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January 8, 2009

Ben-Zvi on Geometric Function Theory

Posted by Urs Schreiber

guest post by David Ben-Zvi


Dear Café Patrons,

In this guest post I want to briefly discuss correspondences, integral transforms and their categorification as they apply to representation theory, a topic that might be called geometric function theory.

This has been one of the fundamental paradigms of geometric representation theory (together with localization of representations and, on a much grander scale, the Langlands program) for at least the past twenty years (some key names to mention in this context prior to the last decade are Kazhdan, Lusztig, Beilinson, Bernstein, Drinfeld, Ginzburg, I. Frenkel, Nakajima and Grojnowski).

These ideas are closely related to topics often discussed in this Café (in particular groupoidification and extended topological field theories), and my hope is to facilitate communication between the schools by focussing on some toy examples of geometric function theory and suppressing techincal details.

I will conclude self-centeredly by describing my recent work Integral transforms and Drinfeld centers in derived algebraic geometry (arXiv:0805.0157) with John Francis and David Nadler, in which we prove some basics of categorified function theory using tools from higher category theory and derived algebraic geometry. (Of course this is out of all proportion to its role relative to the seminal works I glancingly mention or omit, but it provides my excuse to be writing here, so please indulge me.)

We were motivated by a desire to understand aspects of one of the more exciting developments of the past five years, namely the convergence of categorified representation theory (in particular the geometric Langlands program), derived algebraic geometry and topological field theory, which was at the center of last year’s special program at the IAS.

I will not attempt to describe these developments here but refer readers to collected lecture notes on my webpage.

I would also like to apologize in advance for the highly informal and imprecise style, inaccuracies and mis- or un-attributions. Finally I want to thank Urs Schreiber and Bruce Bartlett for their encouragement towards the writing of this post.

[The guest post continues in this pdf file:]

David Ben-Zvi, Geometric function theory (pdf, 8 pages).

See also the following related entries at the nnLab

- Geometric function theory

- Groupoidification

Posted at January 8, 2009 5:02 PM UTC

TrackBack URL for this Entry:   http://golem.ph.utexas.edu/cgi-bin/MT-3.0/dxy-tb.fcgi/1888

40 Comments & 4 Trackbacks

Re: Ben-Zvi on Geometric Function Theory

Enjoyable post, but you skirted a question I’ve had unresolved for awhile now. In reading Edward Frenkel’s expository paper ‘Lectures on the Langland’s Program and CFT’, he mentions Grothendieck’s functions-sheaves dictionary, and then states that the right analog of functions isn’t the category of sheaves, but that of perverse sheaves. However, he doesn’t really go into why this is the right category, aside from it being stable under Verdier duality.

Since I still don’t feel like I understand the significance of perverse sheaves, this claim has always bothered me. In your mind, what propert(y/ies) of perverse sheaves seems to make them the right analog of functions?

Posted by: Gregory Muller on January 8, 2009 8:30 PM | Permalink | Reply to this

Re: Ben-Zvi on Geometric Function Theory

That’s an excellent question but in a sense orthogonal to the point of my post: to get a geometric function theory (i.e., to allow pushforwards with good properties) you shouldn’t take perverse sheaves (or sheaves for that matter) but the full derived category of sheaves (which in nice situations is also the derived category of perverse sheaves). Perverse sheaves are the analogues of very nice functions, but don’t in themselves form a nice function space.

To me perverse sheaves are the right analogs of nice functions since they correspond (in characteristic zero) to holonomic D-modules - ie to nice (maximally overdetermined) systems of differential equations, which you can think of as categorified substitutes for individual solutions of these systems.
Another answer (in any characteristic) is that perverse sheaves are precisely the coefficient systems for cohomology which obey the Morse index theorem on stratified spaces -i.e., local Morse-theoretic measurements of perverse sheaves (aka nearby cycles) live in a single cohomological degree.

Posted by: David Ben-Zvi on January 8, 2009 11:04 PM | Permalink | Reply to this

Re: Ben-Zvi on Geometric Function Theory

Perhaps it’s helpful (at some vague philosophical level) to point out that the *really* nice functions are locally constant sheaves. Perverse sheaves are the ones where you allow nice singularities.

Posted by: Minhyong Kim on January 9, 2009 12:00 PM | Permalink | Reply to this

Re: Ben-Zvi on Geometric Function Theory

What are the chances of that? The same day I use the phrase ‘stable infinity-category’ (in the Lurie sense), so does another David (p. 7 of his notes).

So to pose the question again, does ‘stable’ here mean the same as the Baez-Dolan use, as in sufficiently far down the periodic table?

Posted by: David Corfield on January 9, 2009 9:25 AM | Permalink | Reply to this

Re: Ben-Zvi on Geometric Function Theory

If you look at page 2 of Lurie’s paper on stable (,1)(\infty,1)-categories, you’ll see the prototypical example is the (,1)(\infty,1)-category of infinite loop spaces. These in turn are stable (,0)(\infty,0)-categories — where now ‘stable’ is used in the periodic table sense.

(Lurie calls (,1)(\infty,1)-categories ‘\infty-categories’, and (,0)(\infty,0)-categories are the same as \infty-groupoids, but I’m trying to clarify a certain pattern here.)

So, the question you’re asking amounts to this: is a thing of stable things a stable thing? More precisely: if you take a nice collection of (,n)(\infty,n)-categories, they naturally form an (,n+1)(\infty,n+1) category. If you take a nice collection of stable (,n)(\infty,n)-categories — where I’m using ‘stable’ in the periodic table sense — do they naturally form a stable (,n+1)(\infty,n+1) category?

Modulo the extreme vagueness of this question, I think the answer is ‘yes’. In fact even plain old categories, not ‘stable’, form a stable 2-category (i.e. a symmetric monoidal 2-category).

