The Volume of a Differentiable Stack

September 15, 2008

The Volume of a Differentiable Stack

Posted by David Corfield

The anticipated paper by Alan Weinstein has arrived:

We extend Baez and Dolan’s notion of the cardinality of a discrete groupoid (equal to the Euler characteristic of the corresponding discrete orbifold) to the setting of Lie groupoids. Since this quantity is an invariant under equivalence of groupoids, we call it the cardinality, or volume in the smooth case, of the associated stack rather than of the groupoid itself. In the smooth case, since there is no natural measure like the counting measure in the discrete case, we need extra data to define the volume. This data has the form of an invariant section of a natural line bundle over the stack. Sections of a square root of this line bundle constitute an “intrinsic Hilbert space” of the stack.

Do you get quantities in all dimensions in these stacks? E.g., are there line elements?

Posted at September 15, 2008 3:24 PM UTC

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Re: The Volume of a Differentiable Stack

This is great! Now: roll on the applications!

Posted by: Bruce Bartlett on September 15, 2008 7:19 PM | Permalink | Reply to this

Re: The Volume of a Differentiable Stack

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I like this paper! What’s more, I think that it really has to do with 2-vector spaces, although Weinstein doesn’t say anything along these lines. I’ve tried to explain this below.

Think about Baez-Crans 2-vector spaces as chain complexes of vector spaces with two terms. I don’t know if these things have ‘exterior powers’ in general, but they certainly have a ‘top exterior power’ given by sending $A_1\to A_0$ to $\Lambda^{top}H_0(A)^*\otimes\Lambda^{top}H_1(A)$ . (This should ring a bell with anyone who got to the bottom of page 7.)

Here’s how to define the volume of a manifold:

1) Define the tangent bundle.

2) Take its top exterior power. This is a line bundle on the manifold.

3) Choose a volume form (a never-zero section of the line bundle) and integrate it.

I think it will be possible to recover Weinstein’s definition by extending each of the above steps to stacks, as follows.

1) You have to define the tangent bundle of a stack. This is easily done: take a Lie groupoid representing the stack and put $T$ in front of everything to obtain a new Lie groupoid that represents the tangent stack. This tangent stack is not a vector-bundle over the original stack, but it is a form of ‘Baez-Crans 2-vector bundle’.

For example, consider a quotient stack $X//H$ . Then the tangent stack is $TX//TH$ , and the fibres of the projection map are quotients of the form $T_x X//\mathfrak{h}$ . Here $\mathfrak{h}$ acts on $T_x X$ according to the map $\mathfrak{h}\to T_x X$ given by differentiating the action. We can think of $TX//TH$ as the $2$ -vector bundle with fibres $\mathfrak{h}\to T_x X$ .

2) Take the ‘top exterior power’ of the tangent $2$ -vector bundle to obtain a line bundle over the stack.

3) The required integration should (I think) make sense if the stack is proper. For manifolds you use a partition of unity to integrate over charts. Proper stacks admit partitions of unity, and locally have the form of global quotients, so hopefully similar reasoning will work.

David, I didn’t understand your question. Can you elaborate?

Posted by: Richard Hepworth on September 16, 2008 12:31 AM | Permalink | Reply to this

Re: The Volume of a Differentiable Stack

Can you elaborate?

Aren’t we seeing here a form of higher geometry, where instead of spaces which are locally vector spaces, we have something locally modelled by 2-vector spaces. So, in the presence of various pieces of extra structure, shouldn’t there be an equivalent of Riemannian geometry with the equivalent of differential forms of all degrees, not just the top degree volume form? And then, could there be an equivalent of a geodesic in such a stack?

Posted by: David Corfield on September 16, 2008 8:45 AM | Permalink | Reply to this

Re: The Volume of a Differentiable Stack

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I’d love to see a definition of ‘Riemannian metric on a stack’, but it’s something I don’t currently know the answer to. What I wrote in the previous comment begs the following question, which others here might be able to help with:

Is there a notion of ‘inner product’ on Baez-Crans 2-vector spaces?

Posted by: Richard Hepworth on September 16, 2008 9:03 AM | Permalink | Reply to this

Re: The Volume of a Differentiable Stack

We were discussing that here and below, and then here and below.

Posted by: David Corfield on September 16, 2008 10:26 AM | Permalink | Reply to this

Re: The Volume of a Differentiable Stack

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Richard, back here you said,

Of course, I haven’t told you what a Riemannian metric on an orbifold (or groupoid or stack) is. You can find definitions in various places, but in the case in point these things are all equivalent to putting a $G$ -invariant metric on $X$ .

But now you say

I’d love to see a definition of ‘Riemannian metric on a stack’, but it’s something I don’t currently know the answer to.

Did you mean in the older post that you only know of the definition of a Riemannian metric on a stack for specific kinds of stack, i.e., ones where discrete $G$ acts on a manifold?

Posted by: David Corfield on September 17, 2008 9:52 AM | Permalink | Reply to this

Re: The Volume of a Differentiable Stack

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David wrote:

Did you mean in the older post that you only know of the definition of a Riemannian metric on a stack for specific kinds of stack, i.e., ones where discrete G acts on a manifold?

Exactly. For etale stacks (which include the orbifold ones) the tangent stack is an honest vector bundle, so there’s no problem defining what a metric is. More generally what you have is a 2-vector bundle, and then who knows?

Posted by: Richard Hepworth on September 17, 2008 10:04 AM | Permalink | Reply to this

Re: The Volume of a Differentiable Stack

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David wrote:

Aren’t we seeing here a form of higher geometry, where instead of spaces which are locally vector spaces, we have something locally modelled by 2-vector spaces?

Yes, precisely.

