## April 23, 2012

### Principal ∞-Bundles

#### Posted by Urs Schreiber

I have been speaking about principal ∞-bundles in an $\infty$-topos for a while now, in print starting around the article

Last year Thomas Nikolaus and myself visited Danny Stevenson in Glasgow. We labored day and night for two weeks with the aim to compile an article with a comprehensive account of the matter. As it goes with these glorious plans, they don’t fit the intended time schedule, and so we find ourselves still working on the article these days, always almost finished. At least Danny has meanwhile put a major piece of his on the arXiv, which we are making use of:

See below the fold for an abstract of what we are after now.

While we are not done yet with typing, next week I will speak about the subject (to an audience of topologists). I decided to prepare something like a first-order talk script, possibly usable as a pdf-handout, which gives a concise overview of what it’s all about, what the main theorems are and what the impact on applications is. I still have a bit of time to fine-tune this, but since some feedback can be useful for this, I am hereby posting these notes:

• Principal $\infty$-bundles – Theory and applications, pdf-handout notes.

I’d be interested in hearing whatever comments you might have.

There is a little table on page 5, which indicates something that I am fond of, and which I have been talking about here on the $n$Café in various guises every now and then. It tabulates examples of $\infty$-bundles of smooth moduli $\infty$-stacks and indicates what happens when you interpret these as universal associated coefficient bundles for nonabelian cohomology. Due to size limitations of a “handout”, this table is a small piece of a more extensive table, which is discussed in section 4.4

Along the lines of such a “table of twists” I will probably also give the lectures at ESI in a few weeks.

The theory of G-principal bundles makes sense in any (∞,1)-topos, such as that of topological or of smooth ∞-groupoids. It provides a geometric model for structured higher nonabelian cohomology. For suitable group objects $G$ these $G$-principal ∞-bundles reproduce the theory of ordinary principal bundles, of principal 2-bundles, of gerbes and 2-gerbes, of bundle gerbes and bundle 2-gerbes and generalizes them to higher analogs of arbitrary degree.

We discuss the general abstract theory of principal ∞-bundles, observe that it is induced directly by the ∞-Giraud axioms in an (∞,1)-topos and show the equivalence to the intrinsic nonabelian cocycles, hence their classification by nonabelian cohomology. We discuss that for every extension of ∞-groups there is a corresponding notion of ∞-bundles twisted by a principal ∞-bundle and that these twisted ∞-bundles are classified by the corresponding twisted cohomology.

We give explicit presentations of this theory by model structures of simplicial presheaves and notaby by weakly principal bundles in categories of locally fibrant simplicial sheaves. In the smooth context we discuss in some detail presentations by submersive locally fibrant simplicial smooth manifolds: Lie ∞-groupoids.

Finally we discuss a wealth of examples and applications.

Posted at April 23, 2012 10:17 PM UTC

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### Re: Principal ∞-Bundles

This is nice! Let me ask a very simple question from page 2:

Let $C \coloneqq SmthMfd$ be the category of all smooth manifolds (or some other site, here assumed to have enough points)

What is a point of the site $SmthMfd$? (Why) does it have enough of them?

Posted by: Mike Shulman on April 23, 2012 11:52 PM | Permalink | Reply to this

### Re: Principal ∞-Bundles

What is a point of the site $SmthMfd$?

There is, up to equivalence, one for every $n \in \mathbb{N}$, given by the stalk at the origin of $\mathbb{R}^n \simeq D^n$.

The proof reduces via slicing to the observation that the topos of sheaves over any fixed manifold has enough points, each of which looks like one of these.

Posted by: Urs Schreiber on April 24, 2012 12:28 AM | Permalink | Reply to this

### Re: Principal ∞-Bundles

Okay; that helps a little. I don’t have a very good intuition for what a “point” of a general site looks like, aside from the two special cases of (1) a locale, and (2) the classifying site of a geometric theory (where points are models of the theory in Set).

I guess you would say that as a “space”, this topos consists of countably many “fat points” each of which is fat with a different number of dimensions? I’m assuming that SmthMfd is at least approximately a cohesive site, and as such is supposed to have “the shape of a single point” — but I guess that that is a homotopy-theoretic shape, and it can still have many different actual points as long as they are connected together in a “contractible way”. Is that the right way to think?

Posted by: Mike Shulman on April 25, 2012 6:30 AM | Permalink | Reply to this

### Re: Principal ∞-Bundles

I don’t have a very good intuition for what a “point” of a general site looks like, aside from the two special cases of (1) a locale, and (2) the classifying site of a geometric theory (where points are models of the theory in Set).

So and here we can reduce to the first case. We are asking if two sheaves on $SmthMfd$ have the same value on every single smooth manifold. Since both restrict to sheaves over that given smooth manifold regarded as a topological space, we can now check that they agree there by checking if they agree as objects of a localic topos. So we know we need to check that they agree on every germ of disks around every point of the manifold. But since the two sheaves on this fixed manifold are restrictions of sheaves on all manifolds, their value on any disk is the same, no matter whether we think of this disk now as being a neighbourhood of one point or other in the given manifold. So we finally conclude that the two original sheaves are isomorphic already if they have the same value on all germs of abstract disks.

