Quasicoherent ∞ -Stacks

January 8, 2010

Posted by Urs Schreiber

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This is to tie up a loose end in our discussion of Ben-Zvi/Francis/Nadler geometric $\infty$ -function theory and Baez/Dolan/Trimble “groupoidification”.

Using a central observation in Lurie’s Deformation Theory one can see how both these approaches are special examples of a very general-abstract-nonsense theory of $\infty$ -linear algebra on $\infty$ -vector bundles and $\infty$ -quasicoherent sheaves in arbitrary (“derived”) $\infty$ -stack $(\infty,1)$ -toposes:

the notion of module is controled by the tangent $(\infty,1)$ -category of the underlying site, that of quasicoherent module/ $\infty$ -vector bundle simply by homs into its classifying $\infty$ -functor.

For Ben-Zvi/Francis/Nadler this shows manifestly, I think, that nothing in their article really depends on the fact that the underlying site is chosen to be that of duals of simplicial rings. It could be any $(\infty,1)$ -site. For Baez/Dolan/Trimble it suggests the right way to fix the linearity: their setup is that controlled by the tangent $(\infty,1)$ -category of $\infty Grpd$ itself and where they (secretly) use the codomain bifibration over this, one should use the fiberwise stabilized codomain fibration.

See $\infty$ -vector bundle for a discussion of what I have in mind.

What I am saying here is likely very obvious to somebody out there. I have my suspicions. But it looks like such a nice fundamental fact, that this deserves to be highlighted, and be it in a blog entry. It is just a matter of putting 1 and 1 together. The two central observations are this:

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In our journal club discussion I had remarked that the definition $QC(X)$ of derived quasicoherent sheaves on a derived $\infty$ -stack $X$ that BenZvi/Francis/Nadler use is best thought of as $hom(X,QC)$ in the $(\infty,1)$ -category of $(\infty,1)$ -category-valued $\infty$ -stacks. That this is the way to think about quasicoherent sheaves must be an old hat to some people but is rarely highlighted in the literature.

The only place I know of presently where this is made fully explicit is the $n$ Lab entry on quasicoherent sheaves. As discussed there, at least Kontsevich and Rosenberg have made this almost fully explicit – they say this in side remark 1.1.5 here, in the dual picture where category-valued presheaves are replaced by fibered categories.

The other crucial insight is Lurie’s basic idea from Deformation Theory. This is amazingly elegant, have a look, if you haven’t yet. Lurie shows that for $C$ any $(\infty,1)$ -category, whose objects we here think of as formal duals of test spaces, the tangent $(\infty,1)$ -category fibration $T_C \to C$ is to be thought of as the fibration of modules over the objects of $C$ , generalizing the canonical fibration $Mod \to Ring$ that underlies the classical theory of monadic descent. But strikingly: $T_C$ is effectively nothing but the codomain fibration over $C$ – it differs from that only in that all fibers are stabilized, i.e. linearized in the fully abstract category-theoretic sense. In particular, this says that the Ben-Zvi/Francis/Nadler $\infty$ -stack of derived quasicoherent sheaves $QC : SRing \to (\infty,1)Cat$ is equivalently simply given by assigning $Spec A \mapsto Stab(SRing/A)$ – to any simplicial ring its stabilized overcategory. (This is stated in example 8.6 on page 24 in Stable $\infty$ -Categories.)

I had remarked here in our journal club discussion that Baez/Dolan/Trimble “groupoidification” is like Ben-Zvi/Francis/Nadler theory but with $QC$ replaced by the assignment of overcategories. Now in the light of Lurie’s observation this makes everything fall into place: Baez/Dolan/Trimble groupoidification may be thought of as a shadow of the geometric $\infty$ -function theory induced from the tangent $(\infty,1)$ -category over $\infty Grpd$ itself: instead of pull-pushing bundles of groupoids as they do, in the full theory one would pull-push bundles of abelian $\infty$ -groupoids (and more generally: possibly non-connective spectra).