So, in the presence of various pieces of extra structure, shouldn’t there be an equivalent of Riemannian geometry with the equivalent of differential forms of all degrees, not just the top degree volume form?

I imagine so, but there’s something special about the idea of ‘volume forms’, which isn’t easy (?) to generalize. This has been known for a long time, and Richard alluded to it. If you have finite-length chain complex $V$ of finite-dimensional vector spaces, like this:

$V_0 \rightarrow V_1 \rightarrow V_2 \rightarrow \cdots \rightarrow V_n$

it’s good to form the ‘alternating tensor product of top exterior powers’:

$\Lambda^{top} V_0 \otimes (\Lambda^{top} V_1)^* \otimes \Lambda^{top} V_2 \otimes \cdots$

Why? I think it’s a good idea because this vector space is canonically isomorphic to the alternating tensor product of top exterior powers of homologies:

$\Lambda^{top} H_0(V) \otimes (\Lambda^{top} H_1(V))^* \otimes \Lambda^{top} H_2(V) \otimes \cdots$

Of course both these vector spaces are 1-dimensional, so they’re obviously isomorphic. The point is that they’re canonically isomorphic!

As a result, the ‘alternating tensor product of top exterior powers’ is a very robust invariant of a chain complex. Not merely an isomorphism, but also any equivalence of chain complexes will give an isomorphism between these guys.

I think one buzzword for this circle of ideas is ‘Reidemeister torsion’. It smells like a multiplicative analogue of Euler characteristic. And I know that Alan Weinstein had these ideas in mind when writing his paper.

And then, could there be an equivalent of a geodesic in such a stack?

Before charging ahead, I’d like a definition of ‘inner product’ on a 2-vector space such that any equivalence of 2-vector spaces — not just an isomorphism! — lets you transfer an inner product from one to the other. That’s what’s so good about the ‘volume form’ idea sketched above.

Maybe just an inner product on each homology group? We get something a bit like that in Hodge theory.

Posted by: John Baez on September 18, 2008 11:25 AM | Permalink | Reply to this

Re: The Volume of a Differentiable Stack

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So let’s be horribly concrete and take a baby example of equivalent 2-vector spaces:

$f: \mathbb{R} \to \mathbb{R}, f(x) = 0,$

and

$g: \mathbb{R}^2 \to \mathbb{R}^2, g(x, y) = (0, y).$

What next?

Posted by: David Corfield on September 18, 2008 1:17 PM | Permalink | Reply to this

Re: The Volume of a Differentiable Stack

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So it seems that the answer to my question

Is there a notion of ‘inner product’ on Baez-Crans 2-vector spaces?

is no. Here is my guess.

A metric on a $2$ -vector space $V_1 \to V_0$ is a metric on $V_1$ and a metric on $V_0$ . An equivalence of $2$ -vector spaces with metric is an equivalence of $2$ -vector spaces that induces isometries on homology. A 2-morphism between equivalences is any 2-morphism you like.

Some notes. First, I ought to show that this defines a $2$ -groupoid if it’s going to be of any use. I haven’t done any checking! Second, John’s criterion

any equivalence of 2-vector spaces — not just an isomorphism! — lets you transfer an inner product from one to the other.

is satisfied in a weak sense: there are many metrics on the second $2$ -vector space that make the equivalence into an equivalence of $2$ -vector spaces with metric. I think this is okay. Third, this is only a hair’s breadth from John’s ‘metric on homology’ suggestion, and perhaps the resulting $2$ -groupoids are equivalent. But metrics on homology won’t extend to bundles, since the ranks of the homologies can vary and so don’t define vector bundles.

Posted by: Richard Hepworth on September 18, 2008 3:29 PM | Permalink | Reply to this

Re: The Volume of a Differentiable Stack

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David wrote:

So let’s be horribly concrete and take a baby example of equivalent 2-vector spaces:

$f: \mathbb{R} \to \mathbb{R}, f(x) = 0,$

and

$g: \mathbb{R}^2 \to \mathbb{R}^2, g(x, y) = (0, y).$

What next?

Umm, what do you want me to do next? I would be glad to discuss their top exterior powers, but I’m afraid you want me to invent a concept of ‘inner product’ for 2-vector spaces and figure out how to transfer inner products back and forth using an equivalence between these two 2-vector spaces. I don’t have quite enough energy for the latter…

So, for now, let me just illustrate why the ‘top exterior power’

$\Lambda^{top} V_0 \otimes (\Lambda^{top} V_1)^*$

of a 2-vector space

$V_0 \rightarrow V_1$

doesn’t change under equivalence. Let’s do your example. Let’s see how to get an isomorphism between the top exterior power of this little 2-vector space

$f: \mathbb{R} \to \mathbb{R}, f(x) = 0,$

and the top exterior power of this slightly bigger one:

$g: \mathbb{R}^2 \to \mathbb{R}^2, g(x, y) = (0, y).$

To do this we need to choose an equivalence between the little one and the big one. We can do this by choosing a way to think of the big one as the direct sum of the little one and this one:

$1 : \mathbb{R} \to \mathbb{R}$

This is an essentially trivial 2-vector space. In other words, it’s equivalent to the trivial 2-vector space

$0 : \mathbb{R}^0 \to \mathbb{R}^0$

So: why does adding on an essentially trivial 2-vector space not change the top exterior power?

First, note that when we add vector spaces their top exterior powers multiply:

$\Lambda^{top} (V \oplus W) \cong \Lambda^{top} V \otimes \Lambda^{top} W$

where it’s all-important that this is a canonical isomorphism: just wedge a guy $\Lambda^{top} V$ and a guy in $\Lambda^{top} W$ and you get a guy in $\Lambda^{top} (V \oplus W)$ .

