Journal Club -- Geometric Infinity-Function Theory

April 10, 2009

Journal Club – Geometric Infinity-Function Theory

Posted by Urs Schreiber

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We decided here that it might be fun to go together through the recent Ben-Zvi/Francis/Nadler work on

geometric $\infty$ -function theory

in a kind of online “journal club”. The idea would be to go slowly but surely, step-by-step through the material, discuss it, ask questions about it, understand it, and, crucially, distill the joint insight into entries on the $n$ Lab.

To start with, I have created in parallel to this blog entry here an $n$ Lab entry. The idea is that this blog entry here is the base for the discussion part of our “Journal club”, while that $n$ Lab entry is the base for the write-up part of the undertaking.

We want to sort out questions, strategy and other things that require discussion here on the blog, but want to be sure that all stable insights that we gain in the process will eventually be distilled into $n$ Lab entries, lest all the effort will leave no useful traces.

Therefore, in a combined manner similar to what I tentatively started doing for the book Categories and Sheaves and the book Higher Topos Theory and for a possibly similar joint seminar at Journal Club – $(\infty,1)$ -categories, the idea is that at the entry geometric $\infty$ -function theory we have a bit of background information, literature, summary, overviews, and then crucially a section-by-section list of links to $n$ Lab entries which concern themselves with discussing the relavent keywords.

All this will be very incomplete and preliminary in the beginning, but it may be fun to eventually and incrementrally fill in substance and thereby eventually create a useful web of linked entries that become a useful resource for reading, learning and thinking about this stuff.

Currently, there is some first bit of content at that entry, as much as I could come up with using a bit of virtual time that I didn’t really have. As with the rest of the $n$ Lab, the stuff that is there is not meant to imply to be complete or even necessarily good. The maxime is: incrementally improve.

If you see anything that makes you raise an eyebrow – or if your eyebrows raise because you don’t see something – then you are in the right state for contributing: either add a query box as described at HowTo and drop the rest of us a note on why you are unhappy and what you think should be improved, or – much better – hit the “edit” button on the bottom of the page and implement an improvement. But then, please, if it is anything nontrivial, drop the rest of us a brief note indicating what you did, either in the comment section below or at latest changes.

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Maybe recall that we had a bit of discussion on the first of the Ben-Zvi/Francis/Nadler articles at Ben-Zvi on geometric function theory.

There is also an $n$ Lab entry on geometric function theory.

The title of this “journal club” here is inspired from this, but you should know that the term geometric $\infty$ -function theory is one that I made up when I thought a bit about how to name this project here.

I am not sure how exactly we should start. Personally I’d find it helpful if we could try to disentangle general concepts here from concretely algebraic constructions. For instance the notions of “derived loop spaces” and the like make very general sense in every $(\infty,1)$ -categorical context, while the “collections of $\infty$ -functions” realized as derived quasi-coherent sheaves and D-moules make recourse to specific algebraic realizations. I feel like it would be good to separate conceptually what can be be separated, but then, maybe that’s not a helpful attitude here.

Apart from that, I should maybe point out that both the articles listed at the $n$ Lab entry start with pretty long summary sections, that go through the entire respective article in a heuristic manner. I can recommend strongly to read these summary sections first to get a rough idea for what is going on. The usual kind of question that will arise at various points is: “How on earth might that work in detail?” Such questions might be good starting points for discussion here, and then eventually for distilled answers over at the $n$ Lab.

Posted at April 10, 2009 11:35 AM UTC

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size issues and compact objects

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Since the claim is that

Compact objects are as necessary to this subject as air is to breathe.

it seems sensible to start with discussing these. It’s a bit of a pure technicality which may feel like just a nuisance. In fact, the concept of compact objects sits inside a web of closely related concepts revolving around size issues: accessible categories, presentable categories, categories of ind-objects, all that stuff.

This size issue business is maybe something we don’t want to waste the main part of our energy on, but it gives a good feeling to get this out of the way once, lest we’ll continuously be bother by it.

I don’t feel in the position to give a competent bird’s eye perspective on the entire problem complex yet, but in the process of learning this stuff myself I started filling some material into entries that now contain some crucial ideas and some crucial definitions.

Roughly in logical order, check out

cardinal number

compact object in an $(\infty,1)$ -category

accessible $(\infinity,1)$ -category

presentable $(\infinity,1)$ -category

Still lots of room to improve and expand on these entries. But maybe a start. In any case, I need a break now.

Posted by: Urs Schreiber on April 10, 2009 4:47 PM | Permalink | Reply to this

Block’s version of coherent sheaves

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I feel like it would be good to separate conceptually what can be be separated, but then, maybe that’s not a helpful attitude here.

Likely that what I really would like to think about is a differential-geometric formulation of the $\infty$ -category of coherent sheaves on some smooth $\infty$ -stack along the lines of Jonathan Block’s $L_\infty$ -algebraic description of coherent sheaves.

Posted by: Urs Schreiber on April 10, 2009 6:12 PM | Permalink | Reply to this

Voice in the dark

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Ok. Hands up who is in. It’s Easter weekend, so Urs hopefully gives us a bit of grace.

Posted by: Bruce Bartlett on April 10, 2009 10:32 PM | Permalink | Reply to this

Re: Journal Club – Geometric Infinity-Function Theory

Hi Urs, Bruce,

I want in, but need to go to sleep right now so I can get on a plane in the morning. I haven’t gotten over to the n-Lab yet to see what is written but I will do so next time I am have internet access.

Posted by: Alex Hoffnung on April 11, 2009 6:57 AM | Permalink | Reply to this

Re: Journal Club – Geometric Infinity-Function Theory

Alex, good to have your company. Been a bit under the weather lately, hence the lame response so far from my side.

Posted by: Bruce Bartlett on April 14, 2009 5:12 PM | Permalink | Reply to this

Re: Journal Club – Geometric Infinity-Function Theory

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Alex, Bruce,

thanks for joining in!

I am not sure what you two would consider would be a good start into this project.

I was thinking it would be good to start reading the articles the way one reads a tale to a 5 year old: while after every second sentence there’ll be the question “What does this and that mean?” a rough answer will already be accepted as satisfactory for the first go. Which doesn’t mean that it won’t be asked again in the next run.

So maybe we should just start this way with section 1.1, page 3 “perfect stacks”.

I suggest to read this in stages of increasing ambition. To start with, let’s ignore the “derived” in “derived stack” and see what’s going on.

Also, let’s get rid of the “affine” there, for the time being, just for the sake of it.

So then we are dealing with stacks on stacks on some site $S$ . (The objects of $S$ are called the “affine” spaces, so “affine” here reads: “representable”, these are the test spaces with which we probe our generalized spaces.)

Section 1.1 tells us about a characterization of stacks on stacks in terms of their restriction to representable stacks.

Are you happy with the notion of ind-objects appearing there? The $n$ Lab has some answers. You should maybe ask questions to the extent that you find the answers there lacking. (I should, too.)

No rush. But starting with some questions like “What is going on here?” might be helpful to get us started.

Posted by: Urs Schreiber on April 14, 2009 6:11 PM | Permalink | Reply to this

Re: Journal Club – Geometric Infinity-Function Theory

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Concerning the notion of perfect stacks:

I have expanded the discussion at

$n$ Lab: ind-object

and added remarks about motivation and background and added examples.

I also created

$n$ Lab: ind-object in an $(\infty,1)$ -category

but that so far contains little more than the bare definition.

Posted by: Urs Schreiber on April 15, 2009 9:47 AM | Permalink | Reply to this

Re: Journal Club – Geometric Infinity-Function Theory

Sorry for the delay, I have just returned to the States from Glasgow, but am not quite home yet. So let me try to slow down the conversation a bit by sticking with just the very beginning.

Are you happy with the notion of ind-objects appearing there?

Yes, I am happy with the ind-objects in a category and for ind-objects in an (\infinity -1)-category, the generalization sounds plausible. Although equivalent, I do not think that I understand the first definition on nLab given in terms of classifying right fibrations.

