the enriched setting. Am I missing something?

Posted by: Tim Porter on April 9, 2008 4:40 PM | Permalink | Reply to this

### Re: Categorical Sheaves

I cannot help feeling that somewhere or other the Kan extensions being used need souping up to be ‘homotopy Kan extensions’ especially in the enriched setting. Am I missing something?

I need to have a closer look at examples. If the ordinary Kan-extension of my $\omega$-functor exists, it is hard to imagine how it would not be the right answer to some question related to push-forward. But I am really not done with thinking about this.

One aspect that I need to clarify further is that there should be a secret shift of dimension at work here.

There is this fun toy example, which should be close to your heart, because it connects this to the Yetter model and your formula for the measure on its configuration space:

Let $G_{(2)}$ be a strict 2-group coming from some crossed module $(H \stackrel{t}{\to} G \stackrel{\alpha}{\to} Aut(H))$, let $\Sigma$ be some compact manifold and write $\Pi_2(\Sigma)$ for the strict 2-groupoid obtained by picking a triangulation of $\Sigma$ with $v_0$ vertices and $v_1$ edges and some number of faces, which generate the 2-morphisms in $\Pi_2(\Sigma)$.

Then $conf = hom_{2Cat}(\Pi_2(\Sigma),\mathbf{B}G_{(2)})$ is the “space” (2-groupoid) of fields of the Yetter model.

Let $F : conf \to \mathrm{Cat}$ be a 2-functor on $conf$ which is a direct sum of representable 2-functors.

Postcomposing with Tom Leinster’s category cardinality $|\cdot| : (Cat,\oplus) \to (\mathbb{Q},+)$ we can think of this as a $\mathbb{Q}$-valued gauge-invariant function on configuration space.

$|F(-)| : conf_\sim \to \mathbb{Q} \,.$

I think Tom Leinster’s theorem expresses the cardinality of the left Kan-extension of $F$ along $conf \to \mathrm{pt}$ as $|\int^{conf} F| = |colim F| = \sum_{a \in Obj(conf)} |F(a)| \; d\mu(a) \,,$ where $d\mu : a \mapsto |G|^{-v_0} |H|^{v_0-v_1}$ is the measure on the configuration space which you discuss in your article with João Martins .

So this seems to be an example where we do the ordinary Kan-extension of $Cat$-enriched functors and do get the right answer.

This kind of example made me start thinking that it might be indeed reasonable to just work strictly in the $\omega$-category enriched context.

Posted by: Urs Schreiber on April 9, 2008 7:25 PM | Permalink | Reply to this

### Re: Categorical Sheaves

My worry is that taking Kan extensions is sensitive to the representative spaces’ used within a homotopy type. If things are nice and fibrant/cofibrant’ or free or something, this worry disappears I seem to remember, but otherwise, ????

Posted by: Tim Porter on April 10, 2008 2:36 PM | Permalink | Reply to this

### Re: Categorical Sheaves

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.

Just to be clear, this relates to the first $n$-Café Millenium Prize: to compute the classifying space of charted 2$C$-bundles with

$2C: = Bim(Vect),$

doesn’t it?

No takers? Perhaps we should increase the prize money.

Posted by: David Corfield on April 10, 2008 11:39 AM | Permalink | Reply to this

### Re: Categorical Sheaves

Yes. Only that $Bim(Vect)$ will probably be too simple minded and we will eventually need to look at the classifying space of the bicategory of $X$-algebras and their bimodules, where $X$ is something like “$C^*$” or “von Neumann” or “algebras of local nets of free fermions on the circle” or some other flavor of algebras.

In fact, it should be a parameterized notion of algebra, where the parameter is an elliptic curve. For each elliptic curve there would be a corresponding kind of algebra (“of observables of the CFT on that curve”) and hence a corresponding classifying space for the corresponding elliptic cohomology.

I did talk to 2.5 topologists about this who eventually agreed that computing $|Bimod(something)|$ should be done. I haven’t heard back of further progress along this line for a while, though.

Meanwhile, I am wondering if we need to transfer the problem from $\infty$-categories to topological spaces.

What I’d maybe rather want to do is understand elliptic cohomology not in terms of spectra, but in term of $\infty$-categorical cohomology.

Posted by: Urs Schreiber on April 10, 2008 1:18 PM | Permalink | Reply to this

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