Second: it follows that when we add 2-vector spaces, their top exterior powers multiply… since the top exterior power of

$V_0 \rightarrow V_1$

$\Lambda^{top} V_0 \otimes (\Lambda^{top} V_1)^*$

Third, the top exterior power of an essentially trivial 2-vector space

$V \stackrel{1}{\rightarrow} V$

is canonically isomorphic to $\mathbb{R}$ . Why? Well, it’s this:

$\Lambda^{top} V \otimes (\Lambda^{top} V)^*$

Then, note the tensor product of a 1d real vector space and its dual is canonically isomorphic to $\mathbb{R}$ : just pair a guy in the vector space and a guy in its dual to get a number.

So, we’re done.

(I said something about your ‘horribly concrete’ example, but I had to be more abstract to make it obvious that certain isomorphisms were canonical — that is, basis-independent.)

Posted by: John Baez on September 19, 2008 10:46 AM | Permalink | Reply to this

Re: The Volume of a Differentiable Stack

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Great. That saved me some homework I’d set myself.

But what I don’t see is why the answer to the problem which you knew I was posing - about inner products - should be so difficult, or at least should require the level of energy which you claim to lack.

Is it that you have to delve into perfect tangent complexes, K-theory spectra, etc., as discussed by David Ben-Zvi in his comment?

Really what can be so hard about devising a suitable inner product for

$f: \mathbb{R} \to \mathbb{R}, f(x) = 0?$

(unless, of course, one doesn’t exist).

Is the problem that we don’t know in what a relevant form should take values?

Posted by: David Corfield on September 19, 2008 12:12 PM | Permalink | Reply to this

Re: The Volume of a Differentiable Stack

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I just want to weigh in and say that I believe very few of the things being discussed here have been tackled in any detail. Here is what I know exists:

1) We know what a volume form on a stack is and how to integrate it, thanks to Weinstein’s paper that we’re currently discussing.

2) We know what a tangent stack is: take a Lie groupoid and put $T$ in front of everything. This is well known and appears in Heinloth.

3) We know about Riemannian metrics and vector fields on orbifolds. This is in my paper.

I don’t know of anything else in print, like my suggestion that the tangent stack is a $2$ -vector bundle, David’s question about geodesics …

Posted by: Richard Hepworth on September 18, 2008 3:48 PM | Permalink | Reply to this

Re: The Volume of a Differentiable Stack

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There’s a very general notion of tangent complex to a stack (or higher stack or derived stack etc) which is an object of the derived category (in fact a Lie algebra object in there, in the quasicategorical sense) and one can define the tangent stack as the linear stack associated to this sheaf.

As John explains, if this tangent complex is perfect (in algebraic geometry, this is a condition on being a locally complete intersection) one can consider its determinant line, which is where volume elements live. This is a generalization of the Euler characteristic in the following sense: the tangent complex determines a point in the K-theory spectrum, whose component (projection to $\pi_0$ ) is the Euler characteristic and the determinant line is the projection to the fundamental groupoid ( $\pi_{\leq 1}$ ). (This is explained e.g. in Beilinson’s paper arXiv:math/0610055.)

Given the tangent complex one can do calculus on arbitrary stacks (and their higher and derived brethren) - discuss differential operators, flat connections, characteristic classes, etc. Most typical stacks have perfect tangent complexes, so one can also discuss volume forms, Serre duality etc. Being an algebraic geometer, I don’t know anything about Riemannian geometry in this context though.

On another note, I’m curious if anyone knows a relation between the Baez-Dolan and Weinstein cardinality for stacks and another method of “counting the number of points” on a stack – I’m thinking of Donaldson-Thomas invariants, which are all the rage these days (the latest salvo coming from Kontsevich- Soibelman, building on Behrend and many others) and come from physicists’ attempts to count BPS states (count sheaves of a given type, or points in an appropriate moduli stack). Seems to have a different flavor but would be very interesting to compare (I would guess Behrend’s paper might be the closest point in that literature).

Posted by: David Ben-Zvi on September 19, 2008 5:13 AM | Permalink | Reply to this

Re: The Volume of a Differentiable Stack

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Interesting comments from Beilinson’s paper:

An informal homotopy comment. A peculiar trait of the homotopy theory is the lack of intrinsic language. By default, people resort to a description of the homotopy world as a clever “homotopy localization” of a usual category of rigid objects (topological spaces, simplicial sets, complexes, etc.), referred to as model category. The latter is rather an artificial device, an amber spyglass of the trade, one needs to see objects of the homotopy world. An unsettling quality of this order of things is that all constructions must be performed at the rigidified level, which takes effort and ingenuity, and adds arbitrariness.

Intuitively, a homotopy world is a kind of $\infty$ -category (see Remark below). Lopping off higher homotopies (i.e., replacing the clever localization by a stupid one),one gets a plain category - the homotopy category of the model category. This is a desolate place where no interesting constructions can be performed. The homotopy world is a (yet unnamed) animation of the homotopy category that hovers inbetween the latter and highly non-canonical model categories.

Remark. In [Gr] Grothendieck suggested to perceive homotopy types as $\infty$ -groupoids.6 Unfortunately, a simple intrinsic definition of the concept is not available. Arguably, the common language of category theory may be inadequate for describing the homotopy world.7

6. So the homotopy type of a topological space is its fundamental 1-groupoid whose objects are points, 1-morphisms are paths between points, 2-morphisms are homotopies between paths, etc.

7. A shade of this inadequacy presents already in the plain category theory: while its force lies in the fact that one need not distinguish equivalent categories, it ostensibly asserts that for a (small) category its set of objects is a meaningful notion.