To start with, let’s ignore the “derived” in “derived stack” and see what’s going on.

This sounds like a good idea.

So then we are dealing with stacks on stacks on some site S. (The objects of S are called the “affine” spaces, so “affine” here reads: “representable”, these are the test spaces with which we probe our generalized spaces.)

So stacks on stacks scares me a bit. I have some idea about stacks and I like the idea of test spaces probing generalized spaces, but reading

Section 1.1 tells us about a characterization of stacks on stacks in terms of their restriction to representable stacks.

does not immediately make me feel warm and fuzzy. Maybe someone can say more about why this is what I should have learned from reading this section.

By the way, are we reading just section 1.1 here and then proceeding to 1.2, or are we also trying to understand chapter 3 on perfect stacks right now?

Posted by: Alex Hoffnung on April 20, 2009 6:02 PM | Permalink | Reply to this

Re: Journal Club – Geometric Infinity-Function Theory

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I have to admit, “stacks on stacks” sounds scary to me too.. I would read it as “sheaves (or bundles) on spaces” — in other words, we have a base space, which happens to be a stack, and over it we consider a bundle/sheaf of some kind. (it helps me psychologically to separate the terminology “sheaf” and “stack” this way even though they might end up meaning the same thing in the end)

One can also think though of a bundle or sheaf as being “of geometric origin” - namely we might consider fibrations/spaces over our base space, and then linearize them somehow (by passing to cohomology or functions etc) to get a sheaf on the base.

Finally one more categorified kind of “stack over a stack” is a sheaf of categories, which to each open assigns the category of sheaves of some kind.. though not much on this level of categoricity is used in these papers..

Posted by: David Ben-Zvi on April 20, 2009 11:07 PM | Permalink | Reply to this

Re: Journal Club – Geometric Infinity-Function Theory

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stacks on stacks

That's the sort of thing that one might hope would stabilise. For example, sheaves on sheaves are simply sheaves, in the sense that the (a priori extra-large pre-) topos of sheaves on a given Grothendieck topos $E$ (equipped with the canonical coverage) is in fact equivalent to the (large Grothendieck) topos $E$ itself.

Posted by: Toby Bartels on April 21, 2009 12:10 AM | Permalink | Reply to this

Re: Journal Club – Geometric Infinity-Function Theory

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Sorry for the delay, I have just returned to the States from Glasgow, but am not quite home yet. So let me try to slow down the conversation a bit by sticking with just the very beginning.

Sure, very good, let’s go back to the beginning. Glad to hear from you!

Although equivalent, I do not think that I understand the first definition on nLab given in terms of classifying right fibrations.

I had included what was supposed to be a motivating discussion of this definiton at ind-object, but at one point David Roberts had a complaint and maybe I am mixed up.

So I think the deal is this, but hopefully I’ll be corrected if this is incorrect now:

it so happens that one equivalent characterization for a presheaf

$F : C^{op} \to Set$

to be a filtered colimit of representables is that the comma category $(Y,const_F)$ is filtered and cofinally small. ( $Y$ the Yoneda embedding. See $n$ Lab for links to all terms appearing.)

Now, unless I am mixed up, this comma category is the fibration classified by $F$ , in that this here is a strict pullback of categories

$\array{ (Y,const_F)^{op} &\to& Set_* \\ \downarrow && \downarrow^{U} \\ C^{op} &\to& Set } \,.$

Here $Set_*$ is the category of pointed sets, and $U$ the obvous forgetful functor (the universal $Set$ -bundle).

Namely: this pullback category has over $c \in C$ as objects all those pointed sets, whose underlying set is $F(c)$ . That’s the same as all the elements of the set $F(c)$ . Which is indeed, by Yoneda, the same as the objects of $(Y, const_F)$ over $c$ , namely the presheaf morphisms $Y(c) \to F$ .

Moreover, over a morphism $(c \stackrel{f}{\to} d)^{op}$ in $C^{op}$ the pullback category has those morphisms $((h \in F(c)) \to (g \in F(d))^{op}$ for which $h = g \circ F(f)$ . This is also the same as in $(Y,const_F)^{op}$ .

So a presheaf is an ind-object precisely if the fibration it classifies is filtered and cofinally small.

That first definition of ind-object in an $(\infty,1)$ -category takes this as the primary perspective, I guess.

Posted by: Urs Schreiber on April 20, 2009 9:09 PM | Permalink | Reply to this

Re: Journal Club – Geometric Infinity-Function Theory

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So stacks on stacks scares me a bit.

To unscare yourself, just read “stacks on generalized spaces”. That sounds already rather boring.

That just to raise the comfort level, which doesn’t mean that there is nothing left to be understood.

So, I can’t pretend to be able on the spot to give a comprehensive account, I hope it is clear that I am learning this stuff, too. It’s probably all too clear.

The full answer to what $QC(X)$ is like is supposed to be in the work of Tön-Vezzosi on derived stacks. To get us started, I have now collected at least the relevant literature here:

$n$ Lab: derived stack

So let’s see.

Quasi-coherent modules themselves form an $\infty$ -stack on a site of algebras, this is section 1.3.7, of Homotopical Algebraic Geometry II.

But now quasi-coherent sheaves are considered on things which are themselves generalized spaces.

By the way, are we reading just section 1.1 here and then proceeding to 1.2, or are we also trying to understand chapter 3 on perfect stacks right now?

I am not sure, let’s just keep asking questions and looking for answers and we’ll eventually get somewhere. Because oficially I need to be doing other things, so I won’t find the time to to any major organization tasks here.

Posted by: Urs Schreiber on April 20, 2009 10:20 PM | Permalink | Reply to this

flat differential

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David Corfield kindly created

$n$ Lab: D-module

for us, which we’ll need extensively for the second article.

I have a question to David Ben-Zvi at this point:

the idea is that D-modules are to O-modules as flat vector bundles are to vector bundles.

What is the most precise statement there is along these lines?

But my real question is this:

when I think of differential nonabelian cohomology, classifying $\infty$ -bundles with connection, it works in essence as follows:

begin general nonsense excursion

one considers some $(\infty,1)$ -topos $H$ which comes with a suitable functor $\Pi : H \to H$ and transformation $Id_H \to \Pi$ such that for any object $X$ , the object $\Pi(X)$ behaves like the fundamental $\infty$ -groupoid of $X$ .

Then for any other object $A$ , $A$ -cohomology on $X$ is $H(X,A)$ , while flat differential $A$ -cohomology is $H(\Pi(X),A)$ .

Indeed, when $H$ is $\infty$ -sheaves on $Diff$ and $\Pi$ is induced from the smooth singular simplicial complex functor, one finds that $H(\Pi(X),A)$ classifies $A$ -bundles with flat connection.

And all of (nonabelian) differential cohomology is induced from this: flat $A$ -valued differential forms in $H$ can be realized as the relative cohomology $H_{dR}(X,A) = H_{rel}(\array{X \\ \downarrow \\ \Pi(X)}, \array{* \\ \downarrow \\ \mathbf{B} A})$ , there is a canonical morphism $H(X,A) \to H_{dR}(X, \mathbf{B}A)$ which computes the characteristic forms of an $A$ -bundle and so for any fixed element $F \in H_{dR}(X, \mathbf{B}A)$ the non-flat differential $A$ -cohomology with characteristic form $F$ is the homotopy pullback

$\array{ \bar H^{[F]}(X,A) &\to& * \\ \downarrow && \downarrow^{F} \\ H(X,A) &\to& H_{dR}(X,\mathbf{B}A) } \,.$

Anyway, the upshot is that everything is reduced to an extension problem for the problem of extending a cocycle on $X$ to a cocycle on $\Pi(X)$ .

end general nonsense excursion

Now, it’s clear that this ought to translate somehow into lifts of $O$ -modules to D-modules, by essentially replacing in the above word by word elements in $H(X,A)$ with O-modules and elements of $H(\Pi(X),A)$ with D-modules, if I understand right.