You may remember my motivation for coming to grips with this conceptual framework: for applications in the physics of gauge fields we need smooth differential cohomology and $\infty$ -Lie theory. This doesn’t really take place in $\infty$ -stacks over simplicial rings (only a small part of it does) and it doesn’t take place in $\infty$ -stacks over just $\infty Grpd$ (though important toy models do): it takes place in $\infty$ -stacks over simplicial $C^\infty$ -rings and generally over simplicial objects in sites for smooth toposes. Clearly we want Ben-Zvi/Francis/Nadler geometric $\infty$ -function theory generalized to this context. With $QC$ in hand in this contex, we immediately have $\infty$ -representations of $\infty$ -Lie groups, associated $\infty$ -vector bundles and their pull-push and monadic descent.

And with the above it is clear what one needs to look at: just take the $\infty$ -stack of smooth generalized $\infty$ -vector bundles to be $QC : Spec A \mapsto Stab(SC^\infty Ring/A)$ . And that’s it.

More at $\infty$ -vector bundle.

Posted at January 8, 2010 12:35 PM UTC

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Re: Quasicoherent infty -Stacks

The link in
Using a central observation in Lurie’s Deformation Theory
seems to produce only a cover page

Remind me how to use it? i.e how to get a coherent! single document
hopefully not by clicking on each item in the Contents separately

Posted by: jim stasheff on January 8, 2010 2:04 PM | Permalink | Reply to this

Re: Quasicoherent infty -Stacks

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The arXiv entry of Jacob Lurie’s Deformation Theory is here. That’s the link named “arXiv” on what you call the cover page, right after the article title. If you want to be sure that you see the very latest, download the pdf on his homepage

The (very incomplete) linked keyword list on the “cover page” is intended as a public service for quick reference.

Posted by: Urs Schreiber on January 8, 2010 2:36 PM | Permalink | Reply to this

Re: Quasicoherent infty -Stacks

apologies
I thought you meant that link to get to your summary of
Lurie’s version of def theory

Posted by: jim stasheff on January 8, 2010 3:59 PM | Permalink | Reply to this

Re: Quasicoherent infty -Stacks

Truth in advertising? Lurie is about deformation theory for E-infty algebras. At a glance, I see very little _exposition_ setting the stage in terms of classical deformation theory.

A better source for an up-to-date version of that might be the draft book by Kontsevich and Soibelman -
available from Yan’s home page

Posted by: jim stasheff on January 8, 2010 4:07 PM | Permalink | Reply to this

Re: Quasicoherent infty -Stacks

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Lurie is about deformation theory for $E_\infty$ algebras.

That’s the example worked out. The theory in the first part is entirely general. And it is this generality which the entry here is concerned with.

At a glance, I see very little exposition setting the stage in terms of classical deformation theory.

I found his exposition quite nice. But notice that the discussion here is not really about deformation theory as such, but about a very general notion of modules, which happens to be discussed in an article on deformation theory.

draft book by Kontsevich and Soibelman -

Oh, so you meant this book link in your message recently? How should I know? You just asked me to “include the link from Yan’s homepage”.

Okay, the link is now included here:

$n$ Lab: deformation theory - References - Texbooks

A better source for an up-to-date version of that

Just a comment: the Kontsevich-Soibelman book looks very nice and anyone interested in standard deformation theory should look at that, and not (at first) at Lurie’s article.

But, you see, what Lurie’s article accomplishes, in it’s first part, is that it gives a vastly and vastly more general context for what deformation theory is actually about. He accomplishes this by giving a vastly and vastly more general definition of the notion of module (and derivation, and cotangent complex and…). It is this general-abstract-nonsense perspective, and that only, which I am referring to in my above discussion. The above discussion is not about deformation theory. The above discussion is, if you wish, about how one might set up deformation theory in the context of $\infty$ -Lie theory and other contexts. Lurie’s setup is so general that it provides a prescription for how to proceed with the theory in this case, and in other cases.

But the Kontsevich-Soibelman book is very nice. I’ll have a close look.

