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July 1, 2011

Nikolaus on Higher Categorical Structures in Geometry

Posted by Urs Schreiber

Yesterday Thomas Nikolaus – former colleague of mine in Hamburg – has defended his PhD.

His nicely written thesis

Higher categorical structures in geometry – General theory and applications to QFT

discusses plenty of subjects of interest here; the main sections are titled:

  1. Bundle gerbes and surface holonomy

  2. Equivariance in higher geometry

  3. Four equivalent versions of non-abelian gerbes

  4. A smooth model for the string-group

  5. Equivariant modular categories via Dijkgraaf-Witten theory .

Have a look at his slides for a gentle overview.

Myself, I have to dash off now. Maybe I’ll say a bit more about what Thomas did in these chapters a little later. Or maybe he’ll do so himself…

Posted at July 1, 2011 5:27 PM UTC

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Re: Nikolaus on Higher Categorical Structures in Geometry

Congratulations, Thomas!

Posted by: Tom Leinster on July 1, 2011 6:19 PM | Permalink | Reply to this

Re: Nikolaus on Higher Categorical Structures in Geometry

Here a few more words and links on the content of the thesis:

Chapter 1 is about the notion of connection on a 2-bundle for the case that the structure 2-group is the circle 2-group (\simeq ordinary bundle gerbes) or the (automorphism 2-group) of the ordinary circle group (\simeq the case of “Jand gerbes”: orientifolds). It discusses the higher parallel transport/surface holonomy of these objects over surfaces with boundaries and with defects.

Chapter 2 is about stacks on smooth groupoids, hence (nonabelian) equivariant cohomology, then applied to the previous case of bundle gerbes.

The third chapter discusses four equivalent incarnations of the notion of principal 2-bundle: 1. in terms of nonabelian Cech cohomology, 2. by topological classifying maps 3. as bundle gerbes, 4. explicitly as total “spaces” with principal action.

The fourth chapter discusses a smooth Frechet manifold structure on the topological string group and a smooth 2-group obtained from it.

The fifth chapter discusses aspects of equivariant Dijkgraaf-Witten theory.

And I have to run now to catch a train.

Posted by: Urs Schreiber on July 2, 2011 6:09 AM | Permalink | Reply to this

Re: Nikolaus on Higher Categorical Structures in Geometry

@Tom: Thanks ;)

Also thanks to Urs for this very good summary. If someone is interested I could expand on that or provide more details of the respective chapters. Also if someone finds some errors or has some comments - please let me know.

I should just say that the chapters of the thesis are based on papers with the same name, but with some coauthors, which should be mentioned here:

Chapter 1: “Bundle gerbes and surface holonomy” with J. Fuchs, C. Schweigert, and K. Waldorf

Chapter 2: “Equivariance in higher geometry” with C. Schweigert

Chapter 3: “Four Equivalent Versions of Non-Abelian Gerbes” with K. Waldorf

Chapter 4: “A smooth model for the string group” with C. Sachse, and C. Wockel

Chapter 5: “Equivariant modular categories via Dijkgraaf-Witten theory” with J. Maier and C. Schweigert

Posted by: Thomas Nikolaus on July 2, 2011 10:18 AM | Permalink | Reply to this

Smooth string 2-group

I’ll briefly highlight one aspect of Thomas’s work. Then I make a lengthy comment. Finally I have a question.

The point of section 4 in Thomas’ thesis, the one on the string group, is

  1. to state that there is indeed the structure of a Fréchet Lie group String FrGrp(Frechet)String_{Fr} \in Grp(Frechet) on the topological string group String TopGrp(Top)String_{Top} \in Grp(Top) (based on a model by Stefan Stolz) and

  2. to observe that there is then a Fréchet 2-group coming from a crossed module (Gau^String Fr)(\widehat{Gau} \to String_{Fr}) whose homotopy sheaves are

π 0(Gau^String Fr)Spin \pi_0(\widehat{Gau} \to String_{Fr}) \simeq Spin

π 1(Gau^String Fr)U(1). \pi_1(\widehat{Gau} \to String_{Fr}) \simeq U(1) \,.