If I could get a handle on how to make this statement not just plausible but precise, I’d be quite happy.

For instance, to get started: do people describe

- Deligne cohomology/Cheeger-Simons characters in terms of D-moudles?

If so, what would be a good reference for that?

Posted by: Urs Schreiber on April 17, 2009 5:30 PM | Permalink | Reply to this

Re: flat differential

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To answer your first question: there is a natural notion of flat connection on an O-module, and O-modules with flat connection are precisely D-modules. If the O-module is coherent, it automatically follows (from having a flat connection) that it is in fact a vector bundle, so you have an equivalence {D-modules, coherent as O-modules} = {flat vector bundles}.

To define a flat connection on an O-module there are many alternative ways: you can write it as an operator $\nabla:V\to V\otimes \Omega^1$ satisfying Leibniz and $\nabla^2=0$ , you can write it as an action of vector fields respecting commutators (hence extending to an action of D), etc.

However the way I bet you’ll like most is as follows: a D-module on X (smooth) is an O-module on dR(X), the de Rham space of X – this is an analogue of your functor $\Pi$ I believe. The idea is to rewrite a D-module structure as descent data from X to dR(X) (more precisely, a D-module is a comodule for the dual coalgebra of jets, which is the same as a comodule for a natural comonad on O-modules – a fancy way to say D is the groupoid algebra for the groupoid associated to the equivalence relation “x and y are infinitesimally close”).

The de Rham space (of a scheme - the same ideas work for stacks) is another functor from rings to sets. It is the quotient of X by the equivalence relation (subset of $X\times X$ ) given by the formal neighborhood of the diagonal – this is a nice thing one can do in algebraic geometry that is less natural in differential topology: one has a “canonical neighborhood of the diagonal which retracts to it”.

Another way to say this is that dR is the right adjoint to the functor assigning to any space U:Rings-> Sets the associated reduced space. (if you like this kind of POV the reference is Simpson-Teleman, deRham theorem for stacks, available at http://math.berkeley.edu/~teleman/lectures.html )

Posted by: David Ben-Zvi on April 17, 2009 9:00 PM | Permalink | Reply to this

deRham infinity-stack

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David Ben-Zvi kindly made the following remark:

However the way I bet you’ll like most is as follows: a $D$ -module on $X$ (smooth) is an $O$ -module on $dR(X)$ , the de Rham space of $X$ – this is an analogue of your functor $\Pi$ I believe.

[…]

if you like this kind of POV the reference is Simpson-Teleman, deRham theorem for stacks, available at http://math.berkeley.edu/~teleman/lectures.html

Thanks, David! I hate myself for having taken so long to get back to this, even after you pointed it out again elsewhere, but, yes, this is very close to what I was talking about.

I have started an entry on this:

$n$ Lab: deRham space

That also has provides links to those $\Pi$ -constructions that I mentioned.

It seems to be obvious how both constructions should be related, but I need to think a bit more about the technicalities:

I am thinking of a functor $\Pi(-)$ (path $infty$ -groupoid) that is a Quillen functor on the category of $\infty$ -stacks and which has an infinitesimal version $\Pi^{inf}(-)$ (infinitesimal path $\infty$ -groupoid), which is stil a Quillen functor.

In the Simpson-Teleman article instead the derived functor $\mathbb{R} dR(-)$ of the deRham-space functor on the homotopy category of $\infty$ -stacks is considered. Probably that ought to be the image of $\Pi^{inf}(-)$ under passage to homotopy categories – certainly the results they prove about it support this – but I am not sure yet how to see this in detail.

Posted by: Urs Schreiber on September 16, 2009 2:56 PM | Permalink | Reply to this

Re: flat differential

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Yet another picture, closest I think to your Deligne cohomology question, is to identify D-modules with $\Omega$ -modules - a form of Koszul duality (with all the usual drawbacks of Koszul duality when it comes to “large” objects). Here $\Omega$ is the dga given by the de Rham complex (considered as a complex of O-modules, with morphisms being differential operators). Now $\Omega$ is a resolution of the constant sheaf $C$ , but also has a canonical filtration by truncations (the Hodge filtration). Deligne cohomology is the hypercohomology of these truncated complexes. Is this the kind of relation you want?

(It seems to me that differential cohomology has to do with connections more naturally than flat connections, so not a priori with D-modules, but with the Hodge filtration on $\Omega$ ..?)

Posted by: David Ben-Zvi on April 17, 2009 9:01 PM | Permalink | Reply to this

Re: flat differential

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Thanks for your detailed replies! Let me start by commenting on this remark here:

(It seems to me that differential cohomology has to do with connections more naturally than flat connections, so not a priori with D-modules, but with the Hodge filtration on $\Omega$ ..?)

I am claiming that differential cohomology has to do with the obstruction classes to lifting cocycles to flat differential cocycles.

Let me illustrate this starting from the situation of Deligne cohomology that you mention:

Let $X \mapsto \Pi(X)$ be the fundamental smooth $\infty$ -groupod-assignment as indicated above, and write $\Pi_k(-)$ or equivalently $P_k(-)$ for its $k$ -coskeleton.

Then for every $n \in \mathbb{N}$ the 1-categorical inner hom $[\Pi_k(-), \mathbf{B}^n U(1)]$

is an $n$ -groupoid valued sheaf, which, under Dold-Kan, is a complex of sheaves of abelian groups. Its $\infty$ -stackification gives the differential cohomologies $H(\Pi_k(-), \mathbf{B}^n U(1))$ . One finds

- $H(\Pi_0(-),\mathbf{B}^n U(1))$ – circle $(n-1)$ -gerbes

- $H(\Pi_1(-),\mathbf{B}^n U(1))$ – circle $(n-1)$ -gerbes with “connective structure”

- $H(\Pi_2(-),\mathbf{B}^n U(1))$ – circle $(n-1)$ -gerbes with “connective structure and curving”

- $H(\Pi_3(-),\mathbf{B}^n U(1))$ – circle $(n-1)$ -gerbes with “connective structure and curving and higher curving”

- …

- $H(\Pi_n(-),\mathbf{B}^n U(1))$ – circle $(n-1)$ -gerbes with connection

- $H(\Pi_{n+1}(-),\mathbf{B}^n U(1))$ – circle $(n-1)$ -gerbes with flat connection

and then it stabilizes:

- $H(\Pi_\infty(-),\mathbf{B}^n U(1))$ – circle $(n-1)$ -gerbes with flat connection.

It’s precisely this co-skeleton truncation $\Pi_k$ which corresponds to the Hodge-filtration truncation seen in Deligne cohomology.

More precisely: the Deligne sheaf complex is precisely

$\mathbb{Z}(n+1)_D^\infty \simeq [\Pi_n(-),\mathbf{B}^n U(1)] \,.$

(This is also at $n$ Lab: Deligne cohomology.)

So, the truncation of $\Pi(X)$ to $\Pi_k$ frees the curvature from vanishing. But it turns out to be an illusion that this is the general mechanism by which we get non-flat differential cohomology.

Namely, this truncation strategy fails as soon as the coefficient object $A$ in no longer concentrated in a single degree, as $\mathbf{B}^n U(1)$ is.

For instance, for $String(n)$ a smooth model of the String-2-group, one finds that

$H(\Pi_\infty(X), \mathbf{B}String(n))$

is flat String 2-bundles with connection, but that

$H(\Pi_2(X), \mathbf{B}String(n))$

does not classify general non-flat String 2-bundles with connection. Rather, this classifies those String-2-bundles wih connection whose underlying $Spin(n)$ -1-bundle with connection is flat. (And these then necessarily have trivial underlying String-2-bundles.)

Following a term invented by Breen and Messing we call this phenomenon “fake-flatness”.

So thinking of non-flat diferential cohomology as cohomology on co-skeletal truncations of $\Pi(X)$ is wrong as soon as one moves away from the simplest abelian examples.