Posted by: Urs Schreiber on January 8, 2010 5:03 PM | Permalink | Reply to this

Re: Quasicoherent ∞ -Stacks

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While I certainly subscribe to the philosophy you explain, I don’t think it’s due to Jacob. The modern perspective on tangents=linearization=stabilization which you quote is I believe due to Goodwillie building on Quillen (Jacob certainly ascribes it to Goodwillie and others.. of course he states it in a very elegant way which is only possible given the appropriate language). The statement about QC being stabilization of overcategories of schemes is just restating the classical fact that modules are the linearization of rings. Certainly the idea of thinking about categories of sheaves as linearizations of spaces over a base is old and is for example behind the theory of variations of Hodge structures or more fancily motivic sheaves (as we discussed over at the n-geometry cafe I think)..

Also I’m not sure which assertions in our paper you refer to as holding in such great generality - one needs to make very strong assumptions about the properties of QC(X) to get any traction, and these are available for schemes eg only thanks to a brilliant insight of Thomason (or rather of Trobaugh’s ghost in Thomason’s dream). But if you’re willing to make such assumptions about your spaces, of course everything we actually do is quite formal.

(PS the statement “QC is best thought of as hom(X,QC) in the (∞,1)-category of (∞,1)-category-valued ∞-stacks” is restating the assertion that QC(-) forms a stack, ie descent, or that sheaves are by definition local, no?)

Posted by: David Ben-Zvi on January 9, 2010 5:25 AM | Permalink | Reply to this

Re: Quasicoherent ∞ -Stacks

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David, thanks, I certainly was hoping to get a comment from you.

[…] I certainly subscribe to the philosophy you explain […]

Okay, thanks for the sanity check.

[…] I don’t think it’s due to Jacob. […]

What would be an original source for the statement in example 8.6 of Stable $\infty$ -Categories that $T_{sRings} \simeq Mod_{sRings}$ ? Is this folk lore, as a precise theorem?

The modern perspective on tangents=linearization=stabilization which you quote is I believe due to Goodwillie building on Quillen (Jacob certainly ascribes it to Goodwillie and others

Well, I should maybe say that the point of what I wrote was not so much to announce “Lurie did X” but to announce that “reading Lurie made me see Y”. If you say that it was all in the air I take your word for it, but…

(of course he states it in a very elegant way which is only possible given the appropriate language).

Sometimes obvious structures still never become clear until the right language is found. As somebody once said:

Schläft ein Lied in allen Dingen,
Die da träumen fort und fort,
Und die Welt hebt an zu singen,
Triffst du nur das Zauberwort. #

I think the right language is important here.

You further write:

The statement about QC being stabilization of overcategories of schemes is just restating the classical fact that modules are the linearization of rings.

I appreciate the fact that fiberwise stabilization of codomain fibrations is the evident abstract-nonsense formulation of how $Mod \to Rings$ is fiberwise the abelian category of square-0-extensions of the ring downstairs. With hindsight the story is now very obvious and pleasing. But to make fully explicit the notion of tangent $(\infty,1)$ -category and to demonstrate that it then correctly captures the simplicial and $E_\infty$ -cases still looks like an accomplishment to me.

Well, I take your point that this isn’t shocking news to the experts, and I hear the refrain “category theory is just a language” from those who don’t like it, but I still think Jacob Lurie’s precise description of the situation is of a beauty that deserves to be highlighted.

(as we discussed over at the n-geometry cafe I think)..

Yes, indeed, that was your comment here. I kept coming back to this comment and thought about it. Now it all has become very clear to me, and I thank you for all your help. But somehow I neeeded extra details on how it works in concrete examples to fully absorb the statement.

But if you’re willing to make such assumptions about your spaces, of course everything we actually do is quite formal.

Thanks, yes, that’s what I should have said, more precisely.

PS the statement “ $QC$ is best thought of as $hom(X,QC)$ in the $(\infty,1)$ -category of $(\infty,1)$ -category-valued $\infty$ -stacks” is restating the assertion that $QC(-)$ forms a stack, ie descent, or that sheaves are by definition local, no?)

I’d think that by itself it’s a statement that doesn’t depend on the topology. There it’s just a way to define QC globally from local data.

