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October 20, 2008

Hepworth on 2-Vector Bundles and the Volume of a Differentiable Stack

Posted by Urs Schreiber

guest post by Bruce Bartlett


Recently, Richard Hepworth gave a seminar at Sheffield:

2-Vector Bundles and the Volume of a Differentiable Stack, (pdf, 9 pages).

Abstract: This seminar is an account of Alan Weinstein’s recent paper The Volume of a Differentiable Stack. I’ll explain that differentiable stacks are a generalization of smooth manifolds and that they crop up in many interesting situations, like the study of orbifolds or the study of flat connections. Just as every manifold has a tangent bundle, every stack has a tangent something, and I’ll explain that the something in question is a bundle of Baez–Crans 2-vector spaces. These 2-vector bundles are often horrible compared with vector bundles, but they still admit a ‘top exterior power’. We’ll see that sections of this top exterior power can be treated just like volume forms on a manifold, and in particular can be integrated to define the volume of a stack.

In other words, Richard’s talk is all about the stuff that we were discussing a few weeks ago here at the n-category cafe when Weinstein’s paper came out. Richard kicked off that discussion by saying that Weinstein’s construction for defining the volume of a stack could be placed into a more conceptual framework:

1) Define the tangent bundle of the stack. It is not a vector bundle over the original stack, but rather a form of ‘Baez-Crans 2-vector bundle’. For more details about tangent stacks, see Richard’s recent paper, Vector Fields and Flows on Differentiable Stacks.

2) Take the ‘top exterior power’ of the tangent 2-vector bundle to obtain a line bundle over the stack. Weinstein considered a line bundle Q A=Λ top(TX) *Q_A = \Lambda^{top}(T X)^*, where AA denotes a Lie algebroid. Richard’s whole point is that this line bundle can perhaps be thought of more conceptually as the top exterior power of the tangent 2-vector bundle!

3) A volume form on the stack is a nowhere-vanishing section of this top exterior power line bundle.

4) Integrate!

So, the game of categorification is alive and well in the world of differentiable stacks.

pic

One thing that interests me: do the vector fields on a differentiable stack form a Lie 2-algebra, and can we gain conceptual capital this way?

Posted at October 20, 2008 9:56 PM UTC

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31 Comments & 0 Trackbacks

Re: Hepworth on 2-Vector Bundles and the Volume of a Differentiable Stack

Neat stuff! Everything is starting to gradually fit together!

By the way, someone wrote:

(link to write up of notes, 9 pages)

It would be nice to actually see the link.

Posted by: John Baez on October 21, 2008 4:32 AM | Permalink | Reply to this

Re: Hepworth on 2-Vector Bundles and the Volume of a Differentiable Stack

Sorry, here is the link.

Posted by: Bruce Bartlett on October 21, 2008 9:25 AM | Permalink | Reply to this

Re: Hepworth on 2-Vector Bundles and the Volume of a Differentiable Stack

Only just came back online. Link is fixed now in the above entry.

(Had trouble with internet connection in Göttingen;. After a busy day at the institute, there was no connection to be obtained in the hotel. I posted Bruce’s entry from under a lamppost on the street where I found an unencrypted connection when walking back. Then this morning there was supposed to be WLAN on the train, but that, too, didn’t work. )

Posted by: Urs Schreiber on October 21, 2008 9:47 AM | Permalink | Reply to this

Re: Hepworth on 2-Vector Bundles and the Volume of a Differentiable Stack

One thing that interests me: do the vector fields on a differentiable stack form a Lie 2-algebra,…

There are very general conceptual reasons for thinking that some Lie 2-algebra is lurking around here.

After all, the Lie algebra of the group of diffeomorphisms of a smooth manifold consists of vector fields on that manifold. And this fact should categorify.

In other words: a differentiable stack should have some sort of infinite-dimensional Lie 2-group of symmetries. And, this Lie 2-group should have a Lie 2-algebra.

So far, so good. But then: if there’s any justice in the world, the above Lie 2-algebra should consist of sections of the tangent 2-vector bundle of our stack!

Of course, we can also just ‘fake it’: write down a plausible formula for a Lie 2-algebra structure, and then see if it works.

A combination of ‘faking it’ and conceptual reasoning is usually the quickest way to tackle this sort of problem.

…and can we gain conceptual capital this way?

The fact that vector fields on a manifold form a Lie algebra has proven its worth. So, even without knowing quite how it will be useful, any sort of categorification of this is bound to be useful, if it’s true.

Posted by: John Baez on October 21, 2008 5:28 AM | Permalink | Reply to this

Re: Hepworth on 2-Vector Bundles and the Volume of a Differentiable Stack

Bruce wondered whether 2-vector fields on a differentiable stack form a Lie 2-algebra. I wrote:

… a differentiable stack should have some sort of infinite-dimensional Lie 2-group of symmetries. And, this Lie 2-group should have a Lie 2-algebra.

So far, so good. But then: if there’s any justice in the world, the above Lie 2-algebra should consist of sections of the tangent 2-vector bundle of our stack!

Eugene Lerman reminds me that Richard Hepworth has figured out how to take a vector field on a differentiable stack MM and get ahold of the ‘flow’ it generates — which is apparently a weak action of \mathbb{R} on MM.

So, vector fields on MM seem to be objects in the Lie 2-algebra of the diffeomorphism 2-group of MM.

This strongly suggests that there is justice in the world. It also suggests that Bruce should ask Richard his question! It seems Richard is perfectly placed to settle it.

Posted by: John Baez on October 24, 2008 4:05 PM | Permalink | Reply to this

Re: Hepworth on 2-Vector Bundles and the Volume of a Differentiable Stack

Hello everyone, thanks for all of the interesting discussion!

If vector fields on a stack are a Lie 2-algebra and stacks come from Lie groupoids, then we’re proposing to construct a Lie 22-algebra from every Lie groupoid. I haven’t written out a Lie 2-algebra yet, but I did work out the underlying groupoid — it’s on page 20 of this paper.

Here’s a very explicit example for everyone to sink their teeth into. Let’s think about a global quotient stack [M/G][M/G]. Here GG is a compact Lie group acting smoothly on a manifold MM. The vector fields on this thing form a groupoid (page 33 of the same paper). Its objects are GG-invariant vector fields on MM. An arrow from vector field XX to vector field YY is a smooth equivariant ψ:Mg\psi: M\to g such that Y=X+α(ψ)Y=X+\alpha(\psi). Here gg is the Lie algebra of GG and α:gTM\alpha: g\to TM is the map determined by the action.

Is this groupoid a Lie 22-algebra? (There’s an obvious vector space structure on objects and arrows, as well as an obvious bracket-type-thing.)