So in general one has to phrase differential cohomology differently. I am claiming that the right way to conceive of non-flat differential cocycles is as cocycles in twisted flat differential cohomology, where the twist is the curvature.

This may sound funny, but the point is that there is a general notion of twisted (nonabelian) cohomology and nonflat differential cohomology is precisely that concept applied to the obstruction problem of extending cocycles through $X \hookrightarrow \Pi(X)$ .

In words this is a nice tautology, namely it says that “curvature is the obstruction to flatness”, but with all three terms here given their precise $\infty$ -categorical meaning, its a contentful statement.

This is for instance the technology which we use to describe connections on twisted String-2- and twisted Fivebrane-6-bundles and identify them with the relevant higher gauge fields in heterotic supergravity

So, you see, that’s why I started wondering:

do people consider $\infty$ -sheafy obstruction classes to extendind O-modules to D-modules and realize non-flat differential cocoycle in terms of these obstruction classes?

You argued that

- since D-modules are equivalently $\Omega$ -modules (but that means $\Omega$ -modules in the sense of Jonathan Block’s superconnections, right?)

- and since the truncation of $\Omega$ regarded as a complex of sheaves yields Deligne cohomology

there is in a way a description of Deligne cohomology in terms of D-modules. But I am not sure if I see that there is more to this relation than that differential forms play a role in both cases. For the Deligne complex it is not relevant that $\Omega$ acts on anything, no?

I think using the fact that Jonathan Block’s “superconnections” are the same as what I call representations of the tangent Lie algebroid $T X$ , which is nothing buth the Lie algebroid of $\Pi(X)$ , I can see the relation that I am looking for, but I need to learn or else check with experts to which degree this coincides with what people do in terms of D-modules.

So this is one central question which I am hoping this “journal club” (hm) can eventually help to answer:

to which extent can I take your work on $\infty$ -function theory and replace $\infty$ -categories of quasi-coherent sheaves and/or of D-modules with dg-categories of reps of the tangent Lie algebroid (in the smooth setup).

I keep being a bit confused here, as to whether the derived categories that Block considers should be thought of as modeling quasi-coherent sheaves or D-modules or smothing in between.

Posted by: Urs Schreiber on April 18, 2009 4:13 PM | Permalink | Reply to this

Re: flat differential

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D-modules, $\Omega$ -modules and representations of the tangent Lie algebroid are different names for the same thing.. More precisely the first and third always agree and the middle one is related by Koszul duality, so agrees for “small” (eg coherent) modules. This holds on smooth schemes and stacks (on singular spaces one has to be careful what one means by each of the three). So certainly in a differential topological context one can work with the natural analogs of any of them. The important thing to note though is it’s NOT true in general that they agree with representations of the fundamental $\infty$ -groupoid - they have infinitesimal parallel transport, but it might not exponentiate. The basic example to think about is the module of delta-functions at a point..

As for the relation with Block’s story: by $\Omega$ I mean the de Rham complex, while in his story one is considering a Dolbeault complex. In other words $\Omega$ -modules, with action of de Rham d, correspond to flat connections/actions of vector fields. In his setting he’s looking at dg actions of del-bar, or equivalently of the “Cauchy-Riemann” algebroid, so that flatness along this algebroid is equivalent to holomorphy: dg modules for del-bar are precisely holomorphic vector bundles (dg category thereof). Of course one can consider any Lie algebroid instead of the tangent or Cauchy-Riemann one and get some notion of sheaf.. So to answer your question: Block’s are a version of O-modules, not D-modules.

But in any case the O-module story (in whatever language, Block’s or otherwise) is a story about complex manifolds or algebraic varieties. I’m not sure what you want to substitute for it for just smooth manifolds - the only algebroid you have there in general is the tangent one, so you could consider some version of D-modules, but I don’t know that there’s an interesting category of sheaves to put flat connections on in that setting.

As to Deligne cohomology: the discussion we were having was about constant coefficient cohomology - ie differential cohomology with coefficients in C. So it only immediately relates to the trivial vector bundle as D-module. But you could try to tell the same story with coefficients in any D-module, or better yet with coefficients in any Hodge module (“D-module with Hodge structure”)..

Posted by: David Ben-Zvi on April 18, 2009 4:46 PM | Permalink | Reply to this

smooth or not

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But in any case the O-module story (in whatever language, Block’s or otherwise) is a story about complex manifolds or algebraic varieties. I’m not sure what you want to substitute for it for just smooth manifolds - the only algebroid you have there in general is the tangent one, so you could consider some version of D-modules, but I don’t know that there’s an interesting category of sheaves to put flat connections on in that setting.

Okay, thanks. Sorry for being dense and stubborn on this point. I need to better understand this.

So what would happen with your latest work on “character TFT” if we put it in the smooth context, Lie group $G$ , etc. everything $\infty$ -stacks on $Diff$ etc. The whole thing becomes empty and unintersting then? Why, what fails?

Posted by: Urs Schreiber on April 18, 2009 5:01 PM | Permalink | Reply to this

Re: smooth or not

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I certainly believe there could be interesting function theories on smooth manifolds. for example we could look at representations of the fundamental higher-groupoid (ie the correctly defined “derived category of local systems”) - this really depends only on the underlying homotopy type, not on a manifold.. but already that’s pretty interesting. For example if we look at local systems with sphere coefficients this gives an $\infty$ -category whose K-theory is the Waldhausen K-theory of the underlying topological space, which is a very rich invariant. If we replace sphere coefficients with K-theory coefficients we get something (in the case of a gerbe over a Lie group) closely related to the Freed-Hopkins-Teleman stuff I think.

(It’s interesting to note that if you think of a homotopy type as a constant derived stack, eg over the category of commutative rings, then these locally constant sheaves are exactly the same as [quasi]coherent sheaves on the stack - so there is some direct relation between the two worlds!)

What would be more interesting though for a smooth manifold is to have a richer function theory, not just depending on the homotopy type.. a natural candidate to think about is that of constructible sheaves, which in the real analytic situation is the same as the Fukaya category of the cotangent bundle by a theorem of Nadler, and is also closely related to D-modules. Again one could look at constructible sheaves with various coefficients, like C or sphere or K etc.. I just don’t know anything about how function theory works in this setting.

Posted by: David Ben-Zvi on April 18, 2009 8:44 PM | Permalink | Reply to this

smooth

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I certainly believe there could be interesting function theories on smooth manifolds. for example we could look at representations of the fundamental higher-groupoid (ie the correctly defined “derived category of local systems”) - this really depends only on the underlying homotopy type, not on a manifold.. but already that’s pretty interesting.

Sounds very good!

Do you mind chatting a bit about that correctly defined “derived category of local systems”?

First the fundamental $\infty$ -groupoid itself:

to start with, it should be sufficient to define it on test spaces, i.e.

$\Pi : Diff \to [Diff^{op},\infty Grpd]$

and then Kan-extend it through the inclusion $Diff \hookrightarrow [Diff^{op},\infty Grpd]$ .

I started thinking about this in the context of modellling $\infty Grpd$ by strict globular $\infty$ -groupoids. In that case there is an iterative construction which seems to work nicely:

define $P_1(U)$ to be the smooth 1-groupoid whose sheaf of morphisms is the concrete sheaf of the space of smooth paths in $U$ with siting instant at the ends, as in our articles with John and Konrad. Extend that from Diff to $Sh(Diff)$ .

Then let $P_2$ be similarly the 2-groupoid with sheaf of 2-morphisms the obvious subspace of $P_1(Mor_1(P_1(U)))$ and iterate this way to define $\Pi(U) = lim_n P_n(U)$ .

Alternatively, in the context of simplicial models it should be sufficient to start with the simplicial presheaf

$\Pi(U) : V \mapsto Diff(V \times \Delta^\bullet, U)$

with $\Delta^n \subset \mathbb{R}^n$ the standard $n$ -simplex regarded as a smooth manifold.

The objectwise Kan-fibrant replacement of this would be the smooth fundamental $\infty$ -groupoid proper.