So in fact the real question is not whether every (,1)(\infty,1)-category that’s stable in Lurie’s sense is stable in the periodic table sense — I bet it is. The real question is the converse! But for that I’d need to understand his notion better.

Posted by: John Baez on January 9, 2009 2:34 PM | Permalink | Reply to this

Re: Ben-Zvi on Geometric Function Theory

If you look at page 2 of Lurie’s paper on stable (,1)(\infty,1)-categories, you’ll see the prototypical example is the (,1)(\infty,1)-category of infinite loop spaces. These in turn are stable (,0)(\infty,0)-categories—where now `stable’ is used in the periodic table sense.

There are really three different meanings of “stable” going on here.

  1. The periodic table sense, which I think it is better to call “symmetric monoidal.”
  2. The algebraic topology sense, which refers to spectra.
  3. Lurie’s sense, which means a category of objects that are stable in the spectrum sense.

The first two are related but distinct: a connective spectrum (one whose homotopy groups π k\pi_k vanish for k<0k\lt 0) can be identified with a symmetric groupal \infty-groupoid. An arbitrary spectrum is more like a “Z\mathbf{Z}-groupoid.” So I think that to say, as Lurie does, that the prototypical stable (,1)(\infty,1)-category is the category of “infinite loop spaces” is at best misleading, and at worst false, since an “infinite loop space” usually means a symmetric groupal \infty-groupoid (aka an E E_\infty-space).

I don’t really think the question of comparing the first meaning with the third meaning is well-posed, because being symmetric monoidal is extra structure on an nn-category, whereas being stable in Lurie’s sense is a property of it, a limit-colimit coincidence property. A Lurie-stable (,1)(\infty,1)-category has, I believe, all finite coproducts, and so that gives it one symmetric monoidal structure. Likewise it has all finite products, and that should give it another one. But one might also be interested in Lurie-stable (,1)(\infty,1)-categories equipped with an additional symmetric monoidal structure that is not either of these. (For instance, the smash product on the (,1)(\infty,1)-category of ordinary spectra!)

By the way, Lurie’s use of “stable” is not unique to him; other algebraic topologists use “stable model category” with the same meaning.

Posted by: Mike Shulman on March 11, 2009 4:11 PM | Permalink | Reply to this

Re: Ben-Zvi on Geometric Function Theory

I've used ‘stable’ in the periodic table sense, but I still wouldn't say ‘stable nn-category’ but instead ‘stably monoidal nn-category’. This also helps you distinguish ‘stably monoidal’ from ‘stably groupal’.

Although if you disagree, John, … you can say ‘stable (,1)(\infty,-1)-category’ and the 1-1 will make it a stably groupal \infty-groupoid. That is probably too cute for serious use, however.

Posted by: Toby Bartels on March 11, 2009 11:05 PM | Permalink | Reply to this

Re: Ben-Zvi on Geometric Function Theory

A big thanks to David for this seminal post. (I’ve been away for a while, so I missed this one, but I’m just about back now). He is reaching out a hand to bridge cultures here… or in the words of Simon and Garfunkel, “Take my arms that I might reach you” !

Some quick comments:

  • “More prosaically, the Hecke algebra is precisely what acts on the vector space of HH-invariants in any representation of GG”.

    That’s very helpful… perhaps now I’ll finally remember what it is!

  • “One of the great advances in representation theory (due in large part to some of the names listed at the top) was the realization that many of the algebras of greatest interest can be realized as convolution algebras of the form Fun(X× YX)Fun (X \times_Y X) where X,YX,Y are topological spaces or groupoids (in fact algebraic varieties or stacks) and FunFun is a function theory, such as cohomology and KK-theory.”

    That’s great — but sadly it only reinforces my old bias, that “algebra is what you do if you don’t understand the geometry”.

  • “In fact, results of Müger and Ostrik tell us that the Hecke categories Vect(HG/H)Vect(H\G/H) are all Morita equivalent — i.e. they have equivalent 2-categories of module categories. “

    Could you point out the references? They are two authors whose work I have drawn a lot from, so it would help to build bridges.

  • “However on the level of functions the Morita equivalence fails if we replace finite sets by finite orbifolds…”

    I am a bit disturbed by how a phenomenon present at the decategorified level (that the Hecke algebras of functions on double cosets HG/HH \G /H are not Morita equivalent for different HH) is absent at the categorified level (where they are all equivalent). I know this happens from time to time. Is there a deeper explanation here? Doesn’t it on the face of it suggest that the notion of “categorified function” is not sensitive enough?

  • As far as section 5 goes: I’m interested in the cohomology/Euler characteristic line of thought because it is represents something I feel I can understand. I didn’t follow the logic from paragraph 2 to paragraph 3. In paragraph 2 we seemed to conclude we should assign to XX the cohomology of XX as a graded vector space. Then in paragraph 3 we concluded that we should assign to XX a complex of sheaves. Which complex of sheaves is it? The De Rham complex? (Maybe I’m just being silly here, just got back from holiday!)
  • Finally section 6 — the most important part of the post, since your recent work with Francis and Nadler represents a concrete discussion point which we can learn from at the n-category cafe. I got a little lost about the introduction of stable (,1)(\infty, 1)-categories. But I’ll let that go. Mmm… I think I understood the rough idea here in Section 6 but perhaps I am a little hazy on the concrete applications (you mentioned flag varieties and classifying spaces of groups?) I’m obviously happy about the higher characters featuring.
Posted by: Bruce Bartlett on January 13, 2009 8:44 AM | Permalink | Reply to this

convolution algebras

in re: algebras of greatest interest can be realized as convolution algebras of the form Fun(X× YX)

It took me a while to realize that oo-algebras of conciderable interest are oo-convolution algebras, i.e. string field theory algebras regarded as spaces of functions, sections or forms on a space of strings with product or bracket given by integrating over all DEcompositions of a string into two

Posted by: jim stasheff on January 13, 2009 3:07 PM | Permalink | Reply to this

Re: Ben-Zvi on Geometric Function Theory

The Mueger paper we refer to is arXiv:math.CT/0111205, Ostrik’s is arXiv:math.QA/0202130.