It’s entirely anaogous to the definition of differential forms on presheaves: for $C$ a category and $\Omega^\bullet : C^{op} \to Set$ a notion of differential forms on the test objects in $C$ , we define the differential forms on an arbitrary presheaf by

$\Omega^\bullet(X) := [C^{op},Set](X,\Omega^\bullet) \,.$

For instance for $C = \Delta$ this gives the definition of Sullivan differential forms on simplicial sets. (Just for other readers following our exchange I provide a link to more details on this example.) This doesn’t make use of any Grothendieck topology.

For (ordinary for the moment) quasicoherent sheaves it is a priori really the same. Here $C = Ring^{op}$ or the like and $\Omega^\bullet$ is replaced by $QC : Spec A \mapsto A Mod$ , but then we can define for any presheaf $X$ that $QC(X) := hom(X,QC)$ .

But we can then reverse the question and ask: when does this assignment satisfy descent? When is $\Omega^\bullet$ , when is $QC$ a sheaf, a stack? That’s classically where terms like “effective descent morphism” and Bénabou-Roubaud theorems appear (link again for other readers, I am not suggesting that you don’t know what I am talking about, I am just trying to make myself clear): we have $QC$ given and are now looking for Grothendieck topologies that make it a stack.

Or so I think. Please let me know if I sound like I am mixed up.

Posted by: Urs Schreiber on January 9, 2010 10:42 AM | Permalink | Reply to this

Re: Quasicoherent ∞ -Stacks

One small clarification: the stabilization Stab(sCRing/A) of simplicial commutative rings over A is only equivalent to the category of A-modules if one works relative a base field K of characteristic zero. In general, there is a functor A-mod —> Stab, but this functor is not essentially surjective or full. Its failure to be an equivalence is closely related to the failure of the inclusion of simplicial commutative K-algebras into E-infinity K-algebras to be an equivalence.

This failure highlights the distinction between Quillen’s notion of A-mod(C) for an object A in C, which is the category of abelian group objects in the overcategory C/A, and the stabilization of C/A. Often these notions produce equivalent categories (e.g., if C is E-infinity rings) because stable objects can be strictified to get abelian group objects. But sometimes they can’t, as when C is sCRing. To give another example, if C is the category of spaces and A is a point, then Ab(Spaces) —> Stab(Spaces) is not an equivalence, in part because you have infinite loop spaces that cannot be strictified to be abelian groups on the nose (e.g., BO or BU). Topological abelian groups are pretty special among infinite loop spaces: they’re all products of Eilenberg-MacLane spaces.

(Thanks, by the way, for looking at our stuff so closely; it’s gratifying that you all have taken such an interest in it.)

Posted by: John Francis on January 9, 2010 6:57 PM | Permalink | Reply to this

Re: Quasicoherent oo -Stacks

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One small clarification: the stabilization $Stab(sCRing/A)$ of simplicial commutative rings over A is only equivalent to the category of $A$ -modules if one works relative a base field $K$ of characteristic zero. In general, there is a functor $A mod \to Stab$ , but this functor is not essentially surjective or full. Its failure to be an equivalence is closely related to the failure of the inclusion of simplicial commutative K-algebras into E-infinity K-algebras to be an equivalence.

Ah, thanks for catching that.

I am starting to collect this and other facts at Modules over simplicial rings. Not much there yet right this moment, but it should eventually expand.

Can you point me to more references? I am not sure I have seen the really relevant bits yet, given what you and David are saying here.

(Thanks, by the way, for looking at our stuff so closely; it’s gratifying that you all have taken such an interest in it.)

It’s very cool stuff. I feel we even need to eventually improve the $n$ Lab page on this and give a better impression of what’s really going on, fundamentally. Personally, I was blocked for quite a while by not fully seeing how to generalize your constructions to other sites. Apparently I was just being dense. Now with that out of the way I am hoping to make more progress with fully absorbing this.