Now, what would you imagine representing this on in order to get something like a derived category of local systems? It should probably be some flavor of “ $Ch(Vect)-Mod$ ”.

this really depends only on the underlying homotopy type, not on a manifold.

Yes. By the way, I have seen surprisingly little discussion of this. But maybe I was just looking in the wrong places. One discussion of $\Pi(X)$ in this context is in

Carlos Simpson, A closed model structure for $n$ -categories, internal $Hom$ , $n$ -stacks and generalized Seifert-Van Kampen

Notably, he addresses (p. 2)

$Hom(\Pi(X), A)$

as “nonabelian cohomology”, which I would think should be called “flat differential nonabelian cohomology”. But probably he is precisely being led by the desire to have “cohomology = homotopy invariant”.

(It’s interesting to note that if you think of a homotopy type as a constant derived stack, eg over the category of commutative rings, then these locally constant sheaves are exactly the same as [quasi]coherent sheaves on the stack - so there is some direct relation between the two worlds!)

This I didn’t follow, I am afraid. Would you mind saying this again with slightly more detail? Sorry.

What would be more interesting though for a smooth manifold is to have a richer function theory, not just depending on the homotopy type.

So, the non-flat case.

Concerning the K-theoretic applications you mention:

I was thinking that one should look at the model for K-theory in terms of sheaves of combinatorial spectra as discussed in old article by Kenneth Brown. Then as before evaluating on $\Pi(X)$ should give flat differential K-theory and general differential K-theory should be the corresponding twisted cohomology. But I have yet to think about all of this in any detail.

Posted by: Urs Schreiber on April 20, 2009 9:09 AM | Permalink | Reply to this

Re: smooth

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I think the definition of derived category of local systems / representations of the fundamental $\infty$ -groupoid is easier than that (or it’s my lack of familiarity with the language). Here are a couple of versions:

simplest, given a space $X$ , represented as a simplicial set, one can take a limit (in the $\infty$ -categorical sense) of the $\infty$ -categories of sheaves on the simplices. i.e. if you want to consider sheaves of vector spaces, you put a copy of chain complexes on each 0 simplex and take a limit (or if you’d like this is the totalization of a cosimplicial category). You can replace chain complexes by spectra for example and get the $\infty$ -category of sheaves of spectra.. these are also called twisted cohomology theories/coefficient systems and are studied by May and Sigurdsson I think (I learned all this stuff from my new colleague Andrew Blumberg).
Another way that homotopy theorists look at this is as follows: for $X$ connected we can look at modules for the based loops in $X$ (as an $A_\infty$ -space), with whatever coefficients you like.
Finally, there’s the (somewhat silly) comment I was making about thinking of local systems as coherent sheaves. The point is that when you look at higher stacks on any site you always have a supply of constant stacks (technically the stackification of a constant prestack) with value your favorite space. So you can think of any space as a higher stack on say the category of rings with some standard topology. They’re very far from being algebraic stacks in general, but that’s not so important IMHO. In any case you can apply the general definition of coherent or quasicoherent sheaf on a stack to a space this way.. and what you get is exactly the first definition I gave above of the derived category of local systems.

(one important point: this derived category is NOT the derived category of the abelian category of local systems in general - it feels all the higher homotopy groups of a space, while the latter is only about $\pi_{\leq 1}$ .)

Posted by: David Ben-Zvi on April 20, 2009 11:26 PM | Permalink | Reply to this

reps of the fundamental infinity-groupoid

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David B.-Z., thanks for all this information! I have some questions:

these are also called twisted cohomology theories/coefficient systems and are studied by May and Sigurdsson I think

Thanks for the pointer. I started browsing the book

J. May, J. Sigurdsson Parameterized Homotopy Theory

I had not been aware of their formalization of “twisted cohomology”. It is a pity that “twisted” is such an unspecific term. But since theirs is related to twisted K-theory, it must also be related to our notion.

I need more time to look at this book. Right now I don’t seem to find the construction you were indicating:

one can take a limit (in the $\infty$ -categorical sense) of the $\infty$ -categories of sheaves on the simplices. i.e. if you want to consider sheaves of vector spaces, you put a copy of chain complexes on each 0 simplex and take a limit (or if you’d like this is the totalization of a cosimplicial category)

This begins to sound a lot like an $\infty$ -functor on what I called $\Pi(X)$ to some sort of chain-complexy $\infty$ -cat. What do you mean to put on the 1-, 2-, 3- simplices?

If you have a reference with the deatils handy I’d appreciate a quick pointer, but I can try to track it down myself, of course.

In any case you can apply the general definition of coherent or quasicoherent sheaf on a stack to a space this way. and what you get is exactly the first definition I gave above of the derived category of local systems.

Thanks, that sounds like a very useful connection, indeed. I’ll think about this. Can one go the other way round and define quasi-coherent sheaves on stacks in terms of the first construction?

Posted by: Urs Schreiber on April 21, 2009 1:56 PM | Permalink | Reply to this

Re: reps of the fundamental infinity-groupoid

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Ah, the full text is of course available on Peter May’s website:

J. P. May, J. Sigurdsson, Parametereized homotopy theory (pdf)

reading…

Posted by: Urs Schreiber on April 21, 2009 2:27 PM | Permalink | Reply to this

Re: reps of the fundamental infinity-groupoid

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I just had a very brief exchange with David BZ on this, before he had to run. Since here it’s already late, I’ll be asleep when he comes back online. So I’ll leave this message here.

In that brief exchange David clarified that what was meant in the above comment was the computation of the descent $\infty$ -category of $QC$ ( $\infty$ -sheaf of quasi-coherent sheaves) along a map

$S_0 \hookrightarrow S \,,$

where $S$ is some $\infty$ -stack and $S_0$ its objectwise 0-cotruncation (i.e. inclusion of the presheaf of objects into $\infty$ -groupoid valued presheaf).

Now, here is the message I’d like to leave here, in order to continue discussing this tomorrow:

Namely I am getting the impression that for $S = \Pi(X)$ the fundamental $\infty$ -groupoid of a space $X$ , we are just talking about maps

$Hom(\Pi(X), QC)$

from the $\infty$ -stack $\Pi(X)$ into the $\infty$ -stack $QC$ after all, just expressed in components.

Namely, first of all, we build the simplicial object corresponding to the morphism $X \hookrightarrow \Pi(X)$ . It starts with the homotopy pullback

$\array{ Mor_1(\Pi(X)) &\to& X \\ \downarrow && \downarrow \\ X &\to& \Pi(X) }$

which is just the 1-morphisms $Mor_1(\Pi(X)) = [\Delta^1,X]$ in $\Pi(X)$ , and continues accordingly, so should just be the $SSet$ -valued presheaf $\Pi(X)$ regarded as a simplicial object

$\cdots \to [\Delta^2,X] \to [\Delta^1,X] \to [X] \,.$

I guess we have

$\Pi(X) = hocolim_n ([\Delta^n,X])$

and hence

$\begin{aligned} Hom(\Pi(X), QC) & \simeq Hom( hocolim_n [\Delta^n, X], QC) \\ & \simeq holim_n Hom([\Delta^n, X], QC) \\ & \simeq holim_n QC([\Delta^n,X]) \end{aligned} \,,$

where the starting point is what I was thinking of and the endpoint apparently/maybe what David BZ has in mind.

If that’s what David means, then it’s likely that we have been talking about the same thing in different words.

Maybe just for illustration purposes to reduce all this $\infty$ -mumbo-jumbo to something familiar:

If $\Pi_1(X)$ is the ordinary smooth path 1-groupoid of an ordinary manifold $X$ , regarded as a groupoid valued sheaf, and if $Vect$ (playing the role of QC) denotes the smooth category of vector spaces, regarded as the usual groupoid valued sheaf, i.e. the stack of vector bundles on Diff, say, then

$Hom(\Pi_1(X), Vect)$

is indeed the same as vector bundles with (flat) connection on $X$ (each functor is the parallel transport of one such vector bundle), which is equivalently “path-groupoid equivariant”-vector bundles on $X$ , namely the descent data consisting of

- a vector bundle $E$ on $X$

- a morphism of vector bundles on $Mor_1(\Pi(X)) = paths in X$ between the two pullbacks $d_0^* E \to d_1^* E$

- and the obvious equality on the space of composable paths.