I think the fact that Morita equivalence of convolution algebras holds at the categorified level is just a reflection of the fact that vector bundles are more sensitive than functions - the extra structures you get when you categorify are more sensitive to the geometry. Functions on pt/G are uninteresting, but vector bundles on pt/G are representations of G. This is the idea behind Tannakian theory - you can construct a group out of its category of representations by looking at the automorphisms of the forgetful functor, a fact which does not decategorify well.

As for the question on cohomology/Euler characteristics: pushing forward from X to a point we get a complex of sheaves on a point, which is just a complex of vector spaces, or by passing to cohomology (if we’re over C say) just a graded vector space. So as you say, it is exactly the de Rham complex, which (if we forget about things like ring structure) is equivalent to giving de Rham cohomology. The point I wanted to make there is that the “correct” generalization of graded vector space in families is a complex of sheaves..

Over C (or Q), stable (,1)(\infty,1) categories are the same thing as what are known as “pre-triangulated dg categories”. These are categories, enriched over differential graded vector spaces (ie over complexes), whose homotopy category is triangulated. The main examples are any kind of derived category you may run into – the derived category is a triangulated category, but it comes by passing to H 0H^0 from a dg category (where we have a COMPLEX of Homs, not just a graded vector space).

Posted by: David Ben-Zvi on January 14, 2009 12:17 AM | Permalink | Reply to this

Re: Ben-Zvi on Geometric Function Theory

Ok, got the references, I thought it might be those ones, thanks.

As far as cohomology and sheaves go: one of the points I guess is that I must train myself to always think of the cohomology of a space as a complex which just happens to have zero differential. That is, if we think of the complex as a linear nn-category, then “passing to cohomology” means “passing to a skeleton”. That’s chicken-soup for experts, but it was a revelation to me when I first understood it.

Posted by: Bruce Bartlett on January 17, 2009 2:41 PM | Permalink | Reply to this

Re: Ben-Zvi on Geometric Function Theory

I am wondering how much of the general results in your (David B.Z.’s) work with Francis and Nadler depend crucially on the choice of site as being algebraic.

Do you think you can state more abstractly/generally which conditions you need on a site and on class of derived sheaves on it for the main points of your theorem 1.2 to go through?

Posted by: Urs Schreiber on January 13, 2009 5:33 PM | Permalink | Reply to this

formalization of derived infinity-stacks

On p. 14 in the definition of derived stack (I know its recalled from Toëns work, but I’ll ask the following questions here nevertheless): - why is the stack taken to be valued in TopTop here? Which properties of this coefficient category do you actually need later on?

It seems that a general natural setup for derived \infty-stacks would be

- a coefficient object VV being (symmetric) closed homotopical category

- the site S opS^{op} a VV-enriched category

- derived \infty-stacks objects of the Ho VHo_V-enriched homotopy category Ho CHo_C for C:=[S op,V]C := [S^{op},V] or C:=Sh(S,V)C := Sh(S,V).

In his Higher and derived stacks: a global overview I read Toën’s description as something which ought to fit into this template for V=SimpSetV = SimpSet.

I’d very much enjoy an abstract axiomatization of the story of \infty-stacks in such general terms, so that it becomes easier to see which statements actually depend on concrete realizations (e.g. derived algebras, quasi-coherentness, etc.) and which are general phenomena.

Posted by: Urs Schreiber on January 13, 2009 6:07 PM | Permalink | Reply to this

perfect stacks

I need to get a better intuition for those perfect modules and perfect stacks, p. 17.

For instance, how do perfect stacks relate to derived-representable stacks? Do I understand correctly that Lemma 3.4 and prop 3.17, p. 21 say that the latter are all perfect?

The characterization of perfect in prop. 3.8 currently does not help me much.

How can we say “perfect” if we pass from quasi-categories to model theoretic models? Is there a way?

(And finally a general, half-serious question: are technical terms as wonderful and perfect technically meant to imply the superlatives which they are? Was “nice stack” already taken? Is there a good sense in which “a stack can’t be better than perfect”? Maybe a silly question, but I find myself wondering about this terminology…)

Posted by: Urs Schreiber on January 13, 2009 7:49 PM | Permalink | Reply to this

Re: perfect stacks

I have no idea what a perfect stack is, but if there’s any justice in the world, a ‘perfect module’ should be closely related to a perfect ring. There are a bunch of equivalent definitions of a perfect ring: for example, it’s one for which every module MM has a projective covering, i.e. a projective module PP with a morphism PMP \to M that’s an essential epimorphism.

I’m too lazy to define ‘essential’ here — click the link — nor do I deeply understand the point of this concept. But, I think I’ve run into projective coverings while learning about quiver representations. I think that for ADEADE quivers, at least, the irreducible representations have ‘projective covers’ that are indecomposable representations. If you’ve played with quiver representations, that’ll have some emotional resonance. If not, I guess not.

I don’t see how these ideas are related to ‘perfect stacks’, if at all.

Posted by: John Baez on January 13, 2009 10:58 PM | Permalink | Reply to this

Re: perfect stacks

The terminology “perfect stack” comes from the notion of a perfect complex: this is a complex of sheaves that’s locally equivalent to a finite complex of vector bundles. For example on a smooth variety these are all complexes of coherent sheaves. So things that are not perfect are either “big” (quasicoherent sheaves that are not coherent) or “cohomologically big” — the skyscraper at a singular point of a variety is not perfect (its self-Tor is nonzero in infinitely many degrees).