Posted by: richard hepworth on October 24, 2008 5:05 PM | Permalink | Reply to this

Re: Hepworth on 2-Vector Bundles and the Volume of a Differentiable Stack

Hi,

I think the general answer is this:

for GG a Lie \infty-groupoid and gg its L L_\infty-algebroid, the L L_\infty-algebroid of vector fields on GG is inn(g)inn(g), defined to be the L L_\infty-algebroid whose Chevalley-Eilenberg DGCA is the Weil algebra of gg.

For instance if G=XG = X is a Lie 0-groupoid i.e. a manifold, then the Chevalley-Eilenberg algebra is CE(Lie(G))=C (X)CE(Lie(G)) = C^\infty(X) and the Weil algebra is W(Lie(G))=Ω (X)W(Lie(G))= \Omega^\bullet(X), which is indeed =CE(TX) = CE(T X) the Chevalley-Eilenberg algebra of the tangent Lie algebroid of XX.

This stuff plays a big role in many places. For instance the antifields in BV-formalism are really such tangents to the action \infty-groupoid of the gauge group acting on the space of fields.

Posted by: Urs Schreiber on October 24, 2008 5:16 PM | Permalink | Reply to this

Re: Hepworth on 2-Vector Bundles and the Volume of a Differentiable Stack

I propose that we use Richard’s explicit example he gave above as a good way for Urs, Jim and others to explain the bridge between the Chevalley-Eilenberg/L L_\infty approach and the Lie 2-algebra approach.

Let’s recall Richard’s question:

Here’s a very explicit example for everyone to sink their teeth into. Let’s think about a global quotient stack [M/G][M/G]. Here GG is a compact Lie group acting smoothly on a manifold MM. The vector fields on this thing form a groupoid (page 33 of the same paper). Its objects are GG-invariant vector fields on MM. An arrow from vector field XX to vector field YY is a smooth equivariant ψ:Mg\psi : M \rightarrow g such that Y=X+α(Ψ)Y=X+ \alpha(\Psi). Here g is the Lie algebra of G and α:gTM\alpha : g \rightarrow TM is the map determined by the action.

Is this groupoid a Lie 2-algebra?

Let’s take it as reasonably clear that it does form a Lie 2-algebra. I am beginning to see/remind myself (since I once knew!) how it works in the language of Chevalley-Eilenberg algebras.

What is the 3-cocycle corresponding to this Lie 2-algebra? Is it nontrivial? Can it be written down explicitly in terms of the geometry of MM and GG? Does the L L_\infty/Chevally-Eilenberg approach considerably simplify this kind of computation, as opposed to the “pedestrian” approach of a Lie 2-algebra?

Posted by: Bruce Bartlett on October 30, 2008 2:40 PM | Permalink | Reply to this

Re: Hepworth on 2-Vector Bundles and the Volume of a Differentiable Stack

Let’s recall Richard’s question:

Just a quick reply, more details later:

the Chevalley-Eilenberg algebras of L L_\infty-algebroids corresponding to tangent bundles of action groupoids are the familiar BRST complexes, or rather their BV-versions.

Let ρ:gΓ(TX)\rho : g \to \Gamma(T X) be the action of GG with Lie alghebra gg on the manifold XX. Then the action Lie algebroid has CE-algebra given by

d:(fC (X))ρ()(f)C (X)g * d : (f \in C^\infty(X)) \mapsto \rho(\cdot)(f) \in C^\infty(X) \otimes g^* and d:(tg *)[,] *(t *) d : (t \in g^*) \mapsto [\cdot,\cdot]^*(t^*) with [,][\cdot,\cdot] the Lie bracket on gg. This is the “twisted product” of the Chevalley-Eilenberg algebra of gg with functions on XX.

In BRST-language the tts, the cotangents ot the group action, are the ghosts.

The tangent Lie 2-algebroid of that has, as I said somewhere above, as CD-algebra the Weil algebra of the above (the algebra of superfunctions on the shifted tangent bundle T[1](g×X)T[1](g\times X) in the supermanifold thinking).

That has

d:(fC (X))ρ()(f)+df d : (f \in C^\infty(X)) \mapsto \rho(\cdot)(f) + d f and d:(tg *)[,] *(t *)+σt, d : (t \in g^*) \mapsto [\cdot,\cdot]^*(t^*) + \sigma t \,, where σt\sigma t is tt regarded as sitting in the shifted copy g *[1]g^*[1] of g *g^*.

From just requiring nilpotency of dd this uniquely implies the action of dd on the shifted generators dfd f and σt *\sigma t^*, (the Bianchi identities).

(Here σt *\sigma t^* are dual “antighosts” and forms dfd f are dual “antifields”).

So this is the (CE algebra of the) tangent Lie 2-algebroid of the action groupoid, i.e. the “2-differential forms”, if you like. If you want the Lie 2-algebra of sections as an L L_\infty-algebra you should dualize over C (X)C^\infty(X).

Posted by: Urs Schreiber on October 30, 2008 3:20 PM | Permalink | Reply to this

Re: Hepworth on 2-Vector Bundles and the Volume of a Differentiable Stack

Okay, that’s nice, but what I’d like to see out of this is an explicit geometric description of the 3-cocycle which classifies the Lie 2-algebra.

I suppose it has to do with the cohomology of that complex you get from d 2=0d^2 = 0.

But nevertheless, can you give an explicit geometric description of that cohomology class? We’ve got a Lie group action of GG on MM. What is the 3-cocycle, expressed in terms of ordinary geometric invariants? Is there a nontrivial cocycle?

Posted by: Bruce Bartlett on October 30, 2008 3:51 PM | Permalink | Reply to this

Re: Hepworth on 2-Vector Bundles and the Volume of a Differentiable Stack

what I’d like to see out of this is an explicit geometric description of the 3-cocycle which classifies the Lie 2-algebra.

As a caveat I should amplify again that I haven’t really given you the Lie 2-algebra of vector fields – but instead the tangent Lie 2-algebroid! Notice that this has a CE-algebra over the space XX, whereas a Lie 2-alhebra will will have a CE-algebra over the point.

Let me think about the best way to express the Lie 2-algebra of its sections.

Meanwhile, something about cocycles:

the nice thing about the CE-description of L L_\infty-algebroids is that the CE-algebra is already the complex which computes the L L_\infty-algebroid cohomology. Indeed, this is how most people call it! If you are at a conference and get strange looks when you say “CE-algebra of a Lie algebroid” say instead “the complex that computes Lie algebroid cohomology” and people will nod.

Lie algebra cohomology for a Lie algebra gg (with values in the trivial module) is precisely the cohomology of CE(g)CE(g) (if not with values in the trivial module it is the cohomology of the action Lie algebroid of the action of the module).