The above $\infty$ -mumbo-jumbo would be exactly the same idea, just carried further.

Posted by: Urs Schreiber on April 21, 2009 10:14 PM | Permalink | Reply to this

Re: reps of the fundamental infinity-groupoid

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All of these formulations are different ways of saying “locally constant sheaf” in a derived way, and so we expect them to be all equivalent. Maybe the simplest thing to say is this: the $\infty$ -category of local systems with coefficients in chain complexes (or spectra, or whatever we’d like) is the unique assignment from spaces to $\infty$ -categories which satisfies descent and assigns the $\infty$ -category of chain complexes (or spectra or….) to a contractible space.

The definition of quasicoherent sheaves in general is the same - it’s the unique assignment from schemes (or stacks or functors on rings) to $\infty$ -categories which satisfies descent and assigns to a ring the category of its modules. The reason quasicoherent sheaves are more interesting than local systems is that affine varieties are more interesting than points (and modules over rings are more interesting than just vector spaces).

(More formally we can say that $QC(X)$ is the limit over all affines $Spec(R)$ mapping to $X$ of the $\infty$ -category of $R$ -modules.)

Posted by: David Ben-Zvi on April 22, 2009 1:48 AM | Permalink | Reply to this

Re: reps of the fundamental infinity-groupoid

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David,

yes, I agree, it’s all very simple when one comes down to it.

Still, it seems you found my original comment mysterious while I found yours mysterious until now, when it turns out that we are actually talking about the same thing.

So I thought maybe we could proceed as follows:

I have begun writing a reasonably detailed exposition of some of the issues of relevance here, into our Journal-Club page, under the section headline

Preliminaries / Basics on $\infty$ -stacks .

This is written in a way that is supposedly useful for Bruce, Alex and whoever else is lurking. I’d like to

- ask those co-journalists to have a look and tell me which bits are unclear;

- ask you to have a look and tell me to which extent you can agree with this.

Have to run, a bit in a haste…

Posted by: Urs Schreiber on April 22, 2009 9:46 AM | Permalink | Reply to this

stack hom-spaces in terms of homotopy limits

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I think Alex Hoffnung will come online soon with the first proper Journal Club report (on section 1 of “Integral Transforms”).

Meanwhile, since my declared aim is to distill all discussion we have here into useful, stable, coherent information on the $n$ Lab, as far as possible and feasible, I have added some discussion and explanation, inclusing concrete formulas and potentially helpful links, ect, of the discussion I was having with David Ben-Zvi above to the $n$ Lab.

In addition to the already mentioned bit at the Journal Club page itself, there is now also a description of hom-objects between stacks in terms of (homotopy)limits for the concrete model of $\infty$ -stacks as simplicial presheaves at

$n$ Lab: simplicial presheaves

See the section “properties”.

This is supposed to lead over to the discussion at

$n$ Lab: descent for simplicial presheaves

but there I have for the moment just the rough idea scetched and pointers to the literature.

As always with these entries I have two requests:

- to non-experts: please check how useful you find this and let me know which bits are not understandable;

- to experts: check that I am not mixed up and feel encouraged to add/improve material

Posted by: Urs Schreiber on April 25, 2009 9:35 PM | Permalink | Reply to this

Re: stack hom-spaces in terms of homotopy limits

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I wrote:

This is supposed to lead over to the discussion at

$n$ Lab: descent for simplicial presheaves

I was about to write more on this and its relation to the notion of descent for strict- $\omega$ -catgory valued presheaves, when I noticed that I never fully sorted out the following thing about the relation between homotopy limits and descent for strict- $\infty$ -groupoid-valued presheaves.

Namely, it should be true that Street’s definition of descent for strict- $\omega$ -groupoid-valued presheaves reproduces under the $\omega$ -nerve the notion of descent for simplicial presheaves.

The following table summarizes what I’m going to say now

$\array{ pre \infty-stack & A' : S^{op} \to SSet & A : S^{op} \to Str \omega Grpd \\ descent along cover U_\bullet & lim^W A'(U_\bullet) & lim^O A(U_\bullet) \\ weight & W = N(\Delta/(-)) & O = orientals \\ relation ? & if A' = N \circ A & then lim^W A' \simeq N(lim^O A) ? }$

Street says (a bit implicitly) that for

$U \to X$

a cover in the site $S$ ,

$U_\bullet : \Delta^op \to S$

the corresponding simplicial object, for

$A : S^{op} \to Str \omega Grpd$

a presheaf and

$A(U_\bullet) := A \circ U_\bullet^{op} : \Delta \to Str \omega Grpd$

the corresponding cosimplicial strict $\omega$ -grouoid, that the descent strict $\omega$ -groupoid of $A$ relative to $U$ is the weighted limit

$lim^O A(U_\bullet) = [\Delta, Str \omega Cat](O, A(U_\bullet)) \,,$

where the weight

$O : \Delta \to Str \omega Cat$

is the orientals.

Now, if instead

$A' : S^{op} \to SSet$

is a simplicial presheaf, its descent simplicial set is the homotopy limit in SSet

$holim A'(U_\bullet) \,.$

By that standard formula in terms of weighted limits for $SSet$ -homotopy limits

$\cdots \simeq lim^W A'(U_\bullet) = [\Delta, SSet]( W , A'(U_\bullet))$

in $SSet$ , where the weight $W$ is the assignment of nerves of over-categories

$W : N( \Delta/(-) ) : \Delta \to SSet \,.$

Now, writing

$N : Str \omega Cat \to SSet$

for the $\omega$ -nerve and considering the case

$A' := N \circ A$

that $A'$ is the simplicial version of the strict- $\omega$ -groupoid valued presheaf $A$ , I was hoping I could just use abstract nonsense to show that under the nerve Street’s prescription becomes equivalent to the simplicial homotopy limit

$holim A'(U_\bullet) = holim N(A(U_\bullet)) \,,$

i.e. that

$N ( lim^O A(U_\bullet) ) \simeq holim N(A(U_\bullet)).$

Something close ought to be right, but now I find myself not quite able to derive this.

For the special case where we restrict to 1-stacks along the inclusion

$Grpd \hookrightarrow Str \omega Grpd$

the corresponding statement has been shown in

Sharon Hollander, A homotopy theory for stacks.

Probably the general idea would be to use that $N$ , being a right adjoint, can be commuted with the limit in some sense or other.

Looking at Kelly’s book, section 3.2, I suppose what I am lacking is a notion of preservation of enriched weighted limits under functors that change the enriching category.

Anyway, I am feeling this should have a simple abstract nonsense answer, and that somebody here might be able to help me with that.

Posted by: Urs Schreiber on April 27, 2009 10:44 AM | Permalink | Reply to this

Re: stack hom-spaces in terms of homotopy limits

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Here is one relevant abstract nonsense result. Let $L : C \rightleftarrows D : R$ be a monoidal adjunction between monoidal categories (so $L$ is strong monoidal and $R$ is lax monoidal), let $M$ be a $D$ -category, let $A$ be a $C$ -category and $W:A \to C$ a $C$ -weight, and let $F:L_\bullet A \to M$ be a $D$ -functor, with adjunct $\bar{F}:A \to R_\bullet M$ under $L \vdash R$ . Then

$R( lim^{L_\bullet W} F ) = lim^W (\bar{F}).$

(At least, if the left side exists, then so does the right and they are isomorphic. The existence of the right side needn’t imply the existence of the left. Also, the left side is slightly imprecise notation; it means $\lim^{L_\bullet W} F$ considered as an object of $R_\bullet M$ .) The only published reference I know for this is Dominic Verity’s thesis, “Enriched categories, internal categories, and change of base.”