It is straightforward to see that perfect complexes are precisely the dualizable objects of the derived category of sheaves - these are also known as “rigid” in the symmetric monoidal literature. Thomason realized they were also the compact objects in the derived category (for varieties - more generally for quasicompact quasiseparated schemes..)

A perfect stack is one where there are enough perfect complexes — they generate the full quasicoherent derived category. They are the stacks where Thomason’s observation holds. This means that these are the spaces whose derived categories are algebraically manageable.

Our original terminology was “stack with air”, based on a comment by Neeman quoted as our epigraph —- the idea was to convey that this is something so common you might as well take it for granted, like air, but that you’re in bad shape without. The key point is that basically anything you care about is a perfect stack - eg any of the finite orbifolds, any variety, any moduli stack you might run into in characteristic zero, etc etc.

There is no problem formulating perfection on the model categorical level — it’s really a notion on the underlying homotopy category. But one of the main points is one CAN’T do a lot of algebra, in particular many operations we need, in the model setting - (,1)(\infty,1)-categories have precisely the right amount of structure. The results of Lurie’s DAG 2,3 enable one to really perform (non)commutative algebra with these objects in ways that just don’t work on the level of homotopy categories or of model categories.

Posted by: David Ben-Zvi on January 13, 2009 11:23 PM | Permalink | Reply to this

Re: perfect stacks

Maybe it will help to add that for a commutative ring a perfect complex is the same as a finite complex of free modules (up to equivalence). These are precisely the modules that have duals (for such MM there is an M M^\vee and maps 1MM 11\to M\otimes M^\vee \to 1 where 11 is the unit, satisfying a standard cancellation property). They are also the compact objects: the ones for which Hom(M,)Hom(M,-) is colimit-preserving.

Perfect rings (which John mentions) are very different I think. They’re defined by a homological property which is very very strong — usually objects don’t have projective covers in the category but only as pro-objects. Thus all perfect rings are “finite” (they have finitely many simple modules, or in the commutative case, finitely many prime ideals).

I should also say I don’t view perfect stacks as a super fundamental notion - they’re precisely what’s needed to do algebra with categories of quasicoherent sheaves, given the technology we have available. Jacob has promised better technology —- a better notion of monoidal structure on stable (,1)(\infty,1)-categories, which is “cohomologically completed” —- which will make the main results extend easily to basically any geometric stack.

Posted by: David Ben-Zvi on January 13, 2009 11:39 PM | Permalink | Reply to this

Re: perfect stacks

an example or two of

one to really perform (non)commutative algebra with these objects in ways that just don’t work on the level of homotopy categories or of model categories.

without having to read Lurie
or a particular page ref would be welcome

Posted by: jim stasheff on January 14, 2009 1:30 AM | Permalink | Reply to this

(oo,1) vs model

perform (non)commutative algebra with [(,1)(\infty,1)-categories] in ways that just don’t work on the level of homotopy categories or of model categories.

Could you make this more concrete, maybe in an example? Is this about (,1)(\infty,1)-categories which do not arise by localization of categories with weak equivalences?

I need to understand this better, because I seem to keep running into the opposite phenomenon: homotopical categories with directed homotopies (not every homotopy between morphisms is invertible). Passing from these to their (,1)(\infty,1)-categories by localization loses this essential feature and hence I can do more with the homotpical category than its (,1)(\infty,1)-category in these cases.

But maybe I need to be set straight…

Posted by: Urs Schreiber on January 14, 2009 7:49 AM | Permalink | Reply to this

Re: (oo,1) vs model

Let me mention two aspects of why I feel (,1)(\infty,1) categories are much better suited for algebra than model categories. I am far from an expert in either theory, but rather an end-user, but this is what my expert friends confirm. (Morally I think model categories are to (,1)(\infty,1)-categories as vector spaces with basis are to vector spaces: it’s an extra structure that is extremely useful in computations, and might be how things often arise in nature, but to do abstract algebra one is far better behaved - in particular a given basis/model structure won’t be compatible with lots of natural operations. Other than its importance as a tool for calculations, I am not aware of any essential information contained in a model category that is not captured by the localization to an (,1)(\infty,1)-category.)

The first aspect is internal Hom. We would like to have a decent notion of functor category between two given categories – as well as a notion of tensor product of categories etc. One of the best known drawbacks with triangulated categories is that functors between them don’t form a triangulated category. I am told the same problem exists for model categories. This is non-starter for doing any kind of categorified algebra. For (,1)(\infty,1) categories this is not at all a problem - in fact (various classes of) (,1)(\infty,1) categories form closed symmetric monoidal (,1)(\infty,1)-categories — there is an excellent theory of homs and tensor products for such beasts (due to Lurie – there are very detailed reference and exposition in Section 2 of my paper with Francis and Nadler), on which we rely very heavily.

Perhaps a more profound thing to say is that one of the most powerful tools in category theory (as I’m sure readers here know far better than me) is the theory of adjoint pairs of functors, monads and most crucially the Barr-Beck theorem, giving criteria by which an adjunction allows you to identify one category with (co)modules over a (co)monad in the other. This is behind so many of the applications of category theory. Some examples are the theory of descent (or categorical gluing), the formalism of Tannakian categories, the description of abelian categories as modules over the algebra of endomorphisms of a nice generator, etc — many of the places where (co)algebras appear in geometry can be expressed through Barr-Beck.

So the point is, in the (,1)(\infty,1)-categorical world, Lurie proves a Barr-Beck theorem in DAG 2. In the model world, this doesn’t exist (my expert friends tell me it doesn’t seem one could even formulate such a thing). This means that in the \infty-world we can immediately do all of the great things that one is used to doing with Barr-Beck, while in the model world it is incredibly difficult and requires case-by-case arguments to do each of them (if at all).