The second statement is important: naïvely one might be led to believe that String FrString_{Fr} is already a decent smooth model of StringString. But it is not. Not if one means to have differential cohomology of smooth StringString-principal bundles to come out right.

The way I like to think about this subtlety, in terms of cohesive \infty-toposes, is this:

the classifying space BString topB String_{top} is the homotopy fiber of the first fractional Pontryagin class

12p 1:BSpinB 3U(1) \frac{1}{2}p_1 : B Spin \to B^3 U(1)

in DiscreteGrpd:=GrpdTopDiscrete \infty Grpd := \infty Grpd \simeq Top. One can show that this has, up to equivalence, a unique lift through geometric realization Π:SmoothGrpdDiscreteGrpd\Pi : Smooth \infty Grpd \to Discrete \infty Grpd to a smooth characteristic map of the form

12p 1:BSpinB 3U(1), \frac{1}{2} \mathbf{p}_1 : \mathbf{B} Spin \to \mathbf{B}^3 U(1) \,,

where BSpin\mathbf{B} Spin is the smooth moduli stack for smooth SpinSpin-principal bundles, and B 3U(1)\mathbf{B}^3 U(1) is the smooth moduli 2-stack for smooth circle 3-bundles (aka bundle 2-gerbes).

The “correct” smooth string 2-group is the delooping of the homotopy fiber of this smooth characteristic map

String smooth:=Ωhofib(12p 1). String_{smooth} := \Omega hofib(\frac{1}{2}\mathbf{p}_1) \,.

From this definition it follows immediately by the long exact sequence of geoemtric homotopy groups that

π 0String smoothSpin,π 1String smoothU(1) \pi_0 String_{smooth} \simeq Spin,\;\;\; \pi_1 String_{smooth} \simeq U(1)

in SmoothSpaces=SmoothGrpd 0SmoothSpaces = Smooth \infty Grpd_{\leq 0}.

One can show that both the strict and the weak Lie integration of the 𝔰𝔱𝔯𝔦𝔫𝔤\mathfrak{string}-Lie 2-algebra that are in the literature are indeed models for this abstractly defined String Lie 2-group, and that nonabelian differential cohomology with coeffcients in String smoothString_{smooth} has the properties that originally motivated the search for smooth models of String topString_{top} (details on all this are in section 4.1 of differential cohomology in a cohesive topos ).

So from this prespective one would like to know (or at least I would like to know):

does the Sachse-Nikolaus-Wockel Fréchet 2-group (GauP^String Fr)(\widehat{Gau P} \to String_{Fr}) represent String smoothString_{smooth} in SmoothGrpdSmooth \infty Grpd?

A sufficient condition for this to be true is that the exact sequence of Fréchet crossed modules

(U(1))(GauP^String Fr)(1Spin) (U(1) \to ) \to (\widehat{Gau P} \to String_{Fr}) \to (1 \to Spin)

which they discuss presents a fiber sequence in SmoothGrpdSmooth \infty Grpd.

Last time that I thought and talked about this was when I met Christoph Wockel in Cardiff. I was thinking: present SmoothGrpdSmooth \infty Grpd by the structure of a Brown-“category of fibrant objects” on CartSpCartSp whose fibrations are stalkwise fibrations. Then it should be clear that (GauP^String Fr)(1Spin)(\widehat{Gau P} \to String_{Fr}) \to (1 \to Spin) is a fibration. And thus it follows that its ordinary fiber is also its homotopy fiber.

So finally my question: have you, Thomas, further thought about this? If not, we should try to nail it down.

Posted by: Urs Schreiber on July 3, 2011 5:34 PM | Permalink | Reply to this

Re: Smooth string 2-group

Urs wrote:

A sufficient condition for this to be true is that the exact sequence of Fréchet crossed modules

(U(1)→)→(GauPˆ→StringFr)→(1→Spin)
which they discuss presents a fiber sequence in Smooth∞Grpd.