Posted by: Urs Schreiber on January 10, 2010 5:26 PM | Permalink | Reply to this

Re: Quasicoherent ∞ -Stacks

Urs - I think we agree on all points here. (I apologize for the grumpy tone of my comment - ascribe it to Texas losing the college football championship game..) I don’t know of the proper original references for the kind of linearization results being discussed, but I think it’s all part of the Goodwillie calculus oeuvre (at least I had these ideas explained to me before DAG4 appeared, so I assume the experts understood this). Of course as you know I’m as enthusiastic about what Jacob is doing as anyone, and think it goes far beyond “language” in any dismissive sense, but I do think this yoga of first order calculus (linearization/stabilization/tangents/deformations), very likely in a less general and precise sense, is not new. Of course there’s a basic temptation for newcomers to the field (like me) when there’s a beautiful complete self-contained source, which also has deep new insights, grander scope, more applicable results etc etc etc, to remain ignorant of all previous history of the subject and unintentionally erase everyone who came before. (was there algebraic geometry before Grothendieck??) So I guess the point of my comment is I’m more guilty of this than most, and wanted to balance that when I can.

Posted by: David Ben-Zvi on January 10, 2010 5:15 AM | Permalink | Reply to this

Re: Quasicoherent ∞ -Stacks

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No need to apologize! I am grateful for your reply.

Maybe we were talking past each other a bit. You probably remember that I kept wondering and asking (here, on MO and elsewhere) how one would go about making your geometric $\infty$ -function theory independent of the particular choice of site.

Because I wanted to use it, but need it on a site different from $sAlg^{op}$ .

I saw that up to some assumptions, the only place where you make essential use of that particular site is when saying $Spec A \mapsto A Mod$ , hence when saying $QC(Spec A)$ . I wasn’t sure how I should say that for other sites. I knew that I had to use overcategories as models for cats of modules (there is a remnant of my thinking about this here, which I will in some days remove and replace by the right full answer now) but I happened to have been unaware that simply stabilizing these yields the complete right answer.

Somebody – maybe you – should write up your theory of integral transforms in full generality on arbitrary $\infty$ -sites. I think it’s very beautiful and even more fundamental than your article makes it appear.

Posted by: Urs Schreiber on January 10, 2010 4:12 PM | Permalink | Reply to this

Re: Quasicoherent ∞ -Stacks

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Of course there is a basic temptation for newcomers to the field (like me) when there is a beautiful complete self-contained source, which also has deep new insights, grander scope, more applicable results etc etc etc, to remain ignorant of all previous history of the subject and unintentionally erase everyone who came before.

I know what you mean. I am eager to fill in historical references, but maybe you could help me identifying them. I am in the process of expanding the entry $n$ Lab:module accordingly.

Posted by: Urs Schreiber on January 10, 2010 5:05 PM | Permalink | Reply to this

Re: Quasicoherent ∞ -Stacks

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David wrote:

(PS the statement “QC is best thought of as hom(X,QC) in the (∞,1)-category of
(∞,1)-category-valued ∞-stacks” is restating the assertion that QC forms a stack, ie descent, or that sheaves are by definition local, no?)

Is there a precise statement that $QC(-)$ forms a stack somewhere? And how do you formulate stack precisely for a $(\infty,1)$ -category valued preasheaf.

Just to check that I understand $QC(-)$ right: The restriction of $QC(-)$ to ordinary rings assigns to each ring the $(\infty,1)$ -category of chain complexes of modules over this ring (such that the homotopy category is the classical derrived category), right?. So stack would mean that we can glue toghether chain complexes defined locally… Is there an intuitiv way to think of such a glueing process?

Posted by: Thomas Nikolaus on January 16, 2010 12:49 PM | Permalink | Reply to this

Re: Quasicoherent ∞ -Stacks

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Thomas asked David:

And how do you formulate stack precisely for a (∞,1)-category valued preasheaf.

Just on the general topic:

in principle the technology should be available to model the stack condition for $(\infty,n)$ -category-valued presheaves, for all $n$ (at least as long as descent is taken with respect to just Cech covers and not hypercovers):

namely by combining recent results by Clark Barwick and Charles Rezk:

Clark Barwick has generalized the proof that left Bousfield localization of $SSet_{Quillen}$ -enriched combinatorial model categories at a set of morphisms exists to the case of general tractable $V$ -enriched model categories, in

Clark Barwick, On left and right model categories and left and right Bousfield localization

So this should apply in particular to the localization of the global model structure on $SSet_{Joyal}$ -enriched presheaves at Cech covers.