As a final note, from the viewpoint of an end-user interested in applications, there is now a beautiful well-oiled complete machine (DAG 1-3) for algebra with \infty-categories, which is quite user friendly. (I certainly don’t understand the cogs in the machine but it’s very nice to use). That’s an amazing resource to have, which I’m sure will become absolutely standard.

Posted by: David Ben-Zvi on January 15, 2009 5:07 AM | Permalink | Reply to this

Re: (oo,1) vs model

Hi David,

thanks for the reply!

I should maybe emphasize that I am enjoying your work a lot and that it is great to see how the amazing amount of machinery which Lurie has set up is set to use here in an application very close to my heart.

The kind of questions I am asking here are my way to penetrate the situation, not meant as a criticism on the approach you chose. In parts it is because I am not as versed in the machinery which you use than I should be, in parts it is that I have spent a bit of time developing my own thoughts on some of these matters, some of which I will need to refine, but some of which I am fond of.

So I hope I can use this discussion here to ask a bunch of stupid questions and try to sort out the situation a bit further.

Yes, I think I undertsand what you say about (,1)(\infty,1)-categories and model categories. We’ve talked about this not long ago elsewhere on the blog here.

That question I asked was supposed to be along the lines:

I may have here in my hands a homotopical category (need not be a full model category!) with a notion of left and/or right homotopies between morphisms, where these homotopies are, in general directed. I.e. non-invertible. This directed information must be lost as I pass to the (,1)(\infty,1)-category localization, as, by fiat, an (,1)(\infty,1)-category has only reversible cells in degree greater than 1.

Another thing I am wondering about is this:

when I look at Toën’s impressive body of work, I see that lately he started switching to Segal categories, but that essentially all explicit constructions are done using enrichement over SimpSet and using the model category structure of SimpSet.

In fact, my impression is that Toën’s description of derived \infty-stacks in terms of simplicially enriched techniques fits into the following template, for the case V=SimpSetV = SimpSet

- VV a closed monoidal homotopical category (\to nnLab) ;

- S opS^{op} a VV-enriched site (just Set-enriched in the ordinary case, fully VV-enriched in the derived case)

- an extension of the homotopical structure of VV to something like a local (e.g. stalkwise if SS has enough points) homotopical structure on C:=[S op,V]C := [S^{op},V] or C:=Sh(S,V)C := Sh(S,V) such that CC becomes a VV-enriched homotopical category (\to nnLab).

Then Ho CHo_C is naturally a Ho VHo_V-enriched category. Indeed Ho CHo_C would be a model for the \infty-category of derived \infty-stacks on SS.

It seems that it should be possible to find other realizations of this template. This looks like a decent general picture.

Posted by: Urs Schreiber on January 15, 2009 12:44 PM | Permalink | Reply to this

bundles versus sheaves

It is may be noteworthy that in the Baez-Dolan-Trimble description of groupoidification one pull-pushes generalized bundles of \infty-groupoids through spans, while in Ben-Zvi-et-al. its the sheaves of sections of \infty-vector bundles which are being pull-pushed. (Somewhat morally speaking.)

The difference ought to be just a technical one, in parts due to the general duality between sheaves of sections and bundles, in parts due to the fact that BZ et al consider a structurally vastly more sophisticated setup then BDT have presented so far, in parts due to the emphasis of the algebraic viewpoint in their work

Still, I would like to extract better the common joint abstract nonsense picture encompassing the two realizations (finite groupoids here, derived algebraic \infty-stacks there).

In those notes on “Nonabelian cocycles and their σ\sigma-model QFTs” (\to nnLab) I make a remark on the relation between the picture where we pull push sections and the BDT picture, around p. 15. Not sure if that helps, but something like this should exist.

Posted by: Urs Schreiber on January 13, 2009 9:26 PM | Permalink | Reply to this

Re: bundles versus sheaves

Another difference between the two notions of groupoidification used here is the way the “integral kernel” is encoded. In Ben-Zvi-Francis-Nadler we do

pull-tensor-push

whereas in Baez-Dolan-Trimble we just do

pull-push.

In BZ-F-C a groupoidified linear map is not just a span, but a span with stuff (a coherent sheaf) on its correspondence object.

I am thinking of the way this “stuff” is encoded in the span itself as follows, described for the case of pull-push of ordinary functions.

So let XX be a space (regarded as a category) and let g:XVectg : X \to Vect be the fiber-assigning functor of the trivial vector bundle EXE \to X with fiber the ground field \mathbb{C}.

There are two ways to talk about \mathbb{C}-valued functions on XX then: a) as sections of EE and b) as morphisms from EE to EE. This may seem like a silly distinction on first sight, but both descriptions differ crucially in terms of the abstract arrow-theory which encodes them. In particular, it is the second picture where the fact that functions form a monoid arises from just abstract-nonsense.

The idea is then: when we have a span

XsQtX X \stackrel{s}{\leftarrow} Q \stackrel{t}{\rightarrow} X

and want to pull functions on XX to QQ, tensor them there with a function KK on QQ and then push the result down to functions on XX, the function KK on QQ is secretly actually of nature b) while those on XX are of nature a).

More concretely: a morphism E 1E 2E_1 \to E_2 of vector bundles on QQ comes from a functor

K:Q[I,Vect], K : Q \to [I,Vect] \,,

where I={ab}I = \{a \to b\} is the interval category. If we assume the two endpoints of the interval to always land on the ground field \mathbb{C} then such a KK is precisely a \mathbb{C}-valued function on QQ. It naturally fits into the picture

X×X K X [I,Vect] X g d 0 d 1 g Vect Vect. \array{ && X \times X \\ & \swarrow &\downarrow^K& \searrow \\ X && [I,Vect] && X \\ \downarrow^g &{}^{d_0}\swarrow& &\searrow^{d_1}& \downarrow^g \\ Vect &&&& Vect } \,.