Last time that I thought and talked about this was when I met Christoph Wockel in Cardiff. I was thinking: present Smooth∞Grpd by the structure of a Brown-“category of fibrant objects” on CartSp whose fibrations are stalkwise fibrations. Then it should be clear that (GauPˆ→StringFr)→(1→Spin) is a fibration. And thus it follows that its ordinary fiber is also its homotopy fiber.

Hi Urs,

I have thought a little bit about that. One reason for the notion of “smoothly seperable” Lie-2-group 𝒢\mathcal{G} was to make sure, that the sequence you are mentioning

Bπ 1(𝒢)𝒢π 0(𝒢) B\pi_1(\mathcal{G}) \to \mathcal{G} \to \pi_0(\mathcal{G})

is a fibre sequence. Actually this follows as you point out first from the fact that the right hand map is a fibration and secondly from the fact, that the left hand side is weakly equivalent to its fibre (as you can see e.g. using our Proposition 4.4. or by a direct argument).

But there are some subtleties hidden here: first of all, we are using infinite dimensional manifolds, so one has to be a little bit careful about the Grothendieck Topology: we use submersions on the category of infinite dim. manifolds and I am not sure that this is the same as the usual one on diffeological spaces (resp. presheaves). But apart from that your argument looks fine….

I have always wondered about the fact, that the maps canonically go to B nU(1)B^n U(1). Isn’t this just one model for a K(,n+1)K(\mathbb{Z},n+1). Wouldn’t each other smooth version be equally fine?

Posted by: Thomas Nikolaus on July 3, 2011 8:47 PM | Permalink | Reply to this

Re: Smooth string 2-group

One reason for the notion of “smoothly seperable” Lie-2-group 𝒢\mathcal{G} was to make sure, that the sequence you are mentioning

Bπ 1(𝒢)𝒢π 0(𝒢)\mathbf{B} \pi_1 (\mathcal{G}) \to \mathcal{G} \to \pi_0(\mathcal{G})

is a fibre sequence.

I see.

Actually this follows as you point out first from the fact that the right hand map is a fibration and secondly from the fact, that the left hand side is weakly equivalent to its fibre (as you can see e.g. using our Proposition 4.4. or by a direct argument).

Wait, where are you pointing me to? Is this meant to be prop. 4.4.9?

But there are some subtleties hidden here: first of all, we are using infinite dimensional manifolds, so one has to be a little bit careful about the Grothendieck Topology: we use submersions on the category of infinite dim. manifolds and I am not sure that this is the same as the usual one on diffeological spaces (resp. presheaves).

It would be sufficient to have the following:

for every germ (around the origin) of a smooth map (g 1,g 2): nSpin×Spin(g_1, g_2) : \mathbb{R}^n \to Spin \times Spin and every lift by a germ of a smooth map (g^ 1,g^ 2): nString Fr×String Fr(\hat g_1, \hat g_2) : \mathbb{R}^n \to String_{Fr} \times String_{Fr} there is a germ of a smooth map h: nGau^h : \mathbb{R}^n \to \widehat{Gau} such that (in some convention) (h)g^ 1g^ 2(\partial h) \cdot \hat g_1 \cdot \hat g_2 is a lift of g 1g 2g_1 \cdot g_2.

It seems to me that this is evidently true. But check.

I have always wondered about the fact, that the maps canonically go to B nU(1)\mathbf{B}^n U(1). Isn’t this just one model for a K(,n+1)K(\mathbb{Z},n+1). Wouldn’t each other smooth version be equally fine?

Right, other lifts serve other purposes. The smooth lift of K(,n+1)K(\mathbb{Z},n+1) to B nU(1)\mathbf{B}^n U(1) serves the purpose of inducing a differential obstruction theory given by ordinary differential cohomology, namely by B n1U(1)\mathbf{B}^{n-1}U(1)-principal bundles with connection, aka bundle (n2)(n-2)-gerbes. For the motivating cases of string- and fivebrane structures this is what one needs to get.