Posted by: David Corfield on September 19, 2008 8:57 AM | Permalink | Reply to this

Re: The Volume of a Differentiable Stack

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David, Thanks for that nice comment from the Beilinson paper! I (and presumably many here) would argue that the “as yet unnamed animation” of the homotopy category has actually been named – it lives in the world of $(\infty,1)$ -categories. Namely to a model category we can assign its Dwyer-Kan simplicial localization, which captures all the interesting information of the model category but is canonical; OTOH one can do in it all the interesting constructions one can’t in the desolate homotopy category..

Posted by: David Ben-Zvi on September 19, 2008 2:22 PM | Permalink | Reply to this

Re: The Volume of a Differentiable Stack

So why not do homotopy theory itself? in its own rite?

Posted by: jim stasheff on September 20, 2008 4:05 PM | Permalink | Reply to this

Re: The Volume of a Differentiable Stack

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Hi Jim,

I would be very interested to hear you elaborate on your question - what kind of homotopy theory do you have in mind? I did have just doing homotopy theory in mind, but we might mean that in different senses?

Posted by: David Ben-Zvi on September 20, 2008 5:21 PM | Permalink | Reply to this

Re: The Volume of a Differentiable Stack

I meant homotopy theory as practiced by working homotopy theorists using the language of spaces, maps, homotopies, higher homtopies, etc. even though the objects might be in any category for which the words cn be defined, e.g. dg modules.

What did you have in mind?
jim

Posted by: jim stasheff on September 21, 2008 2:29 PM | Permalink | Reply to this

Re: The Volume of a Differentiable Stack

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I think we’re in agreement – what I was referring to above, and what Beilinson’s comments address (in my view) are exactly what you say, doing homotopy theory, with mapping spaces, higher homotopies etc, but where the ambient category can be either spaces, spectra, dg modules, complexes of sheaves, stacks, representations, etc..

i.e. homotopy theory as a universal language for algebra and geometry, where the usual categories of spaces and spectra are universal initial examples. (In particular they are the units for the tensor product structure on $\infty$ -categories and stable $\infty$ -categories, respectively.)

Posted by: David Ben-Zvi on September 21, 2008 3:11 PM | Permalink | Reply to this

Re: The Volume of a Differentiable Stack

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doing homotopy theory […] but where the ambient category can be either spaces, spectra, dg modules, complexes of sheaves, stacks, representations, etc..

So I guess this is saying: homotopy theory is working internal to any model category.

Posted by: Urs Schreiber on September 21, 2008 8:04 PM | Permalink | Reply to this

Re: The Volume of a Differentiable Stack

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That’s one interpretation, but Beilinson’s point (and Lurie’s and many others, and, by parroting them, mine) is precisely that this is not an optimal one — rather homotopy theory is working internal to an $(\infty,1)$ -category.. you may choose to describe it by “picking bases” or “picking resolutions”, i.e. by choosing a background model category to work in, but the intrinsic data of the homotopy theory is captured precisely by the corresponding $\infty$ -category.

Posted by: David Ben-Zvi on September 21, 2008 9:03 PM | Permalink | Reply to this

Re: The Volume of a Differentiable Stack

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[model categories are] one interpretation, but Beilinson’s point (and Lurie’s and many others, and, by parroting them, mine) is precisely that this is not an optimal one — rather homotopy theory is working internal to an $(\infty,1)$ -category.

Okay, sure. I have this vague recollection of the relation between moel categories and $(\infty,1)$ -categories. Maybe somebody can remind me:

every model category can be turned into an $(\infty,1)$ -category. Right? And the converse should be easy and well known, I guess.

What is the precise form of the relation between model categories and $(\infty,1)$ -categories?

Also, what’s the precise relation between $(\infty,1)$ -categories and categories enriched in a given model category?

Posted by: Urs Schreiber on September 22, 2008 9:48 AM | Permalink | Reply to this

Re: The Volume of a Differentiable Stack

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Am at a bus stop in Zagreb which happens to have free Wifi.

Reading the above thread in total for the first time (I had been travelling and short of time), I realize that I said some repetitve things due to not having followed the previous discussion. Sorry for that.

But I’d still appreciate if anyone could provide answers to the questions I asked in my last comment on the precise nature of the relation between $(\infty,1)$ -categories, model categories and cats enriched in a model category.

Posted by: Urs Schreiber on September 22, 2008 2:03 PM | Permalink | Reply to this

Re: The Volume of a Differentiable Stack

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I could very well have this wrong, but my impression is that Quillen equivalent model categories give rise to the same

(\infty,1)

category. In other words, the

(\infty,1)

category contains the essential information of the homotopy theory.

Posted by: Aaron Bergman on September 22, 2008 4:02 PM | Permalink | Reply to this

Re: The Volume of a Differentiable Stack

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my impression is that Quillen equivalent model categories give rise to the same $(\infty,1)$ category.

I have to remind myself how this construction of an $(\infty,1)$ -category from a model category works. Is it really the same $(\infty,1)$ -category on the nose for two Quillen-equivalent model categories?

I’ve heard that there should be $(\infty,2)$ -categories both of $(\infty,1)$ -categories (certainly) and of model categories (probably?) and that these two are equivalent as $(\infty,2)$ -categories.

On the other hand, what John recalled of Julia Bergner’s work here involves an $(\infty,1)$ -category of all $(\infty,1)$ -categories instead.

Posted by: Urs Schreiber on September 22, 2008 5:43 PM | Permalink | Reply to this

Re: The Volume of a Differentiable Stack

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As far as I understand, model category structures are more refined than $(\infty,1)$ -category structures: the Dwyer-Kan simplicial localization gives a functor from the former to the latter (maybe we want to be working with simplicial model categories to begin with - this is certainly above my paygrade), but it does NOT induce an equivalence. In other words, I don’t believe it’s true that an equivalence of the underlying $(\infty,1)$ categories can necessarily be lifted to a Quillen equivalence of model categories. In particular to get the $(\infty,1)$ category structure you are certainly not using the cofibrations and fibrations, you are simply inverting weak equivalences to get a simplicial set of homs. For a more authoritative POV look at the recent paper by Blumberg and Mandell, arXiv:0708.0206, where some of these issues are discussed.