Here d 0,d 1:[I,Vect]Vectd_0, d_1 : [I,Vect] \to Vect are restriction to the two endpoints of the interval.

Now the claim is that the span of groupoids which encodes the integral kernel given by KK by just pull-push (not pull-tensor-push) is obtained by pull back along this diagram of the universal VectVect-bundle EVectVect\mathbf{E} Vect \to Vect and the universal [I,Vect][I,Vect]-bundle E[I,Vect][I,Vect]\mathbf{E}[I,Vect] \to [I,Vect], which themselves are given by the pullbacks

EVect pt pt [I,Vect] d 0 Vect d 1 Vect \array{ \mathbf{E} Vect &\to& pt \\ \downarrow && \downarrow^{pt \mapsto \mathbb{C}} \\ [I,Vect] &\stackrel{d_0}{\to}& Vect \\ \downarrow^{d_1} \\ Vect }

and

E[I,Vect] pt ptId [I,[I,Vect]] d 0 [I,Vect] d 1 [I,Vect]. \array{ \mathbf{E} [I,Vect] &\to& pt \\ \downarrow && \downarrow^{pt \mapsto Id_\mathbb{C}} \\ [I,[I,Vect]] &\stackrel{d_0}{\to}& [I,Vect] \\ \downarrow^{d_1} \\ [I,Vect] } \,.

This yields a span of (discrete in the present example) groupoids

K *E[I,Vect] g *EVect g *EVect. \array{ && K^* \mathbf{E}[I,Vect] \\ & \swarrow && \searrow \\ g^* \mathbf{E}Vect &&&& g^* \mathbf{E}Vect } \,.

Here g *EVectg^* \mathbf{E}Vect is just the fibration over XX which indeed is the vector bundle EXE \to X represented by the fiber-assigning functor gg.

Now, a section of the original vector bundle EE we can think of (by the same kind of reasoning, actually) as a span

Ψ g *EVect, \array{ && \Psi \\ & \swarrow && \searrow \\ \mathbb{C} &&&& g^* \mathbf{E}Vect } \,,

There is one point of Ψ\Psi sitting over 11 \in \mathbb{C} and over each element in the total space of the vector bundle which is in the image of the corresponding section.

Claim: the action of the integral kernel KK over QQ on the section represented by Ψ\Psi is the composition of these spans

K(Ψ) Ψ K *E[I,Vect] g *EVect g *EVect \array{ &&&& K(\Psi) \\ &&& \swarrow && \searrow \\ && \Psi &&&& K^* \mathbf{E} [I,Vect] \\ & \swarrow && \searrow && \swarrow && \searrow \\ \mathbb{C} &&&& g^* \mathbf{E}Vect &&&& g^* \mathbf{E}Vect }

in that the pullback groupoid K(Ψ)K(\Psi) is over 11 \in \mathbb{C} a collection of points over the total space of the trivial bundle EE which, if you take their groupoid cardinality as fiberwise groupoids over \mathbb{C}, produce the section of EE which is the result of the desired pull-tensor-push operation. Only that the “tensor” part has become implicit in the pull-push.

Posted by: Urs Schreiber on January 14, 2009 9:26 AM | Permalink | Reply to this

addition is highly overrated

I’d like to add that I think of the above description of

- the “monoid sitting on the top QQ of the span which representes the integral kernel”

as

- a monoid of endomorphisms of an \infty-bundle pulled back to QQ

as something that corresponds to the observation John emphasized in a recent entry:

In a vein of enquiry into fundamentals, one may wonder what it is that makes us want to describe the dual of a space as a ring or an algebra. How is all that structure of a ring motivated from general abstract nonsense? For instance, the general duality between space and quantity (\to nnLab) just tells us that the dual of a space is some kind of co-presheaf. That this co-presheaf be monoidal to yield a generalized ring/algebra (\to nnLab) is extra input usually invoked by hand.

Where does it come from, naturally? Why is monoidal crucial, while additive is maybe dispensable? As in the slogan

Addition is highly overrated.

It seems to me the answer to the question is: function algebras etc. used as duals for spaces are really to be thought of as bi-branes ( \to nnLab): spans

XQX X \leftarrow Q \rightarrow X

equipped with a transformation between the two pullbacks of some kind of bundle on XX

Q X X g g Vect. \array{ && Q \\ & \swarrow && \searrow \\ X &&\Rightarrow&& X \\ & ^{g}\searrow && \swarrow_g \\ && \infty Vect } \,.

That’s where the monoidal structure comes from which we are used to in algebraic descriptions of geometry. And that’s also where the action of these things on sections comes from, since a section of this \infty-bundle on XX is similarly

X Id pt σ X g g Vect. \array{ && X \\ & \swarrow && \searrow^{Id} \\ pt &&\Rightarrow^\sigma&& X \\ & ^{g}\searrow && \swarrow_g \\ && \infty Vect } \,.

As an amplification of this point, I find it helpful to look at familiar algebras such as category algebras and group algebras etc. from this perspective. They can be elegantly arrow-theoretically be understood in terms of such bi-branes, as described at the end of nnLab: category algebra. I think various not-so-trivial case arise as less trivial examples of this principle, such as monoidal fusion categories.