But other lifts play a role, too. We can for instance lift K(,4)K(\mathbb{Z},4) to the smooth groupoid B 4\mathbf{B}^4 \mathbb{Z} which happens to have discrete smooth structure. The construction of Cocycles for diff. char. classes that gives

12p 1:BSpincosk 3exp(𝔰𝔬)exp(,[,])B 3()B 3U(1) \frac{1}{2}\mathbf{p}_1 : \mathbf{B}Spin \simeq \mathbf{cosk}_3 \exp(\mathfrak{so}) \stackrel{\exp(\langle -,[-,-]\rangle)}{\to} \mathbf{B}^3 (\mathbb{Z} \to \mathbb{R}) \to \mathbf{B}^3 U(1)

has a slight variant simply by forgetting the \mathbb{R}-factor (which picks up all the “infinitesimal approximation by integration” of the integral cocycle) to

exp(,[,]):cosk 3exp(𝔰𝔬)B 4 \exp(\langle -,[-,-]\rangle) : \mathbf{cosk}_3 \exp(\mathfrak{so}) \to \mathbf{B}^4 \mathbb{Z}

and that’s still a lift of 12p 1\frac{1}{2}p_1 to smooth \infty-groupoids, but with rather different properties now. Its homotopy fiber is not BString smooth\mathbf{B}String_{smooth} but cosk 4exp(𝔰𝔬)\mathbf{cosk}_4 \exp(\mathfrak{so}), I think, which gives after looping a B 2\mathbf{B}^2 \mathbb{Z}-extension of Spin smoothSpin_{smooth}. In the former case there is a homotpy group being filled by a smooth object, in the latter by a discrete object.

An Ωcosk 4exp(𝔰𝔬)\Omega \mathbf{cosk}_4 \exp(\mathfrak{so})-connection is an ordinary Spin-connection on a SpinSpin-principal bundle with vanishing Pontryagin class. There is no differential information on how the Pontryagin class is trivialized, as there is in a StringString-2-connection.

This has its use, or at least has an effect, as we keep going up the Whitehead tower. In the next step there are now four choices: apart from how to lift K(,8)K(\mathbb{Z},8) we can now either consider

16p 2:BStringcosk 7exp(𝔰𝔱𝔯𝔦𝔫𝔤)exp(,[,],[,],[,])B 7U(1) \frac{1}{6}\mathbf{p}_2 : \mathbf{B}String \simeq \mathbf{cosk}_7 \exp(\mathfrak{string}) \stackrel{\exp(\langle -,[-,-], [-,-], [-,-] \rangle)}{\to} \mathbf{B}^7 U(1)

or (since also cosk 7exp(𝔰𝔬)cosk 4exp(𝔰𝔬)\mathbf{cosk}_7 \exp(\mathfrak{so}) \simeq \mathbf{cosk}_4 \exp(\mathfrak{so}))

exp(,[,],[,],[,])):cosk 7exp(𝔰𝔬)B 7U(1) \exp(\langle -,[-,-], [-,-], [-,-] \rangle)) : \mathbf{cosk}_7\exp(\mathfrak{so}) \to \mathbf{B}^7 U(1)

or…, and so on.

These are the only two types of smooth lifts of K(,n+1)K(\mathbb{Z}, n+1) that I have been thinking of. Are there others that would be interesting?

Posted by: Urs Schreiber on July 4, 2011 11:37 AM | Permalink | Reply to this

Re: Smooth string 2-group

Urs wrote:

Wait, where are you pointing me to? Is this meant to be prop. 4.4.9?

Yes, but in our case this is really like breaking a fly on the wheel.