The classification result of Lie 2-algebras which you mention comes from the observation that every Lie 2-algebra is equivalent to one of the form gV[1]g \oplus V[1] with gg an ordinary Lie algebra and VV a module in degree 1. So the CE-algebra of this has as generators tt those of g *g^* and a single one in degree 2, usually called bb. The differential acts on tt as the CE(g)-differential, and bb it has to send to some degree 3-element

d:bμ=μ abct at bt c 3g *. d : b \mapsto \mu = \mu_{a b c} t^a \wedge t^b \wedge t^c \in \wedge^3 g^* \,. Since d 2=0d^2 = 0 we find that dμ=d CE(g)μ=0 d \mu = d_{CE(g)}\mu = 0 and hence that μ\mu has to be a 3-cocycle on the Lie algebra gg.

Posted by: Urs Schreiber on October 30, 2008 4:08 PM | Permalink | Reply to this

Re: Hepworth on 2-Vector Bundles and the Volume of a Differentiable Stack

Urs wrote:

I think the general answer is this:

for GG a Lie \infty-groupoid and gg its L L_\infty-algebroid, the L L_\infty-algebroid of vector fields on GG is inn(g)inn(g), defined to be the L L_\infty-algebroid whose Chevalley-Eilenberg DGCA is the Weil algebra of gg.

How much of this is theorems? I can’t tell if you’re saying “I’ve solved this problem in much more generality, so don’t bother working on it” or just suggesting what the ultimate general solution might look like. The difference matters, especially for people who might want to work on it.

Posted by: John Baez on October 24, 2008 5:49 PM | Permalink | Reply to this

Re: Hepworth on 2-Vector Bundles and the Volume of a Differentiable Stack

Hi John - at least in the algebro-geometric setting, these statements are all theorems. See in particular Theorem 3.2 of Hinich’s paper “dg coalgebras as formal stacks” which performs exactly this kind of integration in a different language. An even more general picture is explained on p.44 of Toen’s Higher and derived stacks, a global overview — at the time it was presented conjecturally but today I’m quite confident it is fairly straightforward.

In the differential topology context the language will be different but the fundamental concepts are the same: given an infinity-groupoid, or higher stack, one can differentiate it at a point to get an L algebra (or globally as an algebroid). Passing to the solutions of the Maurer-Cartan equation recovers the formal completion of the stack in question. This essentially follows from the Koszul duality between Lie and commutative operads and its ∞-version. The duality is given in one direction by passing to the tangent complex and in the other by formally exponentiating (aka passing to solutions of Maurer-Cartan equations, up to a shift).

Posted by: David Ben-Zvi on October 24, 2008 7:47 PM | Permalink | Reply to this

Re: Hepworth on 2-Vector Bundles and the Volume of a Differentiable Stack

How much of this is theorems?

That depends a bit on what you take take as definition.

David Ben-Zvi kindly recalls the algebro-geometric interpretation of this, which he first pointed out to me somewhere here on the blog back when I was running into this from my perspective in terms of “tangent categories” and “inner automorphism (n+1)(n+1)-groups” and so on, as you will recall.

The theorems he mentions are based on some definitions people made and checked to coincide with the simple DGCA definition. But in the end one can turn things around and take the DGCA perspective as fundamental, I suppose. What one really wants is lots of consistency check that, whatever definition one settles on, it reproduces what it should reproduce in special cases.

And that it does. That what I call inn(g)inn(g) for gg an L L_\infty-algebroid and what much of the rest of the world calls T[1]gT[1]g plays the role of the (shifted) tangent to gg (as the notation would suggest ;-) is taken as tautologous on all conferences with either “Poisson” or “BV” or “BRST” or the like in the title.

It reproduces the low dimensional examples one has and lifts them to familiar higher structures. The BV complex being the most striking example. There is a nice formulation of localizations of integrals on finite diemsnioonal manifolds obtained by passing to the shifted tangent bundle, and BV is just that generalized to \infty.

I am in a hurry now, so I will not give more details right this moment. But I’d be more than happy to discuss this stuff. In fact, I keep trying to do just that, for instance last time at Yet another model ω\omega-question.

before I have to run, let me just end by emphasizing, in case it is not clear, what some of the terms David B.-Z. used are in the language I have been using:

When David says “form the space of Maurer-Cartan elements” of gg, then this is moving from the right to the left through the first step of this diagram, this space is S(CE(g))S(CE(g)), the “smooth classifying space of flat gg-valued differential forms”.

The “fomal exponentiation” then is forming the fundamental nn-groupoid of this space, thereby moving all the way to the left, yielding Π n(S(CE(g))). \Pi_n(S(CE(g))) \,. This is the \infty-groupoid GG integrating gg. So one sees that S(CE(g))S(CE(g)) is a smooth K(G,1)K(G,1) or maybe better K(G)K(G) space.

I am doing all this over test domains which are cartesian spaces, thus setting it up in the smooth context. To pass between that and David’s picture it should be true that we simply replace the site CartesianSpaces with Rings opRings^{op}.

Posted by: Urs Schreiber on October 24, 2008 9:21 PM | Permalink | Reply to this

Re: Hepworth on 2-Vector Bundles and the Volume of a Differentiable Stack

I think this is an important conversation to have. Urs has been trying to stress for a while now the whole \infty-groupoid, L L_\infty-algebra, DGCA, BV, \infty-stack, ω\omega-groupoids side of things. (I write a whole bunch of buzzwords like that mainly because I haven’t grokked these things yet, so I just want to make sure I’ve covered all the bases! If I’ve left one out Urs, please tell me.)

I for one really need to get on board with this picture fast, because it is clear that this body of work has the potential of making so much of what some of us enthuse about at the n-category cafe seem so pedestrian and “old hat” :-) Not all of us mind you, because there are some around here who were involved in inventing this technology!

Imagine how silly we probably look to these people, getting excited about the fact that the tangent bundle of a stack forms a Lie 2-algebra, when it has already been done and dusted and well-understood for \infty-stacks!

I have tried to get on board with the ideas Urs has been pushing already, and I commit myself to learning this technology in the near future.

I think it’s very important that we understand just what kind of things are now well understood in that framework, and what kind of things aren’t.

David Ben-Zvi, would it be possible for you to give us your thoughts on what you feel are the outstanding issues at the moment in the game of “categorification”? I want to avoid working on something and having the nagging feeling that there is a whole bunch of people out there who would regard it as trifling. In fact that is already the case to some extent; when I look at Kevin Costello’s paper for instance my heart sinks when I think of what petty things I have been involved with :-)

John, I’d love to hear some wise words from you about this too. I think an important function of a blog like this is so that we can all get a feeling for what is known, what is not known, and most importantly of all, what people actually want to know.