Again, this is precisely Beilinson’s point above – choosing a model structure is analogous to choosing a basis, it’s an auxiliary and difficult-to-work-with choice and does not give rise to an equivalent theory.

These differences really matter for applications: there are several fundamental tools of algebra that are now available (thanks to Jacob Lurie and others) in the $\infty$ -categorical setting, but are not available (and I think false or impossible to formulate) in the model world (or of course in the world of homotopy categories). Chief among these is Jacob’s $\infty$ -categorical version of the Barr-Beck theorem, but there are also the lack of an internal Hom in the world of model categories and other such problems. I’m far from an expert in these foundations, but I am applying these tools for geometry and representation theory and I don’t think one can reasonably get the desired applications in the model world.

Posted by: David Ben-Zvi on September 22, 2008 6:01 PM | Permalink | Reply to this

Re: The Volume of a Differentiable Stack

These remarks have helped me a lot…

Posted by: Bruce Bartlett on September 22, 2008 7:33 PM | Permalink | Reply to this

Re: The Volume of a Differentiable Stack

Excellent! Thanks for this information. I obviously have a lot of reading to do…

Posted by: Richard Hepworth on September 19, 2008 9:58 AM | Permalink | Reply to this

Re: The Volume of a Differentiable Stack

Another reference which I learned about recently (actually I learned about it many years ago, but I had to be reminded) is Minhyong Kim’s paper:

MR1355137 (97d:55012) Kim, Minhyong, A Lefschetz trace formula for equivariant cohomology. Ann. Sci. École Norm. Sup. (4) 28 (1995), no. 6, 669–688,

where the natural measure for a groupoid with finite orbit space and finite isotropy groups is introduced explicitly. Perhaps this is relevant to David B-Z’s question about “point counting”.

I’ll soon be uploading a revised version of my preprint with this reference.

Posted by: Alan Weinstein on September 19, 2008 8:11 PM | Permalink | Reply to this

Re: The Volume of a Differentiable Stack

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David B.-Z.,

thanks for the comment you wrote. Here is how I am looking at the situation. I would be interested in hearing how you think about that:

let $S$ be some site whose objects are simple spaces on which we have a notion of differential forms. A good choice for smooth geometry is $S = CartesianSpaces$ .

Let $Spaces$ be the category of sheaves on $S$ .

Fix your favorite notion of $\infty$ -groupoid and $\infty$ -groupoid internal to $Spaces$ .

This should come with a notion of fundamental $\infty$ -groupoid $\Pi_\infty(X)$ of a space $X$ .

Then:

given an $\infty$ -groupoid $C$ internal to $Spaces$ , we get $|C| \in Spaces$ by setting $|C| : U \mapsto Hom(\Pi_\infty(U),C) \,.$

Moreover, for every $X \in Spaces$ we have the DGCA of forms $\Omega^\bullet(X) := Hom(X,\Omega^\bullet(-)) \,.$ So we can form $\Omega^\bullet(|C|) \,.$ This DGCA plays the role of the Chevalley-Eilenberg algebra of the $L_\infty$ -algebroid corresponding to $C$ or, equivalently, the equivariant forms on the space of objects of $C$ .

The category of representations of these $L_\infty$ -algebroid is typically a derived category. I am hoping that this relates this story to the stuff you mentioned.

Posted by: Urs Schreiber on September 20, 2008 10:37 PM | Permalink | Reply to this

Re: The Volume of a Differentiable Stack

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Urs,

What you write is very similar to the point of view in derived algebraic geometry I was referring to (we think of stacks or higher stacks as sheaves of spaces over a category of test spaces, and calculate forms or tangents by totalizing the corresponding notions over patches). There are a couple of differences however.

First, I think it’s important that the tangent complex (the $L_\infty$ algebra you discuss) and the algebra of differential forms (its Chevalley complex) are intrinsic categorical notions, and don’t depend on already having a class of spaces for which you know the definition of differential forms. This goes back to Quillen’s general approach to homology theories [On the (co)homology of commutative rings], where he defines a very general notion of “abelianization” or “linearization” (which agrees as far as I understand with the first derivative in Goodwillie’s general notion of calculus). In particular the tangent complex and the complex of differential forms arise as Andre-Quillen cohomology and Hochschild homology of functions, respectively.

In practice this means what you said: if you write a stack as a simplicial smooth space (or a singular space as a cosimplicial smooth space), you can calculate its tangent complex from taking the tangent bundles to the simplices and taking the chain complex associated to this (co)simplicial abelian group.

The second difference – which is one reason it’s important to have an intrinsic notion of forms and tangent bundles – is that in algebraic geometry one can’t make do with thinking of stacks or even schemes as functors only on analogs of Cartesian spaces – we have to think of them as functors on all affines (or commutative rings). [Presumably the same problem arises in differential geometry if you want to consider not just manifolds but singular spaces?] So even for an affine scheme we have to figure out what the natural notion of forms or tangents is (namely Hochschild homology and the tangent complex/Andre-Quillen cohomology), by representing them by cosimplicial smooth varieties (at least in characteristic zero).

In any case the tangent complex will be a Lie algebra object of the stable $\infty$ -category of sheaves (i.e. $L_\infty$ -algebra), its enveloping algebra is Hochschild cohomology of the structure sheaf (polyvector fields), and its Chevalley cohomology complex or Koszul dual will be the Hochschild homology (differential forms).