More generally, Charles Rezk has given monoidal model category models for $(n,r)Cat$ for all $n,r \leq \infty$ in terms of Theta-spaces.

This should mean that the left Bousfield localization of $(n,r)Cat_{Rezk}$ -valued presheaves at Cech covers exists and models the corresponding stacks.

What I just said is also collected briefly at model structure on homotopical presheaves.

Posted by: Urs Schreiber on January 16, 2010 3:48 PM | Permalink | Reply to this

Re: Quasicoherent ∞ -Stacks

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Thomas - that’s right, QC assigns to a ring the $\infty$ -category of chain complexes of modules (i.e. the refinement of the unbounded derived category of the ring). It is covariantly functorial in morphisms of rings (by tensoring up of modules), ie contravariant in affine schemes (by pullback). For a general scheme or stack we can define its QC as the limit of $QC(R_i)$ over all $Spec(R_i)\to X$ .

The statement that QC is a stack in some topology - e.g. etale or flat - is the assertion that for $U\to X$ a cover in this topology, with associated Cech simplicial object $U_\bullet$ (whose $k$ -simplices are the fiber products $U\times_X U\times_X \cdots U$ ) we can calculate $QC(X)$ as the totalization (limit) of the cosimplicial $\infty$ -category $QC(U_\bullet)$ . This is an $\infty$ -version of the usual descent statements for quasicoherent sheaves. It follows from Lurie’s comonadic Barr-Beck theorem once we verify that etale or flat covers satisfy the corresponding Barr-Beck criteria – apparently this is written up in detail in the forthcoming DAG 7, but similar statements can be found in Toen-Vezzosi.

As to its intuitive meaning, it’s just a statement that you can glue complexes – given say two Zariski opens U and V and complexes of sheaves on each, if you are given isomorphisms on the overlaps you can define a sheaf on the union. For a general $U\to X$ you need to consider higher gluing data on all the n-fold intersections.

This is actually something we’re all very familiar with if you replace sheaves with cochains. You can ask, in what sense are cochains on a topological space X local with respect to a cover? the answer is given eg by the Cech-de Rham spectral sequence (in the case of forms). Namely to define the complex of cochains on X you take a limit (totalization) of the Cech cosimplicial object, given by cochains on n-fold intersections. Of course cohomology of X is not local in this sense, you have to go to the cochain level. Likewise the derived category of sheaves is not local, but once you refine it to the corresponding $\infty$ -category the same gluing/descent picture holds.

Posted by: David Ben-Zvi on January 16, 2010 9:31 PM | Permalink | Reply to this

Re: Quasicoherent ∞ -Stacks

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Hi David, thanks for the detailled answer. Now I’m still more looking forward to the release of DAG7 ;)

is the assertion that for $U \to X$ a cover in this topology, with associated Cech simplicial object $U_\bullet$ (whose k-simplices are the fiber products $U\times_X \times U \times_X \cdots U$ ) we can calculate QC(X) as the totalization (limit) of the cosimplicial $\infty$ -category $QC(U_\bullet)$ .

Okay, thats clear from an abstract point of view. Nevertheless I don’t know how to compute such a limit over cosimplicial $\infty$ -categories. Is it the homotopy colimit which we can compute using the model strucutre on $\infty$ -categories (depending on your model of $\infty$ -categories)? Or do we have to be a little bit more carefull because we are in a $\infty$ -bicategory? It would be good to have an explicit formula for the Descent- $\infty$ -categories.

Posted by: Thomas Nikolaus on January 17, 2010 4:59 PM | Permalink | Reply to this

Re: Quasicoherent ∞ -Stacks

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Is it the homotopy colimit which we can compute using the model strucutre on ∞-categories (depending on your model of ∞-categories)? Or do we have to be a little bit more carefull because we are in a ∞-bicategory? It would be good to have an explicit formula for the Descent-∞-categories.