Posted by: Bruce Bartlett on October 25, 2008 11:59 AM | Permalink | Reply to this

Re: Hepworth on 2-Vector Bundles and the Volume of a Differentiable Stack

Bruce wrote: what people actually want to know

yes, that’s crucial as opposed to developing beyond the point of no return

jim the provocateur

Posted by: jim stasheff on October 25, 2008 2:47 PM | Permalink | Reply to this

Re: Hepworth on 2-Vector Bundles and the Volume of a Differentiable Stack

I strongly agree with Bruce and encourage others here to answer his questions. I was planning to start writing up some work on 2-vector bundles on monday, but maybe I’d be wasting my time.

Or maybe I should start climbing the ladder to infinity? In fact, Urs, is there any accessible reading you can recommend on Bruce’s list of buzzwords? I should gen up before coming to Higher Structures. (David Ben-Zvi has kindly recommended some articles to me already.)

Posted by: Richard Hepworth on October 25, 2008 2:50 PM | Permalink | Reply to this

Re: Hepworth on 2-Vector Bundles and the Volume of a Differentiable Stack

Once I thought “we here” (whoever feels subsumed under this) are, while each gnawing on one different piece of a big bone, involved in the same kind of program.

This program being characterized roughly by these ingredients:

- in a differential geometric setup which is general enough to subsume smooth mapping spaces as well as “discrete” geometries

- understand “higher gauge theory” aka \infty-bundles with smooth connection

- and the extended quantum field theories obtained from these by quantization.

The differential geometric setup we took to be the “diffeological” one, where we look at sheaves over test domains.

The “higher connection” aspect we took to be in terms of parallel transport nn-functors.

I started modelling the “\infty” here on “ω\omega”, since that seems to be both nicely tractable as well as closely related to the tools we need in applications, thanks mainly to the great work by Ronnie Brown and his school on crossed complexes.

On the differential side it was also clear that Lie-\infty-algebras are there. It is mainly counting 1 and 1 together to reformulate the stuff on Lie-\infty theory which exists out there (essentially Sullivan, Getzler, Henriques, Ševera, also Weinstein, Zhu and others) which exists in the literature and which is mainly done in a simplicial setup to the world of ω\omega-categories internal to diffeological spaces. This is just what I am doing.

Posted by: Urs Schreiber on October 25, 2008 5:37 PM | Permalink | Reply to this

Re: Hepworth on 2-Vector Bundles and the Volume of a Differentiable Stack

Bruce,

Thanks for the question. I won’t presume to lecture the n-category cafe about categorification, and there are a lot of questions of interest and important aspects of categorification that are regularly discussed here. I feel though there could be a lot more interaction between the different schools. In particular, a lot of closely related ideas to things discussed here are also studied in the context of algebraic geometry and geometric representation theory, and I feel there’s a language barrier that’s preventing the interaction. I’d be glad to do my part and try to explain e.g. what I think about (arXiv:0805.0157), which deals (albeit it in a somewhat unfriendly language) with issues closely related to topics of interest here (such as groupoidification).

As for questions of interest there are a lot of them coming from geometric representation theory. I think one of the big open areas is giving a structure theory for say semisimple groups acting on categories (usually these are derived categories, so should be treated with some infinity-care). Lusztig in the early 80s categorified the notion of character of a representation [character sheaf - a special G-equivariant sheaf on G], and Boyarchenko-Drinfeld have been developing - in a “toy” setting of unipotent groups - a full representation theory based around this notion. It should go very roughly like this: the 2-category of categorical representations of a group G decomposes according to the structure of the G-equivariant derived category of G. The latter is a braided tensor category, and it decomposes as a “direct integral” of elementary (indecomposable) pieces, which are modular tensor categories. These are called L-packets of representations. Each L-packet contains a collection of “irreducible character sheaves”, analogs of irreducible characters, which interact in a complicated way.

This is very sketchy but I think one of the central goals in geometric representation theory is to have a detailed picture of this nature for semisimple Lie groups and then loop groups. This is in some sense what the geometric Langlands is trying to get at — for example this is the structure behind Frenkel-Gaitsgory’s “local geometric Langlands” conjecture.

These questions also arise from studying 3 and 4 dimensional topological field theories going down to a point or the circle. So of course this relates to other trends of categorifications, extending TFTs further and further down. In fact I believe much of representation theory (over C or R) can be profitably viewed through the lens of extended TFT. However to really impact the questions of interest for representation theory and algebraic geometry, one needs to consider these TFTs in a homotopical or derived context – the categories of interest are not semisimple categories, but complicated derived categories, and the TFT language needs to reflect that. That kind of TFT is brilliantly developed in the Costello paper you quote, and related works of Kontsevich-Soibelman, and now the exciting work of Hopkins-Lurie on the Baez-Dolan hypotheses. But there’s a lot more to do to connect these foundations to geometry and representation theory, and a lot of promising questions. But I’ve been going on too long.

Sorry for the rant!
David

Posted by: David Ben-Zvi on October 26, 2008 2:32 PM | Permalink | Reply to this

Re: Hepworth on 2-Vector Bundles and the Volume of a Differentiable Stack

Sorry for the rant!

No, it’s good. No need to apologize. We need more of that kind of discussion.

(Sorry for this otherwise non-substantive comment, but I will have to run now. More tomorrow.)

Posted by: Urs Schreiber on October 26, 2008 11:14 PM | Permalink | Reply to this

Re: Hepworth on 2-Vector Bundles and the Volume of a Differentiable Stack

Bruce wrote:

John, I’d love to hear some wise words from you about this too.

I guess you’re asking about infinitely categorified geometry?

I think it’s clear we need a theory of weak \infty-categories internal to any sufficiently nice category, including categories that make nice contexts for topology, differential geometry and algebraic geometry. I think it’s clear we need to relate this to a theory of \infty-stacks. It’s crucial that these theories be as simple as possible, and it’s crucial that they be clearly explained — so everyone can use them.

David Ben-Zvi and Urs Schreiber have described aspects of this program quite nicely. It’s a huge program whose consequences will occupy mathematicians for at least a century.

But (on a more personal note) I’m not sure I want to work on it. There are plenty of smart people involved already. You know the list. Their numbers will only increase. Keeping up with them would be a full-time job. I don’t have time. And what would be the point? After all, they’ll eventually sort everything out, regardless of what I do. What’s the point in chasing a train that’s already pulled out of the station, just to push it along faster? I’d rather think about other stuff — weird stuff that nobody cares about except me. That’s why I got into nn-categories in the first place.

Posted by: John Baez on October 27, 2008 3:57 AM | Permalink | Reply to this

Re: Hepworth on 2-Vector Bundles and the Volume of a Differentiable Stack

What’s the point in chasing a train that’s already pulled out of the station, just to push it along faster? I’d rather think about other stuff — weird stuff that nobody cares about except me. That’s why I got into nn-categories in the first place.