It would be sufficient to have the following:

for every germ (around the origin) of a smooth map (g1,g2):ℝn→Spin×Spin and every lift by a germ of a smooth map (g^1,g^2):ℝn→StringFr×StringFr there is a germ of a smooth map h:ℝn→Gauˆ such that (in some convention) (∂h)⋅g^1⋅g^2 is a lift of g1⋅g2.

Yes, that is true, since locally the extension looks like a product. Good.

These are the only two types of smooth lifts of K(ℤ,n+1) that I have been thinking of. Are there others that would be interesting?

Well, first of all it is good to know how B nB^n\mathbb{Z} behaves. I was just wondering, in your argumentation this choice seems to be the “uncanonical” one. Everything else is more or less forced. What is for example about PU(H)PU(H) as a model for K(,2)K(\mathbb{Z},2)?

Posted by: Thomas Nikolaus on July 4, 2011 2:50 PM | Permalink | Reply to this

Re: Smooth string 2-group

Wait, where are you pointing me to? Is this meant to be prop. 4.4.9?

Yes, but in our case this is really like breaking a fly on the wheel.

Okay, thanks, I was just trying to make sure that I knew what you are thinking of. But so this statement of 4.4.9 – overkill or not here – is about being a fiber sequence after geometric realization, no? It seems here we are after it being a fiber sequence before geoemtric realization.

What is for example about PU(H)P U(H) as a model for K(,2)K(\mathbb{Z},2)?

Oh, I see, sure. Right, for instance the extension of Fréchet groups

Gau(P)String FrSpin Gau(P) \to String_{Fr} \to Spin

with Gau(P) whPU(H)Gau(P) \simeq_{wh} P U(H) should directly yield a fiber sequence of smooth groupoids BGau(P)BString FrBSpin\mathbf{B}Gau(P) \to \mathbf{B}String_{Fr} \to \mathbf{B}Spin. What I can’t see is what the classifying morphism BSpin???\mathbf{B}Spin \to ??? for this could be. (?) Or how to build any lift of 12p 1\frac{1}{2}p_1 involvin PU(H)P U(H). Naively it would have to go to “B 2PU(H)\mathbf{B}^2 P U(H)”, which of course does not exist.

Hm, maybe I should think about this more. Do you have an idea?

So, anyway, there are in general many lifts of a given characteristic map through Π:SmoothGrpdGrpd\Pi : Smooth \infty Grpd \to \infty Grpd and the choice of which one to use depends on purpose and application.

For instance there is always the discrete lift: for c:XK(,n+1)c : X \to K(\mathbb{Z}, n+1) any morphism whatsoever, we have

Disc(c):Disc(X)Disc(K(,n+1)) Disc(c) : Disc(X) \to Disc(K(\mathbb{Z}, n+1))

in SmoothGrpdSmooth \infty Grpd and Π(Disc(c))c\Pi (Disc(c)) \simeq c, by the fact that (ΠDisc)(\Pi \dashv Disc) and that DiscDisc is full and faithful. This always exists, but is of course maximally uninteresting, again by full-and-faithfulness of DiscDisc: it is just the image of the situation in TopGrpdTop \simeq \infty Grpd under the embedding as discrete smooth \infty-groupoids.

Posted by: Urs Schreiber on July 4, 2011 3:52 PM | Permalink | Reply to this

Re: Smooth string 2-group

Okay, thanks, I was just trying to make sure that I knew what you are thinking of. But so this statement of 4.4.9 – overkill or not here – is about being a fiber sequence after geometric realization, no? It seems here we are after it being a fiber sequence before geoemtric realization.

I was refering to the first part of the proposition. This proof is in the appendix and is surprisingly complicated… However, I was just saying that this could be used to see that the fibre of the map 𝒢π 0(calg)\mathcal{G} \to \pi_0(\calg) is weakly equivalent to π 1(𝒢\pi_1(\mathcal{G}. But thats really not a big deal…

I have to think about the other thing first…

Posted by: Thomas Nikolaus on July 4, 2011 9:30 PM | Permalink | Reply to this

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