Posted by: David Ben-Zvi on September 21, 2008 6:47 AM | Permalink | Reply to this

Re: The Volume of a Differentiable Stack

Andre-Quillen versus Hochschild?

I would say rather Andre-Quillen for commutative associative algebras in any characteristic 0
(= Harrison in characteristic 0) versus
Hochschild for associative algebras in any characteristic 0

Posted by: jim stasheff on September 22, 2008 2:27 PM | Permalink | Reply to this

Re: The Volume of a Differentiable Stack

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Jim,

Please correct me if I get this wrong, but as I understand it you may consider a commutative associative algebra either as a commutative algebra or as an associative algebra, and calculate its (co)homology in either category. In the first you get Andre-Quillen (co)homology, which gives the tangent and cotangent complexes, while in the second you get its Hochschild homology and cohomology. Quillen proves a general version of the Hochschild-Kostant-Rosenberg theorem, which says (with maybe some hypotheses – maybe I want to make sure I’m in a situation with perfect tangent complex, but probably not) that the symmetric algebra of the (shifted) tangent complex is the Hochschild cohomology, and the symmetric algebra of its dual is Hochschild homology (here I just mean as complexes, ignoring higher structures which are also well understood).

in particular differential forms (Hochschild homology of functions) are the symmetric algebra of the shifted cotangent complex, ie Andre-Quillen homology. Does that sound right? (as you say in characteristic zero we can work with the Harrison complex to calculate A-Q, but I think what I wrote holds in any characteristic).

Posted by: David Ben-Zvi on September 22, 2008 5:42 PM | Permalink | Reply to this

Re: The Volume of a Differentiable Stack

What’s more, I think that it really has to do with 2-vector spaces, although Weinstein doesn’t say anything along these lines. I’ve tried to explain this below.

Richard this sounds brilliant. Now that I am in Glasgow and have a chance I will ask John to respond to this ASAP :-)

Posted by: Bruce Bartlett on September 18, 2008 9:37 AM | Permalink | Reply to this

Re: The Volume of a Differentiable Stack

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In thinking about inner products and 2-vector spaces, is it worth remembering that when John wanted to categorify the Hilbert space concept, he took Hilb, the category of Hilbert spaces as a prime example of a 2-Hilbert space? Then $Hom(-,-)$ served as the categorified inner product.

With a KV 2-vector space, $Vect^n$ , we could have the inner product as the space of matrices of linear maps between components.

But the ‘ground field’ of a BC 2-vector space hasn’t seemed enough to generate an interesting inner product.

Posted by: David Corfield on September 17, 2008 4:36 PM | Permalink | Reply to this

Re: The Volume of a Differentiable Stack

I’ve loved Kapranov-Voevodsky 2-vector spaces ever since I learned about the Doplicher-Roberts(-Baez) theorem. Is there any similar ‘killer application’ for Baez-Crans 2-vector spaces? Although I know what they are, I don’t really know what people use them for.

Posted by: Jamie Vicary on September 17, 2008 11:55 PM | Permalink | Reply to this

Re: The Volume of a Differentiable Stack

So there are 2 kinds of 2-vector spaces?
enlighten me!

And there’s a problem with metrics/norms on
2-vector spaces? or only on 2-vector bundles?

Posted by: jim stasheff on September 18, 2008 2:29 AM | Permalink | Reply to this

Re: The Volume of a Differentiable Stack

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At least two!

Baez-Crans (BC) 2-vector spaces are categories internal to $\mathbf{Vect}$ , which makes them also symmetric 2-groups.

Kapranov-Voevodsky (KV) 2-vector spaces are categories ( $k$ -linear) equivalent to $\mathbf{Vect}_k^n$ .

Then there’s Urs’ favourite, something like modules for some monoidal category i.e. not just $\mathbf{Vect}$ , I can’t remember the details, so I’m sure he’ll step in here.

(Also something about categories of bimodules I’m not going to dig for)

Posted by: David Roberts on September 18, 2008 5:59 AM | Permalink | Reply to this

Re: The Volume of a Differentiable Stack

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Concerning the general concept of 2-vector spaces:

one general approach is to say that a 2-vector space is a module category $N$ over some monoidal category $C$ , where both $N$ and $C$ are required to be suitably “linear”, for instance be abelian categories.

Then:

Baez-Crans 2-vector spaces are modules over the monoidal category $Disc(k)$ , the discrete category over the ground field $k$ .

Kapranov-Voevodsky 2-vector spaces are module categories over the monoidal category Vect. But just very, very special such module categories. There is a chain of inclusions of 2-categories $KV2Vect \hookrightarrow Bimod \hookrightarrow 2Vect \simeq Vect-Mod \,,$ which exhibits $KV2Vect$ as the full sub 2-category of Bimod on those algebras which are direct sums of the ground field.

There is something on this in the appendix of QFT from FQFT and also in the section of 2-vector parallel transport in my last article with Konrad Waldorf.

Posted by: Urs Schreiber on September 20, 2008 10:08 PM | Permalink | Reply to this

Re: The Volume of a Differentiable Stack

Can someone rephrase that for an ancient homotopy theorist? e.g. what beyond specifying a graded vector space V= V_0 \oplus V_1 with a morphism V_0 –> V_1
does either school require to say we have a 2-vector space?

same question for an \infty-vector space as
V= V_0 \oplus V_1 \oplus V_2 ….
where now the morphisms V_n –> V_n+1
must compose to 0

Posted by: jim stasheff on September 21, 2008 2:36 PM | Permalink | Reply to this

Re: The Volume of a Differentiable Stack

Urs outlines different versions of 2-vector spaces here.

Posted by: David Corfield on September 18, 2008 8:36 AM | Permalink | Reply to this

Re: The Volume of a Differentiable Stack

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Jamie wrote:

Is there any similar ‘killer application’ for Baez-Crans 2-vector spaces?