John (and James Dolan), you will always be the spiritual foundation of my viewpoints. No-one can explain it like John! Insert song here: The only one who could ever reach me, was the son of a preacher man.

Once I drove through Las Vegas, where there really is just one game in town: gambling. I stopped and took a look. I saw the big fancy casinos. I saw the glazed-eyed grannies feeding quarters into slot machines, hoping to strike it rich someday. It was clear: the odds were stacked against me. But, I didn’t respond by saying “Oh well - it’s the only game in town” and starting to play.

Instead, I left that town.

Posted by: Bruce Bartlett on October 29, 2008 1:02 PM | Permalink | Reply to this

Re: Hepworth on 2-Vector Bundles and the Volume of a Differentiable Stack

Thanks for the kind words, Bruce. I’m in a foul mood, and can use all the cheering up I can get. I have three papers that absolutely need to be finished soon — but I can’t seem to find the time, because I’m teaching three courses and looking after 5 grad students. All I really want to do is think about new things, but I’m stuck under all these commitments. So, I’m starting to dream of retiring early and spending my autumn years playing music, writing This Week’s Finds and tending the garden.

It should get a bit better after this quarter ends.

Posted by: John Baez on October 30, 2008 6:00 PM | Permalink | Reply to this

Re: Hepworth on 2-Vector Bundles and the Volume of a Differentiable Stack

I apologize for taking so long to respond here. David, thanks a lot for explaining your viewpoints.

I should say that I certainly noted Integral Transforms and Drinfeld Centers in Derived Algebraic Geometry when it came out, and I remember getting to about page 4 or so and thinking “Wow this is all so related to n-category cafe stuff and what I’ve been doing” before I ran out of steam.

I first heard about “Luztig’s character sheaves” when I read the paper on 2-characters by Ganter and Kapranov, and also from Petro Polesello at his talk at the 2-groups workshop in Barcelona.

The programme you have outlined is great and certainly very close to the stuff I’ve been doing; the main difference of course being that I’ve been working in the “trivial” case of group actions on semisimple categories and their 2-characters, mainly because Chern-Simons theory has been my spiritual home. But I hope to try and bridge the gap soon.

I should also say that, from perusing the seminar schedule at the University of Texas at Austin, there have been some really interesting talks lately!! It would have been great for instance to hear what Freed had to say in his talk Rozansky-Witten, and I wonder what Rozansky spoke about in Rozansky-Witten down to points.

However to really impact the questions of interest for representation theory and algebraic geometry, one needs to consider these TFTs in a homotopical or derived context – the categories of interest are not semisimple categories, but complicated derived categories, and the TFT language needs to reflect that.

I agree that this is a crucial point and I have been won over in this regard; not in the least part because Simon Willerton my supervisor has been involved in this stuff (I and II).

I think Urs and John will be able to say more on this point, but I have roughly noticed at least five groups of people involved in the categorification game, coming from different traditions:

  • “Baez-Dolan” n-category cafe people.
  • Homotopy theory people.
  • Algebraic geometry people.
  • Symplectic geometry, Poisson geometry, Lie groupoids people.
  • Khovanov homology people.

I know I, um, speak for all of us when I say that none of us are guilty of not knowing enough about what our neighbours are doing… it’s those jolly people over the fence who don’t know what we’re doing! Seriously though, we need to communicate more.

Perhaps we could ask David to make some guest posts explaining the context and results from his paper to an n-category cafe audience. On the other hand, perhaps I am the only (cough!) n-category cafe patron who is not yet up to speed on homotopical algebraic geometry!

Posted by: Bruce Bartlett on October 29, 2008 12:54 PM | Permalink | Reply to this

Re: Hepworth on 2-Vector Bundles and the Volume of a Differentiable Stack

Now I have the time to give a more detailed reply to the question

How much of this is theorems? #

It seems there is some room for interpretation what exactly “this” refers to, so I will say various things and ask you to get back to me with further questions in case I left out soemthing you wanted me to comment on.

So, I’ll talk about this diagram

and what I know about it for sure and what I expect to be true about it and what evidence I have.

I am setting this up in a smooth \infty-context using the model: Spaces=Sheaves(CartesianSpaces) Spaces = Sheaves(CartesianSpaces) and Groupoids=ωGroupoids(Spaces). \infty Groupoids = \omega Groupoids(Spaces) \,. Notice that an ω\omega-groupoid internal to spaces is equivalently a sheaf with values in ω\omega-categories ωGroupoids(Spaces)Sheaves(CartesianSpaces,ωGroupoids(Sets)), \omega Groupoids(Spaces) \simeq Sheaves(CartesianSpaces, \omega Groupoids(Sets)) \,, which makes the contact to the stacky point of view that started this thread: a “smooth \infty-groupoid” in this context, i.e. an ω\omega-groupoid internal to SpacesSpaces is a rectified \infty-prestack, if one insists on this point of view. This is sticking with our strategy to work internal to diffeological spaces, which, I think, is a good idea. Many things are treated as stacks for which this is overkill. I think \infty-Lie theory is such a case. But whoever reads this here and wants to have proper stacks instead of prestacks is free to apply stackification to the following statements.

Finally, L L_\infty, the category of L L_\infty-algebroids, is, by definition, the category dual to the category of qDGCAs (quasi-freee DGCAs): those coming from a chain complex gg of C (X)C^\infty(X)-modules as A=( C (X) g *,d)A = ( \wedge_{C^\infty(X)}^\bullet g^*, d). We think of these as Chevalley-Eilenberg-algebras of the corresponding L L_\infty-algebroids, which yields the contravariant functor CE:L qDGCACE : L_\infty \to qDGCA.

The above diagram arose as the result of my attempts to put into a single systematic picture the existing approaches to \infty-Lie theory which are out there. For the direction from right to left this is

a) Sullivan’s age old idea underlying rational homotopy theory

b) Getzler’s re-interpretation of that in terms of Lie theory (as summarized here).

Sullivan and Getzler take a DGCA AA and form the simplicial set/space A\langle A \rangle whose set/space of nn-simplicies is the set/space of DGCA maps from AA to forms on the standard nn-simplex:

A n:=Hom DGCA(A,Ω (Δ n)). \langle A \rangle_n := Hom_{DGCA}(A, \Omega^\bullet(\Delta^n)) \,.

All I observed is that it may be helpful understand this, trivially, as a result of a 2-step process:

first we form S(A)SpacesS(A) \in Spaces given by S(A):UHom DGCA(A,Ω (U)) S(A) : U \mapsto Hom_{DGCA}(A, \Omega^\bullet(U)) and then

second we take the fundamental \infty-groupoid of that space. For Getztler \infty-groupoid means Kan complex and fundamental \infty-groupoid means singular simplicial complex. Interpreting Π (S(A))\Pi_\infty(S(A)) this way and using the Yoneda lemma produces the above expression A=Π (S(A)). \langle A\rangle = \Pi_\infty(S(A)) \,.