In our case, we would like to take $C$ = simplicial sets, $D = Str\omega Cat$ , $R = N$ and $L$ its left adjoint, $A = \Delta$ (regarded as a simplicially enriched category) and $W$ the canonical cosimplicial simplicial set, so that $L_\bullet W = O$ the orientals. Then, if this result applied, it would tell us that

$?? \quad N( lim^O A(U_*) ) = lim^W N(A(U_*)).\quad ??$

However, in this case $L$ is not strong monoidal, not even for the Crans-Gray tensor product. (Note that it is strong monoidal for 1-categories. So there is a genuine difference between $n=1$ and $n\gt 1$ .) It’s possible that using the Crans-Gray tensor product and the fact that all your strict $\omega$ -categories are $\omega$ -groupoids would make $L$ ‘monoidal up to homotopy’ in a strong enough sense to make this result true up to equivalence, but I don’t know.

A less serious issue is that the canonical cosimplicial simplicial set $W$ is different from $N(\Delta/(-))$ ; a $W$ -weighted limit is the ‘totalization’ $Tot A'(U_*)$ of a cosimplicial object (the dual of geometric realization). This is not ‘the’ homotopy limit as usually defined, but it is weakly equivalent to it for Reedy fibrant cosimplicial objects (18.7.4 in Hirschhorn’s book). The nerve of a cosimplicial strict $\omega$ -groupoid will not be Reedy fibrant in general (not even the nerve of an arbitrary cosimplicial 1-groupoid is Reedy fibrant), but it might be if you started with a ‘Reedy fibrant’ cosimplicial strict $\omega$ -groupoid. And you probably want your cosimplicial strict $\omega$ -groupoid to be some sort of fibrant anyway; otherwise the (strict) $O$ -weighted limit isn’t going to be homotopically sensible. (The ‘homotopy limit’, meaning $N(\Delta/(-))$ -weighted limit, also isn’t really sensible unless its input is fibrant enough—hence why I prefer to call it the uncorrected homotopy limit.)

Posted by: Mike Shulman on April 27, 2009 5:52 PM | Permalink | Reply to this

Re: stack hom-spaces in terms of homotopy limits

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So is the $\omega$ -nerve strong monoidal

$N : (Str \omega Grpd, \otimes_{CransGray}) \to (SSet, \times)$

This question also appears in the discussion box in the entry $n$ Lab: Crans-Gray tensor product.

Posted by: Urs Schreiber on April 27, 2009 6:26 PM | Permalink | Reply to this

Re: stack hom-spaces in terms of homotopy limits

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I don’t think that either the $\omega$ -nerve, or its left adjoint, is strong monoidal up to isomorphism even on $\omega$ -groupoids, which is what you would need for this result to apply verbatim. It may be strong monoidal up to equivalence, but then you would probably need a fancier result about ‘homotopical enrichment.’

Posted by: Mike Shulman on April 27, 2009 6:28 PM | Permalink | Reply to this

Re: stack hom-spaces in terms of homotopy limits

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A couple things. Firstly, you don’t actually need $L$ to be strong monoidal for the result I quoted as long as $A$ is unenriched.

Claim: if $C$ and $D$ are cosmoi, $L:C \rightleftarrows D:R$ is a monoidal conjunction (i.e. $R$ is lax monoidal, and therefore $L$ is oplax monoidal), $A$ is an unenriched category, $M$ is a $D$ -category, and $W:A\to C$ and $F:A\to M$ are functors such that $lim^{L W} F$ and $lim^W R_\bullet(F)$ exist (the former being a weighted limit in the $D$ -category $M$ and the latter in the $C$ -category $R_\bullet(M)$ ), then $lim^{L W} F \cong lim^W R_\bullet(F).$

Proof: Let $X\in M$ ; then $\begin{aligned} M_0(X, lim^{L W} F) &\cong D^A_0 ( L W-, M(X, F-))\\ &\cong C^A_0 ( W-, R(M(X,F-)))\\ & = C^A_0 ( W-, R_\bullet M(X,R_\bullet(F) -))\\ &\cong M_0(X, lim^W R_\bullet(F)). \end{aligned}$ Apply Yoneda. $\Box$

The nerve functor $N$ clearly preserves cartesian products, so it is strong monoidal and this result applies if we consider $Str\omega Cat$ with its cartesian monoidal structure. If we wanted to use the Crans-Gray-Brown-Higgins structure, though, it’s not clear to me that $R=N$ is even lax (which is what you need to define $R_\bullet M$ ).

However, I also realized that this result, as nice as it is, is not quite what you are looking for. It says that the $L W$ -weighted limit of $F$ in the $D$ -category $M$ is the same as its $W$ -weighted limit when we regard $M$ as a $C$ -category via $R$ . However, in our case $M=D$ and we want to compare not limits in $D$ not just to limits in $D$ regarded as a $C$ -category via $R$ , but their images in $C$ under $R$ . For this it suffices if the adjunction $L:C \rightleftarrows D:R$ becomes a $C$ -enriched adjunction $L:C \rightleftarrows R_\bullet D:R$ (since right $C$ -adjoints preserve limits), which concretely means that the canonical map $R(D(L X,Y)) \to C(X, R Y)$ is an isomorphism. This seems certainly false to me for the nerve if we give $Str\omega Cat$ its cartesian structure; let $Y$ be some $\omega$ -category and $X=\Delta^1$ and look at the 1-simplices on each side.

Posted by: Mike Shulman on April 27, 2009 7:10 PM | Permalink | Reply to this

Re: stack hom-spaces in terms of homotopy limits

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Ah, I hadn’t appreciated before that the left hand side was the limit in $R_\bullet M$ , even though you said so explicitly. That’s indeed not what we need (being a simplicially enriched version of $\omega$ -Cat for our example).

Concerning the cartesian monoidal structure:

This seems certainly false to me for the nerve if we give $Str \omega Cat$ its cartesian structure;

Okay. The cartesian monoidal structure on $\omega Cat$ really shouldn’t play much of a role here.

So it seems to stand or fail with the extent to which $N : Str \omega Grpd \to SSet$ preserves monoidal structure.

which concretely means that the canonical map

$R(D(L X ,Y )) \to C(X, R Y)$

is an isomorphism.

Just to make it explicit to facilitate following this, in our case this is

$N(\omega Grpd(F X ,Y )) \to SSet(X, N(Y))$

for $N$ the $\omega$ -nerve and $F$ the free $\omega$ -category on a simplicial set, $Y$ some strict $\omega$ -groupoid and so I guess we need to allow/force ourselves to take $X$ to be a Kan simplicial set.

I have to call it quits now, but I’d like to come back to this tomorrow.

Posted by: Urs Schreiber on April 27, 2009 8:49 PM | Permalink | Reply to this

Re: stack hom-spaces in terms of homotopy limits

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I guess we need to allow/force ourselves to take $X$ to be a Kan simplicial set.

If we’re working with ordinary enriched categories here, then $X$ has got to be allowed to be any simplicial set. But I would guess that that won’t matter much, since we are mapping out of it, rather than into it.

Just to make extra clear where the question is at the moment, this canonical map $R(D(L X,Y))\to C(X,R Y)$ doesn’t even exist (at least, I don’t know how to construct it) until you know that $R$ is lax monoidal. So if we’re discarding the cartesian monoidal structure, then as you say the real question is to what extent $N$ takes the tensor product of strict $\omega$ -categories to the cartesian product of simplicial sets.

Posted by: Mike Shulman on April 28, 2009 12:30 AM | Permalink | Reply to this

Re: stack hom-spaces in terms of homotopy limits

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Actually, given an adjunction $L:C\rightleftarrows D:R$ between monoidal categories, any of the following maps determine the other two uniquely by mate correspondence:

$R(D(L X,Y)) \to C(X, R Y)$
$L(X\otimes X') \to L X \otimes L X'$
$R Y \otimes R Y' \to R(Y\otimes Y')$

Moreover, of the first two, one is an isomorphism iff the other is. So to get the limit-preservation that you want (at least, to get it from ordinary enriched category theory), we really would need the left adjoint of the nerve to be strong monoidal after all.