We introduced them in order to define Lie 2-algebras (categorified Lie algebras). We showed that every simple Lie algebra gives rise to a 1-parameter family of Lie 2-algebras. Danny Stevenson and Urs Schreiber then helped us find the Lie 2-groups corresponding to these Lie 2-algebras. These 2-groups turn out to be closely related to the math of string theory. Later Urs Schreiber, Jim Stasheff and Hiram Sati figured out the role of these Lie 2-algebras in topology and higher gauge theory, while Danny and I classified 2-bundles with these Lie 2-groups as ‘gauge group’.

I’d say this stuff is the ‘killer app’. More generally, as David hinted earlier, the point of so-called ‘Baez–Crans 2-vector spaces’ is that they show up naturally as the tangent spaces of Lie groupoids, or differentiable stacks.

So, we should definitely try to take all our favorite concepts from differential geometry and boost them up a notch, with these 2-vector spaces taking the place of ordinary vector spaces.

And not just one notch, either! Urs and his collaborators have done a vast amount with $n$ -categorified differential geometry, so I’m only scratching the surface in my summary here.

Posted by: John Baez on September 18, 2008 10:53 AM | Permalink | Reply to this

Re: The Volume of a Differentiable Stack

Is there such a thing as a ‘Lie category’, so we can get the Leinster characteristic into the game?

Is there even a ‘Lie monoid’? Hmm, just a few Google hits.

Posted by: David Corfield on September 18, 2008 11:26 AM | Permalink | Reply to this

Re: The Volume of a Differentiable Stack

A simple example of a Lie category consists of all the linear maps between fibres of a vector bundle. Other examples would be given by the action of Lie monoids on manifolds. (The heat equation flow could be a nice infinite dimensional example.)

In the general definition, I think that one wants the source map to be a submersion, but this need not be true of the target map.

Posted by: Alan Weinstein on September 21, 2008 6:31 AM | Permalink | Reply to this

Re: The Volume of a Differentiable Stack

David almost wrote:

Is there such a thing as a ‘Lie category’?

In my work I call them ‘smooth categories’, but they’re usually called ‘differentiable categories’.

In fact, Ehresmann’s ‘categories differentiables’ were among the first examples that led him to the general theory of internal categories.

For just 25 bucks you can buy this old book:

EHRESMAN, CHARLES. Categories Differentiables et Geometrie Differentielle. 90 pp. Plus errata. Universite De Montreal Department de Mathematiques. 1961. Self wraps, pages stapled at spine, good condition.

Posted by: John Baez on September 23, 2008 1:49 AM | Permalink | Reply to this

Re: The Volume of a Differentiable Stack

So are ‘Lie groupoid’, ‘smooth groupoid’ and ‘differentiable groupoid’ synonymous? All seem to be in use. Three synonyms would seem a little excessive.

Posted by: David Corfield on September 23, 2008 9:34 AM | Permalink | Reply to this

Re: The Volume of a Differentiable Stack

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So are ‘Lie groupoid’, ‘smooth groupoid’ and ‘differentiable groupoid’ synonymous? All seem to be in use. Three synonyms would seem a little excessive.

Usually ‘Lie groupoid’ is taken to be “groupoid internal to $Manifolds$ with some condition on the regularity of source and target map”.

Using the convenient category of smooth spaces one would tend to say that a “smooth groupoid” is one internal to diffeological spaces.

The literature is still not clear about which concept is the “right” one. It all depends on how one looks at $\infty$ -Lie theory: in the end we will want a nice relation between “smooth $\infty$ -groupoids” and $L_\infty$ -algebras and, I think, the right notion of “smooth groupoid” will be the one that makes this work.

I think $\infty$ -groupoids internal to smooth spaces, $\infty Groupoids(SmoothSpaces)$ is a good idea, because in terms of them one can look at $\infty$ -Lie theory as two pairs of functors $\infty Groupoids(SmoothSpaces) \stackrel{\leftarrow}{\to} SmoothSpaces \stackrel{\leftarrow}{\to} qDGCAs \stackrel{\simeq}{\leftarrow} L_\infty Algebroids$ where going from right to left is Lie integration and going from left to right is Lie differentiation.

Using $SmoothSpaces$ , this is general enough to work in general and provide a nice coherent $\infty$ -Lie theory. In concrete special cases we can then ask if a given construction happens to factor through more tame objects in $SmoothSpaces$ along the chain of inclusions $Manifolds \hookrightarrow FrechetManifolds \hookrightarrow SmoothSpaces \,.$

For instance, when we start with an ordinary Lie algebra on the right, then it is a theorem that the 1-groupoid internal to $SmoothSpaces$ we obtain by chasing this from right to left through this diagram is actually the familiar one-object groupoid $\mathbf{B}G$ internal to $Manifolds \subset SmoothSpaces$ with $G$ the familiar simply connected Lie group integrating the Lie algebra.

But already when we start with a Lie 1-algebroid, it is not always true that the smooth 1-groupoid obtained by chasing it from right to left through this diagram is a groupoid internal to manifolds, see Integrability of Lie brackets.

There are proposals, in particular by Chenchang Zhu, to conceive these “Lie” groupoids not internal to manifolds in a stacky fashion. The $\infty$ -groupoids internal to $SmoothSpaces$ can be regarded as certain “rectified $\infty$ -stacks”. So I guess it will all come together in the end. But the jury on that is still out.

Posted by: Urs Schreiber on September 23, 2008 12:35 PM | Permalink | Reply to this

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Re: The Volume of a Differentiable Stack

Presumably given a 2-vector space, we could look for the sub-2-group of the 2-group of transformations which preserves volume (and/or orientation(s)?). Something like an equivalent of the special linear group.