In my setup I use a different model for \infty-categories, namely ω\omega-categories, and hence a different model for the fundamental \infty-groupoid, namely Π ω(X)\Pi_\omega(X), the strict smoot fundamental \infty-groupoid whose kk-morphisms are thin-homotopy classes of smooth images of the standard kk-disk in XX. So I form instead Π ω(S(A)) \Pi_\omega(S(A)) and its truncations Π n(S(A)) \Pi_n(S(A)) obtained by taking equivalence classes of nn-morphisms.

This is just a technicality, but I state it to make clear in what precisely the results are which I know. I think this is nice, because it produces results in a form which is close to stuff familiar in differential geometry and physics, largely because in the background we have Brown-Higgins theorem which says that Π ω(X)\Pi_\omega(X) is always equivalently encoded in a crossed complex of groups.

Okay, so here are the first results:

Let gg be an ordinary integrable Lie algebroid. Write CE(g)CE(g) for its Chevalley-Eilenberg DGCA (“the DGCA which computes Lie algebroid cohomology”). Then:

Theorem: Π 1(S(CE(g)))\Pi_1(S(CE(g))) is precisely the Lie groupoid integrating gg that is discussed in the literature, nicely summarized by Cranic and Fernandes.

In particular, if gg is an ordinary Lie algebra then Π 1(S(CE(g)))=BG\Pi_1(S(CE(g))) = \mathbf{B}G is the one-object 1-groupoid coming from the simply connected Lie group GG integrating GG.

(Here by “is precisely” I mean: the groupoid internal to sheaves which one obtaines is representable by the groupoid internal to manifolds that Crainic-Fernandes consider.)

This is proven by simply unwrapping (or maybe: “wrapping” ;-) what they are doing and checking that it amounts to precisely the computation of Π 1(S(CE(g)))\Pi_1(S(CE(g))), using the well known equivalences between Lie algebroid morphisms and the dual DGCA maps.

If the Lie algebroid gg is not integrable, then Π 1(S(CE(g)))\Pi_1(S(CE(g))) still exists as a Lie groupoid internal to Spaces, but now it is no longer living in ManifoldsSpacesManifolds \hookrightarrow Spaces. We are precisely in the situation then that Chenchang Zhu considers in her articles #. I think it is clear that she essentially takes Π 1(S(CE(g)))\Pi_1(S(CE(g))) and stackifies its pre-stack of morphisms. I have not taken the time to write that statement up though, so you may want to feel cautious here.

Next there are the following integration results using this strict ω\omega-categoriecal integration

Theorem

1) For g=b n1u(1)g = b^{n-1} u(1) the L L_\infty-algebra of (n1)(n-1)-fold shifted U(1)U(1) we have Π n(S(CE(b n1u(1))))=BB n1. \Pi_n(S(CE(b^{n-1} u(1)))) = \mathbf{B} \mathbf{B}^{n-1}\mathbb{R} \,.

(This is useful as an ingredient in many other computations. In particular, this facilitates understanding how \infty-Lie integration relates the model structures on DGCAs and ωGroupoids(Spaces)\omega Groupoids(Spaces) #)

For g μ 3g_{\mu_3} the weak String Lie 2-algebra with normalized canonical 33-cocylce μ 3\mu_3, we have Π 2(S(CE(g μ 3)))BString BCSS(G)BString Mick(G). \Pi_2(S(CE(g_{\mu_3}))) \simeq \mathbf{B} String_{BCSS}(G) \simeq \mathbf{B} String_{Mick}(G) \,. Here in the middle is our strict model of the String 2-group, on the right is a similar model using Mickelsson’s cocycle and the equivalences are 2-anaequivalences of strict 2-groupoids.

This is a joint result with Danny Stevenson. It shows among other things that the Π ω\Pi_\omega-integration procedure reproduces correctly the strictified versions of the known results, in particular with regard to André Henriques’ work. But moreover, this provides the starting point for eventually understanding Brylinki-McLaughlin’s construction of Čech cocycles as the computation of the obstruction of lifting GG-bundles String-2-bundles. #

These examples (which in one way or other are known in the literature) serve the check that going from right to left through the above diagram is really \infty-Lie integration in that it reproduces in low dimensions what it ought to reproduce.

Next we want to understand the general \infty-Lie theory and then the differentiation procedure, to assign to each Lie nn-groupoid its tangent Lie nn-algebra.

So. One of the points of my 2-step reformulation is to make manifest that it consists of what should be two consecutive adjunctions.

If you look at the diagram above, you see that the step on the right is obtained by homming into/out of the “object of infinitesimal paths”, namely Ω :=UΩ (U)=CE(TU)\Omega^\bullet := U \mapsto \Omega^\bullet(U) = CE(T U), the tangent Lie algebroid.

Thanks to what Todd Trimble taught us, we know that the pair of maps obtained by such an “ambimorphic object” (deprecated original term: “schizophrenic”) constitutes an adjunction, being a generalization of Stone duality.

Now, I observe that the operation on the left also comes from homming into/out of an ambidestrous object, namely the “object Π ω\Pi_\omega of finite” paths.

I think therefore that also the left pair of maps is an adjunction. This is kindergarten stuff for people like Todd.

Another thing which I have not shown but which I think is true, is that given these two adjunctions now, the process of going from left to right through this diagram is essentially what Ševera proposed in L L_\infty-algebras as jets of simplicial manifolds.

Again, I can check the \infty-Lie differentiation process in low dimensions explicitly to see that it gives the right result

Theorem

1) For XX the discrete groupoid over space, we have Ω (K(X))=Ω (X)=CE(TX). \Omega^\bullet(K(X)) = \Omega^\bullet(X) = CE(T X) \,.

(So we do reproduce the Lie algebra of vector fields of ordinary spaces).

2) For GG a Lie group, we have Ω (K(G))=Ω (S(CE(g))). \Omega^\bullet(K(G)) = \Omega^\bullet(S(CE(g))) \,. (So we do reproduce the Lie algebra of a Lie group up to going once back and forth through the adjunction).

2) For GG a strict Lie 2-group coming from the differential crossed module gg, we have Ω (K(G))=Ω (S(CE(g))) \Omega^\bullet(K(G)) = \Omega^\bullet(S(CE(g))) (so, again, we do reproduce 2-Lie differentiation up to going once back abd forth through the adjunction).

The proof of 1) is obvious, 2) and 3) are reformulations of theorems we proved in the context of parallel 2-transport #. These proofs follow an obvious pattern and it is clear how to continue them to higher dimensions. But this hasn’t been written up.

Also, the published proofs consider this just for Lie nn-groups, not for Lie nn-groupoids (though we do have material on those which didn’t make it our paper). But I think the proofs go through pretty much verbatim also for the groupoid case. This, too, needs more attention.