Posted by: Mike Shulman on April 28, 2009 4:33 PM | Permalink | Reply to this

Re: stack hom-spaces in terms of homotopy limits

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we really would need the left adjoint of the nerve to be strong monoidal after all.

Mike, thanks for all your input.

I was busy this morning with something else and now spent quite a bit of time with bringing the entry on descent in new shape and now pushing it to the point where our discussion here kicks in.

I am almost there. Right now I’ll write entries on Reedy fibrancy, Bousfield-Kan map and totalization versus homotopy limit to give a discussion for when descent may be expressed in terms of $\Delta$ -weighted limits and hence in terms of familiar gluing data.

So, that delays me a bit, but I’ll get back to you then.

I am feeling we must be missing something here. I am pretty sure that for $A$ an $\omega$ –groupoid valued presheaf which is in the image of the Dold-Kan map from abelian complexes, i.e. a prsheav of abelian crossed complexes, Street’s descent does reproduce the right sheaf Cech hypercohomology of complexes of abelian sheaves. I suppose this must have been his main motivation, too, though I am wondering why he never talks about it (in the articles that I have seen).

But also the $holim A(U^\bullet)$ -description does reproduce ordinary abelian sheaf cohomology, as recalled at $n$ Lab: abelian sheaf cohomology, so some relation should be there.

So I am not sure what’s going on, given that, as you emphasize, the obvious guess for how the relation works seems to fail. But I’d like to find out.

Posted by: Urs Schreiber on April 28, 2009 5:16 PM | Permalink | Reply to this

Re: stack hom-spaces in terms of homotopy limits

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I came across these slides, which look like they might have something relevant for our discussion here:

Dominic Verity, Weak complicial sets and internal quasi-categories

Slide 60 is about the nerve/realization adjunction for strict $\omega$ -categories and stratified simplicial sets

where the nerve

$N : Str \omega Cat \to Strat$

is the ordinary $\omega$ -nerve

$Str \omega Cat \to SSet$

postcomposed with choosing the canonical stratification.

There is a tensor product on $Strat$ , which, as slide 60 asserts, is such that it makes the left adjoint

$F : Strat \to Str \omega Cat$

strong monoidal.

But unless I am mixed up slide 61 asserts that this tensor product on Strat is just the ordinary cartesian tensor product on $SSet$ equipped with a suitable stratification.

Another potentially relevant remark for the discussion here is on slide 76, which asserts that some $\omega$ -thing is Reedy cofibrant. But I am too tired now to see what exaxtly is happening at that point

Posted by: Urs Schreiber on April 28, 2009 11:36 PM | Permalink | Reply to this

Re: stack hom-spaces in terms of homotopy limits

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I really need to go home now and get some sleep. But I am thinking now that perhaps the more relevant question to be looked at is rather how to nicely characterize cosimplicial strict $\omega$ -groupoids

$A(U_\bullet) : \Delta \to Str \omega Grpd$

such that under the nerve they become Reedy fibrant cosimplicial simplicial sets ( $n$ Lab entry on Reedy fibrancy goes here, tomorrow), so that we know at least that

$holim N(A(U_\bullet)) \simeq lim^{\Delta} N(A(U_\bullet)) \,.$

Because this is gonna be a help for component computations only in low dimensions anyway, and there evaluating the right hand of the above explicitly is not much more tedious than doing it for the $Str \omega Cat$ version $lim^O A(U_\bullet))$ that we are not sure about if it is equivalent in general.

Okay, good night. :-)

Posted by: Urs Schreiber on April 27, 2009 9:23 PM | Permalink | Reply to this

Re: smooth

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You don’t have to go to a “nonflat” setting to sense more than the homotopy type - that’s what constructible sheaves/D-modules are for.. they’re both “flat” objects but for which parallel transport is only infinitesimal. Namely having a flat connection on something not finite dimensional does not in general give a monodromy action of the fundamental groupoid – you need the connection to integrate not just to every finite order but beyond. That’s not guaranteed by flatness. That’s why D-modules are much more interesting than local systems.. and there’s a purely topological (though not homotopy-invariant) version of them, constructible sheaves. So I guess my feeling is that that’s the (or “a”) natural “function theory” to consider in your setting of smooth manifolds, namely sheaves which are locally constant along a Whitney stratification..

Posted by: David Ben-Zvi on April 20, 2009 11:31 PM | Permalink | Reply to this

Re: flat differential

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In what can for all practical purposes be read as a direct followup/reply to the discussion here, David Speyer starts posting a series on cohomology in terms of local systems et al. at Three ways of looking at a local system: Introduction and connection to cohomology theories.

Posted by: Urs Schreiber on April 20, 2009 10:28 PM | Permalink | Reply to this

Re: flat differential

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Zoran Škoda kindly created $n$ Lab: regular differential operator and polished the overview blurbs at D-module and local system.

Should we assign similar contributions as homework to the Journal-Club participants?

To get started with the first section, we need $n$ Lab entries at least with a paragraph

“Idea” – stating the central idea and meaning in words

“Definition” – stating at least briefly something like a precise definition to some approximation

“References” – at least one pointer to page and verse for more details

on the following keywords:

- quasicoherent sheaves

- (infinity,1)-categorical quasicoherent shaves

- perfect stack

Bruce? Alex? Anyone else?

I had started collecting some potentially relevant literature at derived stack.

$n$ Lab submissions signed as “Anonymous Coward” are just as fine, if that helps. And don’t try to be anything like perfect on the first go, lest the rest of us won’t have anything left to do.

Posted by: Urs Schreiber on April 21, 2009 10:12 AM | Permalink | Reply to this

Re: Journal Club – Geometric Infinity-Function Theory

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Behind the scenes, by means of massively parrallel Google Chats, we managed to organize the beginning of what is intended to become a systematic Journal-Club schedule.

The idea is that we’ll “assign” sections to volunteers, and that one such volunteer presents, each week on Monday, a “report” on his or her encounter with that section.

For a few more comments on this idea, see the new section “Timeline” at our $n$ Lab-entry.

There you also find a list with assignments of sections, as far as organized so far.

The plan is that Alex Hoffnung starts next Monday, April 27, with section 1, “Introduction”.

Then we’ll see what happens. Currently we are still looking for volunteers for sections 4, 5 and 6 of “Integral transforms”. Let us know if you are interested.

Posted by: Urs Schreiber on April 21, 2009 6:13 PM | Permalink | Reply to this

Read the post Journal Club -- Geometric Infinity-Function Theory -- Week 1
Weblog: The n-Category Café
Excerpt: Journal Club on Geometric oo-Function theory part I: Introduction to integral transforms.
Tracked: April 27, 2009 4:59 PM

Read the post Journal Club -- Geometric Infinity-Function Theory -- Week 2
Weblog: The n-Category Café
Excerpt: Preliminaries for the discussion of geometric infinity-function theory: higher categories, higher sheaves, higher algebra, higher traces and what it all means.
Tracked: May 5, 2009 3:53 PM

Read the post Journal Club -- Geometric Infinity-Function Theory -- Week 3
Weblog: The n-Category Café
Excerpt: This week in our Journal Club on [[geometric ∞-function theory]] Bruce Bartlett talks about section 3 of "Integral Transforms": perfect stacks. So far we had Week 1: Alex Hoffnung on Introduction Week 2, myself on Preliminaries See here for...
Tracked: May 11, 2009 11:20 PM

Read the post Journal Club -- Geometric Infinity-Function Theory -- Week 4
Weblog: The n-Category Café
Excerpt: Chris Brav reviews technical details about tensor products and integral transforms of quasi-coherent sheaves on perfect stacks.
Tracked: May 18, 2009 7:23 AM

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April 10, 2009