Finally: why such endomorphisms of \infty-bundles on bi-branes? Answer: I think because these are the natural operations on probes of spaces with an \infty-bundle on it, much again in the general philosophy of space and quantity:

the space XX with background field gg XgVect X \stackrel{g}{\to} \infty Vect is probed by test objects Σ\Sigma by homming into it

[Σ,X][Σ,g][Σ,Vect], [\Sigma,X] \stackrel{[\Sigma,g]}{\to} [\Sigma, \infty Vect] \,, but if we regard the Σ\Sigmas not in isolation but as parts of a “hyperstructure” as Nils Baas would say, namely as sitting in multi-cospans usually called cobordisms, for instance

Σ in out Σ in Σ out \array{ && \Sigma \\ & {}^{in}\nearrow && \nwarrow^{out} \\ \Sigma_{in} &&&& \Sigma_{out} }

then probing our target space + bundle by these naturally yields spans of cocycles

[Σ,X] [Σ,g] [Σ in,X] [Σ,Vect] [Σ out,X] [Σ in,g] [Σ out,g] [Σ in,Vect] [Σ in,Vect]. \array{ && [\Sigma, X] \\ & \swarrow &\downarrow^{[\Sigma,g]}& \searrow \\ [\Sigma_{in},X] &&[\Sigma, \infty Vect]&& [\Sigma_{out},X] \\ \downarrow^{[\Sigma_{in},g]} &\swarrow &&\searrow & \downarrow^{[\Sigma_{out},g]} \\ [\Sigma_in, \infty Vect] &&&& [\Sigma_in, \infty Vect] } \,.

The transformations in the above diagrams arise from this by taking Σ in=Σ out\Sigma_{in} = \Sigma_{out} and Σ=Σ inI\Sigma = \Sigma_{in} \otimes I, where II is the interval object (\to nnLab) in the given context. So these are those “time evolutions in the σ\sigma-model” where no real time evolution takes place, but just some auxiliary transformations (such as T-duality (\to blog) for instance (this was the blog entry where I first mentioned the above bi-brane yoga) or other groupoidily dualities).

Posted by: Urs Schreiber on January 14, 2009 2:18 PM | Permalink | Reply to this

Re: bundles versus sheaves

Urs wrote:

Another difference between the two notions of groupoidification used here is the way the integral kernel is encoded. In Ben-Zvi-Francis-Nadler we do

pull-tensor-push

whereas in Baez-Dolan-Trimble we just do

pull-push.

I noticed this too, and wondered about it. Just for sets there is no difference: In B-D-T you have a span with summit S, and legs X, Y which gives by the universal property of products a map from S to XxY, and the pushforward of the identity on S to XxY gives precisely the integral kernel K that BZ-F-N tensor with.

So in this case you can do everything with integral kernels on the product, instead of arbitrary spans. I suspect that this is true in more general situation, but I haven’t checked that.

Posted by: Maarten Bergvelt on January 15, 2009 4:16 PM | Permalink | Reply to this

Re: bundles versus sheaves

In B-D-T you have a span with summit S, and legs X, Y which gives by the universal property of products a map from S to XxY, and the pushforward of the identity on S to XxY gives precisely the integral kernel K that BZ-F-N tensor with.

Yes, exactly!

I tried to give a formalization of what’s happening here in a comment above: there I indicated how I think we can understand the B-D-T-spans as the spans of total spaces of (higher generalized) bundles which are classified by the endotransformations whose compnents give the (higher generalized) function that BZ-F-N tensor with. (This is the idea I started describing in simple terms in the entry “Exercise in groupoidification” #).

At the risk of sounding weird, I am thinking that this is important for understanding what the role of the quasicoherent sheaves in BZ-F-N is more abstractly, i.e. in a context where we forget that the site we are working over is algebraic. It is the assumption of an algebraic siite which gives us the sheaf AMod AA \mapsto Mod_A for free, so that we might tend not to think about where this really comes from, and where its monoidal structure really comes from.

Posted by: Urs Schreiber on January 15, 2009 4:31 PM | Permalink | Reply to this

Re: bundles versus sheaves

This is “always” true – more precisely, it’s true for any theory which satisfies the projection formula, in particular any reasonable theory of sheaves (or anything I would call a “function theory”). Namely, given a correspondence XZYX\leftarrow Z\rightarrow Y if we push forward the structure sheaf (or fundamental class) of ZZ to X×YX\times Y and use it as an integral kernel this is the same as using ZZ for pull-push. So one formalism is a special case of the other (not all integral kernels come this way, though one might say all “really nice ones” do - in sheaf theory, these are usually called sheaves “of geometric origin”, and there are deep conjectures trying to characterize which things are of this type.)

Posted by: David Ben-Zvi on January 15, 2009 5:50 PM | Permalink | Reply to this

Re: bundles versus sheaves

if we push forward the structure sheaf (or fundamental class) of ZZ to X×YX \times Y and use it as an integral kernel this is the same as using ZZ for pull-push

Ah, thanks for this bit of information! Do you have a reference for this, and for the

deep conjectures trying to characterize which things are of this type

?

Is this known/doable over non-algebraic sites, too?

Posted by: Urs Schreiber on January 15, 2009 9:50 PM | Permalink | Reply to this

Re: bundles versus sheaves

At least for the former, it’s obvious – just draw the diagram – if your product obeys a nice projection formula. And, honestly, shouldn’t your product always obey a nice projection formula?

Posted by: Aaron Bergman on January 15, 2009 10:31 PM | Permalink | Reply to this

Re: bundles versus sheaves

And, as to the latter question,

Is this known/doable over non-algebraic sites, too?

the answer, as David has said, surely depends on whether you have at your disposal a suitably rich category of “nice functions” (where “nice” means possessing pullbacks, push-forwards and products).

Posted by: Jacques Distler on January 15, 2009 10:46 PM | Permalink | PGP Sig | Reply to this

Re: Ben-Zvi on Geometric Function Theory

Thanks, David Ben-Zvi, for writing this post!

I feel bad for not responding to it sooner, since it seems like you’re attempting to build a much-needed bridge between the nn-category crowd and the derived algebraic geometry crowd. I’ve been very busy — and also, I admit, a bit intimidated by the amount of stuff one must understand to truly comprehend the work of:

… Kazhdan, Lusztig, Beilinson, Bernstein, Drinfeld, Ginzburg, I. Frenkel, Nakajima and Grojnowski.