It wouldn’t have to preserve volume in each term of the chain complex, just the overall volume form.

Posted by: David Corfield on September 22, 2008 9:07 AM | Permalink | Reply to this

Re: The Volume of a Differentiable Stack

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The comments are getting a bit too tightly nested up above, so I’ll go down here to tackle Urs’ questions about model categories versus $(\infty,1)$ -categories.

Urs wrote:

Every model category can be turned into an $(\infty,1)$ -category. Right?

Right. The classic version of this result uses the oldest version of $(\infty,1)$ -categories: simplicially enriched categories. These are categories enriched over the category of simplicial sets — or roughly speaking, categories where you’ve got a simplicial set of morphisms between any two objects.

(Simplicially enriched categories were the first and in some ways the simplest approach to $(\infty,1)$ -categories.)

In 1980, Dwyer and Kan found two different ways to turn a model category into a simplicially enriched category. I think these are usually called the ‘Dwyer–Kan simplicial localization’ and the ‘hammock localization’. But, they proved these two methods give weakly equivalent simplicially enriched categories.

Here are the relevant papers. As usual for Dwyer and Kan, they’re frighteningly elegant:

William G. Dwyer and Daniel M. Kan, Simplicial localizations of categories, J. Pure Appl. Algebra 17 (1980), 267-284.
William G. Dwyer and Daniel M. Kan, Calculating simplicial localizations, J. Pure Appl. Algebra 18 (1980), 17-35.
William G. Dwyer and Daniel M. Kan, Function complexes in homotopical algebra, Topology 19 (1980), 427-440.

Urs wrote:

And the converse should be easy and well known, I guess.

Umm, that doesn’t sound right. I think there’s more information in a model category than just its underlying simplicially enriched category.

The reason is that to build the simplicially enriched category we only need the ‘most important portion’ of our model category. We need the very nice objects (those that are both fibrant and cofibrant), we need the morphisms between these, and we need to know which of those morphisms are weak equivalences. So, don’t shoot me if I’m wrong, but I’m pretty sure you can cook up Quillen inequivalent model categories that give weakly equivalent simplicially enriched categories.

Also, I’ve never heard of a way to take an arbitrary simplicially enriched category and promote it to a model category! That sounds tough.

So, I think we should view model categories as a convenient trick for constructing and working with $(\infty,1)$ -categories — but not ‘the same thing’.

I have to remind myself how this construction of an $(\infty,1)$ -category from a model category works.

Again: there are two famous constructions, which give weakly equivalent answers.

Is it really the same $(\infty,1)$ -category on the nose for two Quillen-equivalent model categories?

If by ‘the same on the nose’ you mean equal or isomorphic, the answer is no.

Instead, Quillen equivalent model categories give weakly equivalent simplicially enriched categories.

Posted by: John Baez on September 23, 2008 12:23 AM | Permalink | Reply to this

Re: The Volume of a Differentiable Stack

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Here are the relevant papers.

Thanks. Yesterday night I took the time to read and re-read Julia Bergner’s A survey of $(\infty,1)$ -categories which you pointed to here, and where these things are all mentioned.

Here are a couple of questions I have:

first a remark: I found it noteworthy that the article has the term “ $(\infty,1)$ -category” essentially only in the title and in the introduction, while everything else is a discussion of model categories. The answer given to the question “What is an $(\infty,1)$ -category?” seems to be: “An object in any one of the following specific four Quillen-equivalent model categories.”

(Simplicially enriched categories were the first and in some ways the simplest approach to $(\infty,1)$ -categories.)

Which brings me back to a question I asked before:

it seems to me (but that might be wrong) that the point of simplicially enriched categories is that they are categories enriched in a model category. Namely the model category structure on the category of all simplicially enriched categories (hence the structure which knows that these simplicially enriched categories are supposed to be models for $(\infty,1)$ -categories), as discussed by Bergner on page 5, crucially uses the model category structure of $SimpSet$ Hom-wise.

I am wondering therefore: given a monoidal model category $C$ other than $SimpSet$ , under which conditions does it induce a model category structure on $C$ -enriched categories? And, if $C$ is Quillen equivalent to $SimpSet$ , will the model structure on $C-Cat$ be Quillen equivalent to that on $SimpSet-Cat$ ?

In particular, what is the situation for $C = Sheaves(Manifolds)$ ? After all, the idea seems to be that in general models for $(\infty,1)$ -categories are categories enriched in “spaces”.

And once I am at this poitn: here $C$ -enriched categories are a special case of categories internal to $C$ . What about model category structures on $Cat(Sheaves(Manifolds))$ (categories internal to sheaves on manifolds) and their relation to the model category structure on simplicially enriched categories?

And the converse should be easy and well known, I guess.

Umm, that doesn’t sound right. I think there’s more information in a model category than just its underlying simplicially enriched category.

Thanks, I understand this now, also thanks to David Ben-Zvi’s comments. It seems that the difference is mainly that the information about fibrations and cofibrations is ignored by its “underlying” simplicially enriched category.

My question: is this maybe all the difference?

In other words: I suppose Dwyer-Kan localization works for any category with a system of weak equivalences (not necessarily equipped with a compatible system of fibrations and cofibrations). Is it true then that categories equipped just with systems of weak equivalences are “the same” as simplicially enriched categories?

In yet other words: is the “choice of basis” which David Ben-Zvi says is what passing to a model category is like, is this just the choice of fibrations and cofibrations? Or is it more?

Posted by: Urs Schreiber on September 23, 2008 2:12 PM | Permalink | Reply to this

Read the post Hepworth on 2-Vector Bundles and the Volume of a Differentiable Stack
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Tracked: October 21, 2008 4:40 AM

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