The fact that we don’t get the Lie nn-algebra but something “larger” (forms on the classifying space) back from this general-nonsense way to \infty-Lie differentiation is a familiar phenomenon, already from rational homotopy theory, and appearing also in Ševera’s work. In rational homotopy theory the central theorem says that the canonical injection AΩ (S(A)) A \hookrightarrow \Omega^\bullet(S(A)) is a weak equivalence. I haven’t even tried to prove the analog of this for the context we are talking about (in rational homotopy theory this is a pretty nontrivial thing). But given that everything here is just a reformulation of basic ideas of rational homotopy theory, really, this is one statement which ought to be true. So this is a weak spot at the moment from the general point of view of \infty-Lie theory. But as far as I know, this is just as open in the formulation by Ševera (at least last time I checked, we’ll see how much progress there has been meanwhile).

Finally, I think I can prove the following, which is not tto hard:

Theorem: For XX a concrete space (i.e. a diffeological space) we have K(Π ω(X))=XK(\Pi_\omega(X)) = X.

So on concrete spaces the left adjunction yields an isomorphism. (Notice however that all the spaces of the form S(CE(g))S(CE(g)) are not concrete.)

Posted by: Urs Schreiber on October 27, 2008 3:33 PM | Permalink | Reply to this

Re: Hepworth on 2-Vector Bundles and the Volume of a Differentiable Stack

jargon again

what is an example of a space that’s not concrete?

Posted by: jim stasheff on October 30, 2008 2:29 PM | Permalink | Reply to this

Re: Hepworth on 2-Vector Bundles and the Volume of a Differentiable Stack

jargon again

what is an example of a space that’s not concrete?

Here “Space” is jargon for “sheaf on a site such as manifolds” and “concrete space” is jargon for the jargon “concrete sheaf”, or for “quasi representable sheaf”: a sheaf which is not necessarily quite representable, but which still has an underlying set X sX_s, in that the set it assigns to a test domain UU is naturally a subset of maps of sets from UU to X sX_s.

Details are here.

Examples for non-concrete spaces are smooth classifying spaces of flat L L_\infty-algebra valued forms: those sheaves which to test domain UU assign the set of flat forms with values in an L L_\infty-algebra gg. S(CE(g)):UΩ flat (U)=Hom DGCA(CE(g),Ω (U)). S(CE(g)) : U \mapsto \Omega^\bullet_{flat}(U) = Hom_{DGCA}( CE(g), \Omega^\bullet(U) ) \,. In particular, since all these forms are at least 1-forms, there is a single such form on the point U=ptU = pt. Accordingly all the spaces of the form S(CE(g))S(CE(g)) have a single concrete point. But still they may have many higher-dimensional curves.

For instance the space S(CE(b 4099u(1))) S(CE(b^{4099}u(1))) (otherwise known as the space of closed 41004100-forms) has

- a single point

- a single curve

- a single surface

- a single kk-dimensional hypersurface for all k<4100k \lt 4100.

- lots of hypersurfaces of dimension 4100.

So this space cannot be a topological space equipped with a smooth structure of sorts. It’s not that concrete. Still, it qualifies as a smooth space for many purposes. So in our jargon it’s a non-concrete space.

Posted by: Urs Schreiber on October 30, 2008 4:19 PM | Permalink | Reply to this

Re: Hepworth on 2-Vector Bundles and the Volume of a Differentiable Stack

Thanks for the detailed reply, Urs. I only partially understand it, but it’s a beautiful picture. I dream of having time to learn this stuff well enough to work on it. I’m not sure I ever will, and that makes me very sad, because this stuff — or something similar — clearly lies at the heart of 21st-century geometry.

Here’s a question or two.

You say you’re using ‘ω\omega-groupoids’. By this do you mean strict ω\omega-categories with strict inverses at all levels? One could also use strict ω\omega-categories with weak inverses.

My worry is that either way, these are ‘not sufficiently general’ — for example, ignoring ‘smooth’ aspects for now, not sufficiently general to model arbitrary homotopy types.

I know you’ve already pondered this. I guess you know that in 1991 Kapranov and Voevodsky once wrote a paper claiming that globular strict ω\omega-categories with weak inverses were sufficiently general to model arbitrary homotopy types. I guess you know that Carlos Simpson wrote a paper analyzing and in some sense disproving this claim.

Now, it’s easy to see why you’d prefer globular strict ω\omega-categories to Batanin’s globular weak ω\omega-categories, at least in the short term: even though they’re ‘insufficiently general’, they’re a lot simpler to understand and work with. So, they make a good testing ground for ideas that should be weakened later.

But if you want to think about \infty-groupoids, I don’t see why you’d prefer globular strict ω\omega-categories with strict or weak inverses to Kan complexes. The former are probably not sufficiently general; the latter clearly are sufficiently general. And, Kan complexes are probably easier to work with — at least after one learns all the techniques people have developed for them.

In short, I see Kan complexes and globular strict ω\omega-categories as good contexts for doing something less general than full-fledged \infty-category theory. But globular strict ω\omega-categories with weak or strict inverses seem strictly worse (pardon the pun) than Kan complexes.

Of course other considerations may push you towards them…

Posted by: John Baez on October 29, 2008 5:11 PM | Permalink | Reply to this

Re: Hepworth on 2-Vector Bundles and the Volume of a Differentiable Stack

Thanks for your message!

You say you’re using ‘ω\omega-groupoids’. By this do you mean strict ω\omega-categories with strict inverses at all levels?

Yes, exactly. I am trying to follow the standard terminology, in particular Street’s original use of ω\omega-category as strict \infty-catgegory in his “Algebra of oriented simplices” and Ronnie Brown’s and his school’s use of ω\omega-groupoid for the \infty-categorical things equivalent to crossed complexes.

Further, I adopted the habit of saying weak ω\omega-groupoid for an ω\omega-category (ω\omega=strict composition!) in which each cell is an ω\omega-equivalence.

My worry is that either way, these are ‘not sufficiently general’ — for example, ignoring ‘smooth’ aspects for now, not sufficiently general to model arbitrary homotopy types.

I know you’ve already pondered this. I guess you know that in 1991 Kapranov and Voevodsky once wrote a paper claiming that globular strict ω\omega-categories with weak inverses were sufficiently general to model arbitrary homotopy types. I guess you know that Carlos Simpson wrote a paper analyzing and in some sense disproving this claim.

Yes. It was to order my thoughts on questions like this that I wrote the literature summary titled Semistrict \infty-categories. As far as I understand, Simpson didn’t just disprove the claim, but made the crucial observation that it should be true that one can have all compositions strict – but allow the identites to be weak.