But your blog entry looks quite readable! So I’ll print it, stick it under my pillow, and pull it out whenever I need something to ponder.

I’m actually optimistic that someday I’ll understand some of this stuff, because Jim Dolan and I have been studying algebraic geometry for the last year or so, and it’s starting to bear fruit. Your blog entry will speed up this process.

Posted by: John Baez on January 13, 2009 11:12 PM | Permalink | Reply to this

Re: Ben-Zvi on Geometric Function Theory

I’m starting adding related material to the nnLab.

For them moment I have at nnLab: homotopy limit as an example included details of the computation of the homotopy limit

holim( pt pt BG)G holim \left( \array{ && pt \\ && \downarrow \\ pt &\to& \mathbf{B}G } \right) \simeq G

which appears on p. 3 of David B.-Z.’s intro article.

The free loop space (\to nnLab) example

holim( X Id×Id X Id×Id X×X)ΛX holim \left( \array{ && X \\ && \downarrow^{Id \times Id} \\ X &\stackrel{Id \times Id}{\to}& X \times X } \right) \simeq \Lambda X

is discussed at nnLab: span trace, there with an emphasis on the point of view of ΛX\Lambda X as the homotopy trace on the identity span on XX.

Posted by: Urs Schreiber on January 30, 2009 6:10 PM | Permalink | Reply to this

Re: Ben-Zvi on Geometric Function Theory

Now there are also the details of the computation of

holim( BH BH BG)H\\G//H holim \left( \array{ && \mathbf{B}H \\ && \downarrow \\ \mathbf{B}H &\to& \mathbf{B}G } \right) \simeq H\backslash \backslash G // H at nnLab: homotopy limit.

Posted by: Urs Schreiber on February 2, 2009 1:04 PM | Permalink | Reply to this

Re: Ben-Zvi on Geometric Function Theory

On the train back home I typed a message to Todd Trimble. Since I am in a hurry now I’ll just post a link to the pdf here now:

Letter to Todd Trimble: Spans of monoidal Trimblean ω\omega-categories

This comes from within the context that I am thinking about here but can be read independently.

Gotta run now…

Posted by: Urs Schreiber on January 30, 2009 10:13 PM | Permalink | Reply to this

Re: Ben-Zvi on Geometric Function Theory

Thanks for writing me, Urs – it’s always nice to get letters!

I’m a little more distracted than usual because on top of the usual family commitments, I’m sick right now, and my mind feels a little sluggish. I’ll carve out some time soon though.

Posted by: Todd Trimble on January 31, 2009 12:31 PM | Permalink | Reply to this
Read the post Dendroidal Sets and Infinity-Operads
Weblog: The n-Category Café
Excerpt: Notes from a talk by Ieke Moerdijk on dendroidal sets, with a few remarks on presheaves on the category of posets.
Tracked: February 6, 2009 1:02 AM
Read the post Question on Geometric Function Theory
Weblog: The n-Category Café
Excerpt: On the choice of "nice generalized geometric functions" in a general context of oo-stacks.
Tracked: February 27, 2009 2:34 PM

Re: Ben-Zvi on Geometric Function Theory

David Benzvi and David Nadler have a paper out today – The Character Theory of a Complex Group. Judging from the abstract, there’s plenty to get your teeth into over the Easter break.

Posted by: David Corfield on April 9, 2009 10:49 AM | Permalink | Reply to this

Re: Ben-Zvi on Geometric Function Theory

Wow.

Posted by: Bruce Bartlett on April 9, 2009 11:47 AM | Permalink | Reply to this

Re: Ben-Zvi on Geometric Function Theory

If we were good and put this blog to some good use, we’d start some kind of journal club on this Ben-Zvi categorical TFT stuff, chapter by chapter.

If only the day had more hours. I didn’t even manage to get back to David with comments on his draft… :-/

But okay, since there is still the night, too, maybe I can set up something.

BTW, with a little bit of luck I’ll be able to make it to the Northwestern TFT meeting. Any other Café-dweller who’ll be there? It has the golden speaker list™ concerning \infty-QFT.

Posted by: Urs Schreiber on April 9, 2009 12:07 PM | Permalink | Reply to this

Re: Ben-Zvi on Geometric Function Theory

BTW, with a little bit of luck I’ll be able to make it to the Northwestern TFT meeting. Any other Café-dweller who’ll be there?

Yes!

Posted by: Mike Shulman on April 9, 2009 9:31 PM | Permalink | Reply to this

Re: Ben-Zvi on Geometric Function Theory

Ok, darn it. Not allowing me to laze around, eh? I’ll be in for the journal club studying the Ben Zvi, Nadler et al latest papers. Should we start with the previous one? I suggest you open up a journal club post for that first one, maybe the first two sections. While we’re about it, how about another journal club studying lurie’s work on the cobordism hypothesis?

Perhaps we can run it like a real journal seminar. Ask Jacques to make a ‘journal club’ category on the right hand side of the main page. Interested parties will be asked to prepare a section and explain it in their own words, mailing their entries to urs, David or john. It will be like a normal page with comments except the schedule for the ‘mini section posts” will appear in the main blog entry area.

Posted by: Bruce Bartlett on April 10, 2009 7:19 AM | Permalink | Reply to this
Read the post Journal Club -- Geometric Infinity-Function Theory
Weblog: The n-Category Café
Excerpt: A place to discuss and learn about the work by Ben-Zvi/Francis/Nadler on geometric infinity-function theory and its application in infinity-quantum field theory.
Tracked: April 10, 2009 12:15 PM
Read the post Spans in 2-Categories: A Monoidal Tricategory
Weblog: The n-Category Café
Excerpt: A paper constructing a monoidal tricategory of spans.
Tracked: December 16, 2011 6:33 AM

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