## 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 $String_{Fr} \in Grp(Frechet)$ on the topological string group $String_{Top} \in Grp(Top)$ (based on a model by Stefan Stolz) and

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

$\pi_0(\widehat{Gau} \to String_{Fr}) \simeq Spin$

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

The second statement is important: naïvely one might be led to believe that $String_{Fr}$ is already a decent smooth model of $String$. But it is not. Not if one means to have differential cohomology of smooth $String$-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 $B String_{top}$ is the homotopy fiber of the first fractional Pontryagin class

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

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

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

where $\mathbf{B} Spin$ is the smooth moduli stack for smooth $Spin$-principal bundles, and $\mathbf{B}^3 U(1)$ 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

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

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

$\pi_0 String_{smooth} \simeq Spin,\;\;\; \pi_1 String_{smooth} \simeq U(1)$

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

One can show that both the strict and the weak Lie integration of the $\mathfrak{string}$-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 $String_{smooth}$ has the properties that originally motivated the search for smooth models of $String_{top}$ (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 $(\widehat{Gau P} \to String_{Fr})$ represent $String_{smooth}$ in $Smooth \infty Grpd$?

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

$(U(1) \to ) \to (\widehat{Gau P} \to String_{Fr}) \to (1 \to Spin)$

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

Last time that I thought and talked about this was when I met Christoph Wockel in Cardiff. I was thinking: present $Smooth \infty Grpd$ by the structure of a Brown-“category of fibrant objects” on $CartSp$ whose fibrations are stalkwise fibrations. Then it should be clear that $(\widehat{Gau P} \to String_{Fr}) \to (1 \to Spin)$ 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 $\mathcal{G}$ was to make sure, that the sequence you are mentioning

$B\pi_1(\mathcal{G}) \to \mathcal{G} \to \pi_0(\mathcal{G})$

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 $B^n U(1)$. Isn’t this just one model for a $K(\mathbb{Z},n+1)$. 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 $\mathcal{G}$ was to make sure, that the sequence you are mentioning

$\mathbf{B} \pi_1 (\mathcal{G}) \to \mathcal{G} \to \pi_0(\mathcal{G})$

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 $(g_1, g_2) : \mathbb{R}^n \to Spin \times Spin$ and every lift by a germ of a smooth map $(\hat g_1, \hat g_2) : \mathbb{R}^n \to String_{Fr} \times String_{Fr}$ there is a germ of a smooth map $h : \mathbb{R}^n \to \widehat{Gau}$ such that (in some convention) $(\partial h) \cdot \hat g_1 \cdot \hat g_2$ is a lift of $g_1 \cdot g_2$.

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

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

Right, other lifts serve other purposes. The smooth lift of $K(\mathbb{Z},n+1)$ to $\mathbf{B}^n U(1)$ serves the purpose of inducing a differential obstruction theory given by ordinary differential cohomology, namely by $\mathbf{B}^{n-1}U(1)$-principal bundles with connection, aka bundle $(n-2)$-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 $K(\mathbb{Z},4)$ to the smooth groupoid $\mathbf{B}^4 \mathbb{Z}$ which happens to have discrete smooth structure. The construction of Cocycles for diff. char. classes that gives

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

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

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

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

An $\Omega \mathbf{cosk}_4 \exp(\mathfrak{so})$-connection is an ordinary Spin-connection on a $Spin$-principal bundle with vanishing Pontryagin class. There is no differential information on how the Pontryagin class is trivialized, as there is in a $String$-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 $K(\mathbb{Z},8)$ we can now either consider

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

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

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

or…, and so on.

These are the only two types of smooth lifts of $K(\mathbb{Z}, n+1)$ 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 $B^n\mathbb{Z}$ 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 $PU(H)$ as a model for $K(\mathbb{Z},2)$?

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 $P U(H)$ as a model for $K(\mathbb{Z},2)$?

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

$Gau(P) \to String_{Fr} \to Spin$

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

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

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

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

in $Smooth \infty Grpd$ and $\Pi (Disc(c)) \simeq c$, by the fact that $(\Pi \dashv Disc)$ and that $Disc$ is full and faithful. This always exists, but is of course maximally uninteresting, again by full-and-faithfulness of $Disc$: it is just the image of the situation in $Top \simeq \infty Grpd$ 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 $\mathcal{G} \to \pi_0(\calg)$ is weakly equivalent to $\pi_1(\mathcal{G}$. 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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