If we do suppose, as I indicated above, that the (non-hypercomplete) $(\infty,2)$ -topos of $(\infty,1)$ -stacks on a site $C$ is modeled by the $SSet_{Joyal}$ -enriched model category obtained as the left Bousfield localization of $[C^{op},SSet_{Joyal}]_{proj}$ at the set of morphisms of the form $C(\{U_i \to V\}) \to V$ for $\{U_i \to V\}_i$ a covering family in the site and $C(\{U_i\})_n = (\coprod_i U_i)^{\times_V n-1}$ the Cech nerve simplicial presheaf of the cover, then the answer is clear:

an $(\infty,1)$ -stack will be a fibrant object in the left localization, which is (as you know well, but I just say it for the record and for other readers) is a simplicial presheaf $A$ that is objectwise Joyal-fibrant (i.e. quasi-category-valued) and such that for $Q(C(\{U_i \to V\})) \to Q(V)$ a cofibrant replacement we have that

$[C^{op},SSet](Q(V),A) \to [C^{op},SSet](Q(C(U_i)),A) =: Desc(\{U_i\},A)$

is a weak equivalence in $SSet_{Joyal}$ . This enriched hom may be expressed in terms of ordinary limits in $SSet$ , the homotopy information is all in the cofibrant replacement.

In the familiar case of $SSet_{Quillen}$ -valued presheaves we know that the representable $V$ is already cofibrant (obvious because acyclic Quillen fibrations are surjetive on 0-cells) and that $C(\{U_i\})$ is cofibrant (from Dugger’s work, summarized here).

So one way to answer your question would be to understand whether this cofibrancy fails with $SSet_{Joyal}$ -valued presheaves, and what the cofibrant replacement is, instead.

It is still true in $SSet_{Joyal}$ that acyclic fibrations are surjective on 0-cells. So representables are still cofibrant. Im am not sure right now whether also Cech nerves of covering families of representables are still cofibrant. I would expect so, but not sure yet about the proof.

Posted by: Urs Schreiber on January 18, 2010 7:33 AM | Permalink | Reply to this

Re: Quasicoherent ∞ -Stacks

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David wrote:

similar statements can be found in Toën-Vezzosi

Just for the record and the sake of other readers: the theorem in question — that derived quasicoherent sheaves form an $\infty$ -stack in a suitable sense – is

Toën-Vezzosi, Homotopical algebraic geometry II: Geometric stacks and applications, Theorem 1.3.7.2 on page 96

But notice: their statement refers not to the $(\infty,1)$ -categories of quasicoherent sheaves, but just to their cores, their maximal $\infty$ -groupoids. This is in definition 1.3.7.1 on the same page.

Accordingly, their notion of $\infty$ -stack is just the ordinary (hypercomplete) one (Joyal-Jardine), only that they generalize this from 1-categorical sites to $(\infty,1)$ -categorical sites in the obvious way:

Toën-Vezzosi, Homotopical algebraic geometry I: Topos theory, Theorem 1.0.4, page 4.

Posted by: Urs Schreiber on January 18, 2010 1:09 PM | Permalink | Reply to this

Re: Quasicoherent ∞ -Stacks

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I started listing some of the pertinent definitions and propositions in Toën/Vezzosi’s discussion at:

quasicoherent $\infty$ -stack.

Posted by: Urs Schreiber on January 18, 2010 6:02 PM | Permalink | Reply to this

Re: Quasicoherent ∞ -Stacks

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I have added to the entry some remarks on flat $\infty$ -vector bundles and D-modules:

Flat $\infty$ -vector bundles / D-modules

Posted by: Urs Schreiber on January 9, 2010 1:17 PM | Permalink | Reply to this

Re: Quasicoherent ∞ -Stacks

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As Thomas kindly points out, there is this recent reference here, coming from a similar motivation as discussed above:

Dominic Joyce, Algebraic geometry over $C^\infty$ -rings.

Posted by: Urs Schreiber on January 13, 2010 7:42 PM | Permalink | Reply to this

Re: Quasicoherent ∞ -Stacks

For more on descent - lots more - see

1. arXiv:1001.1556 [ps, pdf, other]
Title: A general framework for homotopic descent and codescent
Authors: Kathryn Hess

Posted by: jim stasheff on January 15, 2010 7:36 PM | Permalink | Reply to this

Re: Quasicoherent ∞ -Stacks

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By the way, this had already been labified, together with other references and further links, here:

higher monadic descent

I would have enjoyed a brief remark on how this proposal relates to Lurie’s definition of $(\infty,1)$ -monads and the monadicity theorem in this context. Does anyone know?