And maybe it does have something to do with smoothness:

how would one build a fundamental \infty-groupoid of a space XX which has strict composition but weak units? As Thomas observed # to get strict composition one needs to divide out at least something like self-homeomorphisms of the parameter spaces (the disks which we map into XX). So let’s do that.

One sees: since homeomorphisms cannot shrink the domain of a constant map [0,1]X[0,1] \to X to a point it should follow that with this construction precisely those cells have strict units at their boundary which have sitting instants there. I.e. those which are represented by maps D kXD^k \to X constant in a neighbourhood of the boundary. The others do not.

If true, it means that it’s the assumption of sitting instants that is too strong if one wants to capture arbitrary homotopy types by their fundamental \infty-groupoids. But of course paths without sitting instants have “kinks”: they are in particular not smooth in general (in that they may not have smooth composites). (I am just saying this for other readers following us.)

So this is one thing I thought about. Another is that it should be pointed out that the left pair of morphisms in that diagram I keep pointing to ωGroupoids(Spaces) K()Spaces \omega Groupoids(Spaces) \stackrel{\to^{K(-)}}{\leftarrow} Spaces does not have the usual geometric realization functor going from left to right. It’s that issue we talked about a couple of times before #: the functor KK here from smooth ω\omega-groupoids to Spaces takes an ω\omega-groupoid BG\mathbf{B}G with a single object coming from the ω\omega-group GG not to the classifying space of GG-bundles, but to a smooth model of a K(G,1)K(G,1) (or K(G,1)K(G,1\cdots \infty) or whatever the notation is. I just say K(G)K(G), letting the group know in which degrees it is concentrated). And not even that: it really takes BG\mathbf{B}G to the space which acts like a smooth model for K(G^)K(\hat G) for G^\hat G some kind of “simply connected version” of GG in some higher sense.

So this is not the usual setup of the “homotopy hypothesis”. But a bit different. For a while I started addressing this scenario as the “smooth homotopy hypothesis”, in need for a term, but I stopped doing that.

Another thing to notice is: all this \infty-Lie theory business is really, as remarked before, a variation on the theme of rational homotopy theory. So we are not talking about all topological spaces here, but about rational spaces. Maybe that matters?

But if you want to think about \infty-groupoids, I don’t see why you’d prefer globular strict ω\omega-categories with strict or weak inverses to Kan complexes.

For a couple of reasons:

- it seems to be sufficient for the purpose.

- it allows me to use the theory of crossed complexes which gives plenty of nice direct relation to concrete formula known in differential geoimetry and physics.

- one shouldn’t underestimate this: there is additional information obtained when passing to a strictified structure. It means you really extract information and go beyond just abstract nonsense. The weak integration of the String Lie 2-algebra for instance, the integration to a Kan complex, can be described in two lines. It doesn’t yet exhibit the information one wants. Then one has to work on it, for instance by strictifying (or doing other things with it, of course). The \infty-Lie integration process which I described is integration with built-in strictification. The idea is to get out something which you can directly carry over too the physics or differential geometry department to use there.

- it allows me to seamlessly build on what we have been doing: parallel nn-transport in low dimensions using higher categories – globular higher categories – using the theorems we proved in that globular context. I don’t know how you see it, but I am really still trying to finish what we started doing: understanding higher connections. The \infty-Lie business is just a means to an end for me: I use the \infty-Lie integration to construct nonabelian differential cocycles, such as String 2-bundles with connection, their twisted version and their associated Chern-Simons bundles with connection. The idea is: apply the functor Π ωS\Pi_\omega \circ S to the entire article I wrote with Hisham and Jim and obtain higher bundles with connection.

- lastly: I am wondering about this: a strict \infty-category is really a presheaf on a geometric category (a globular set, like simplicial sets are presheaves on the simplicial category) equipped with extra structure: the composition operations. We can “free” this extra structure by forming free ω\omega-categories, for instance free on simplicial sets.

It seems to me that the functor F:SimpSetωCatF : SimpSet \to \omega Cat, F(S)= nS nO(Δ n)F(S) = \int^n S^n \cdot O(\Delta^n) is faithful. (Is that wrong?) That means all kinds of simplicial constructions can be used in the context of ω\omega-categories. In particular from hypercovers etc. one gets Čech-ω\omega-groupoids and the like.

To summarize this:

Main reason for using strict \infty-models: practical reasons. But on the other hand: I don’t yet find myself running against a wall this way. If I saw there is something I’d want to obtain which can’t be obtained with strict \infty-categories, I’d have a better reason start thinking about an alternative.

Posted by: Urs Schreiber on October 29, 2008 6:16 PM | Permalink | Reply to this

Re: Hepworth on 2-Vector Bundles and the Volume of a Differentiable Stack

Well, you guys are opening up a can of worms here in my understanding of these things. I’m going to ask some really silly elementary questions, so that I get them behind me once and for all and become a better person for it.

Place yourself in my position: I have just rewatched John’s slides on the homotopy hypothesis, and skimmed through the Lectures on nn-categories and cohomology.

  • Homotopy 2-types are classified by their fundamental weak 2-groupoids.

Question: What is a good reference for this?

  • A weak 2-groupoid is the same thing as a strict 2-groupoid together with an associator.

I’ve said that slightly differently to normal — I haven’t pushed the whole thing down to 2-groups or crossed modules, because I want to make direct contact with what Urs is talking about.

Question: Give me a nice explicit example of two homotopy 2-types XX and YY which have the same underlying strict path 2-groupoids Π 2(X)\Pi_2 (X) and Π 2(Y)\Pi_2 (Y) but have different associators.

I think having this example close at hand is crucial to getting to grips with what Urs is talking about.

I think part of the whole problem comes down to nomenclature, a pet bugbear of mine which I’ve been frustrated by for ages.

We should distinguish between weak higher groupoids and strong higher groupoids.

A weak higher groupoid is something like a Kan complex. Sure it works… but you can’t directly get your hands on the “extra information” eg. associators. In Urs’s words, you can’t directly carry it over to the physics or differential geometry department to use there.

A strong higher groupoid is a globular type thing. It consists of the data of a strict higher groupoid plus extra information. I suspect it is extremely misleading to muse about whether we should be having `sitting instants’ or `weak inverses’ as in Urs’s comments above. I don’t think those things should matter. What matters it that we should have extra information. We are not weakening things, we are strengthening them! I know I am preaching to the converted here, but we have to be careful because the terminology is so bad that it can cause us to “backslide” conceptually unless we are extremely vigilant!

Posted by: Bruce Bartlett on October 30, 2008 2:05 PM | Permalink | Reply to this

Re: Hepworth on 2-Vector Bundles and the Volume of a Differentiable Stack

fake formula aka an ansatz?

Posted by: jim stasheff on October 25, 2008 2:44 PM | Permalink | Reply to this

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