Posted by: Urs Schreiber on January 15, 2010 8:01 PM | Permalink | Reply to this

Re: Quasicoherent ∞ -Stacks

I’m impressed with how rapidly you integrate new material into the n-Lab, Urs!

I suppose I’m best equipped to say something about how the framework I propose relates to Jacob’s work.
:-)

I hadn’t looked at Jacob’s work on monadicity before yesterday, when I saw the link in your post. I’ve skimmed through it now, and it looks to me as if our approaches are quite different. In particular the concepts I work with are considerably less sophisticated than those Jacob introduces, perhaps because our goals were different.

Most of the results in my paper are stated in terms of monads on simpicially enriched categories such that the underlying endofunctor of the monad is a simplicial functor, and the multiplication and the unit of the monad are simplicial natural transformations. Not terribly sophisticated notions, perhaps, but powerful enough for my purposes. My goal wasn’t to develop the ultimate theory of descent but rather to set up a minimally complicated framework that was general enough to describe a wide range of particular descent theories that are relevant in homotopy theory and their related spectral sequences.

The theory of derived (co)completion that is also in that paper developed naturally in parallel with the descent theory, as it provides the language in which to express the criterion for homotopic (co)descent and to interpret the associated spectral sequences.

I’ll have to think about the relationship between Jacob’s monadicity theorem and my criteria for homotopic (co)descent, since it’s not immediately clear to me.

Posted by: Kathryn Hess on January 16, 2010 10:02 AM | Permalink | Reply to this

Re: Quasicoherent ∞ -Stacks

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Thanks for the reaction, Kathryn!

I’m impressed with how rapidly you integrate new material into the $n$ -Lab, Urs!

This one was joint with Zoran Škoda. He had kindly alerted me of your article when it came out, and then we decided to split off an entry on higher monadic descent from the one on monadic descent that we had been working on earlier, to record this reference and other things we happened to know of.

I suppose I’m best equipped to say something about how the framework I propose relates to Jacob’s work.

Right, i was sort of hoping you would see my comment. :-)

Most of the results in my paper are stated in terms of monads on simpicially enriched categories such that the underlying endofunctor of the monad is a simplicial functor, and the multiplication and the unit of the monad are simplicial natural transformations. Not terribly sophisticated notions, perhaps, but powerful enough for my purposes.

Possibly that captures the fully general setup already if one in addition ensures that the simplicial category that the monad acts on is sufficiently well resolved (e.g fibrant, cofibrant in the pertinent model structures)?

Posted by: Urs Schreiber on January 16, 2010 10:50 AM | Permalink | Reply to this

Re: Quasicoherent ∞ -Stacks

$MathML-enabled post (click for more details).$

Possibly that captures the fully general setup already if one in addition ensures that the simplicial category that the monad acts on is sufficiently well resolved (e.g fibrant, cofibrant in the pertinent model structures)?

Interesting question! The general slogan is that weak functors can be replaced by strict functors between fibrant-cofibrant objects, but weak transformations can’t necessarily be replaced by strict ones. For instance, this is why the Gray tensor product is useful: it’s designed to handle strict functors and weak transformations.

It is true sometimes, though, that there’s also a good notion of “fibrant-cofibrant functor” between which weak transformations can be replaced by strict ones; the most natural way I know of involves a bar/cobar construction. But this requires the target of the functors to be sufficiently complete or cocomplete, in the strict sense, and hence probably also in the weak $(\infty,1)$ -sense. But even if the target $(\infty,1)$ -category is (co)complete, it might be difficult to arrange a strict model for it which is both strictly (co)complete and fibrant. Of course, every topological category is fibrant, but for a simplicial category to be fibrant it it has to be locally Kan, and the prototypical locally Kan simplicial category, namely $Kan$ itself, is not complete or cocomplete in the strict sense.

But possibly there’s some other trick that I don’t know about.

Posted by: Mike Shulman on January 16, 2010 2:56 PM | Permalink | Reply to this

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January 8, 2010