I guess you would say that as a “space”, this topos consists of countably many “fat points” each of which is fat with a different number of dimensions?

Yes, that’s exactly how I think of it. The “fat cohesive point” that smooth geometry is modeled on is something like the inductive limit over all germs of abstract smooth $n$-disks. Which intuitively makes perfect sense, doesn’t it?

I’m assuming that SmthMfd is at least approximately a cohesive site

Yes, exactly. The precise statement is that $CartSp_{smooth} \hookrightarrow SmthMfd$ exhibits a dense subsite, and that $CartSp_{smooth}$ is an $\infty$-cohesive site. That’s one reason for going on about that site $CartSp_{smooth}$. For the purposes of sheaves over it it is equivalent to $SmthMfd$, but it has some better properties as compared to the latter.

as such is supposed to have “the shape of a single point”

Yes, every cohesive $\infty$-topos is pointlike in the following respects (many of which imply each other):

• it is locally and globally $\infty$-connected and local (by definition);

• it has the shape of the point;

• it has homotopy dimension 0;

• it has cohomology dimension 0;

• it is hypercomplete.

However, it does not necessarily need to have “a unique $\infty$-topos point”, up to equivalence.

— but I guess that that is a homotopy-theoretic shape, and it can still have many different actual points as long as they are connected together in a “contractible way”. Is that the right way to think?

Yes, I guess so. I haven’t thought about how to formally think of the fact that there may be many different “topos points” in a cohesive $\infty$-topos.

Let’s see. So I think here the thing is that all these points “sit inside each other” like a Matryoshka doll. Not by invertible morphisms, though. Still, there is this inductive system of points of $Smooth \infty Grpd$. I should think about formalizing that somehow only the single colimiting point of this system is the “actual” single point that we would like to see. Hm…

Posted by: Urs Schreiber on April 25, 2012 9:24 AM | Permalink | Reply to this

### Re: Principal ∞-Bundles

Thanks! Of course, in general the real question to ask about the points of a topos is, what is the (higher) category of its points, rather than just the set or (higher) groupoid of them. Can you say precisely what the category of points of Smooth∞Grpd is?

Still, there is this inductive system of points of Smooth∞Grpd.

The category of points of a 1-topos has filtered colimits (B3.4.8); is an analogous statement true for $(\infty,1)$-toposes?

I should think about formalizing that somehow only the single colimiting point of this system is the “actual” single point that we would like to see.

In the 1-topos case, traditionally people have thought about “topos homotopies” as geometric morphisms $\mathcal{E}\times Sh([0,1]) \to \mathcal{F}$. Since there is a map from $Sh([0,1])$ to $Sh(\Sigma)$ where $\Sigma$ is the Sierpinski space, and the latter is the incarnation of the interval category as a topos, any natural transformation gives rise to a “topos homotopy”, so that even noninvertible maps between topos-points induce paths between “points” of the “topos homotopy type”. Does that have an analogy from the $\Pi_\infty$/shape-theory perspective on topos homotopy types?

For example, I believe we can say that $\Pi_\infty$ of an $(\infty,1)$-presheaf topos $\infty Gpd^C$ is the $\infty$-groupoid reflection of $C$. I’m not sure what more general precise statement could be made though.

Posted by: Mike Shulman on April 25, 2012 3:48 PM | Permalink | Reply to this

### Re: Principal ∞-Bundles

Of course, in general the real question to ask about the points of a topos is, what is the (higher) category of its points, rather than just the set or (higher) groupoid of them. Can you say precisely what the category of points of Smooth∞Grpd is?

It feels like it should be possible to use the quadruple of adjoint functors on a cohesive topos to show that the category of points in it is contractible in some sense by doing some retract yoga or something. But I don’t see it yet.

What I think can be checked easily over a cup of coffee (now that I had one) is that $Smooth \infty Grpd$ has one point $p(\infty)$ which alone is “enough points” and which is the inductive limit over all the $p(n)$ that are stalks at the origin of the $n$-disk.

I have to run now to catch a train. But I would like to come back to thinking about what can be said generally and specifically about topos points of a cohesive topos. That’s an evident fundamental question that I haven’t thought enough about to date.

Posted by: Urs Schreiber on April 25, 2012 5:46 PM | Permalink | Reply to this

### Re: Principal ∞-Bundles

It feels like it should be possible to use the quadruple of adjoint functors on a cohesive topos to show that the category of points in it is contractible in some sense

According to C3.6.1(vi), if a topos $\mathcal{E}$ is local, then for every topos $\mathcal{G}$ the hom-category $Topos(\mathcal{G},\mathcal{E})$ has an initial object, and thus a (weakly) contractible nerve. Thus the category of “$\mathcal{G}$-points of $\mathcal{E}$” is (weakly) contractible for any $\mathcal{G}$, not just $\mathcal{G}=Set$.

(There was some vaguely amusing banter at Swansea about whether calling a category “contractible” should mean that it has a weakly contractible nerve, or that it is equivalent as a category to the terminal one.)

What I think can be checked easily over a cup of coffee (now that I had one) is that Smooth∞Grpd has one point p(∞) which alone is “enough points” and which is the inductive limit over all the p(n) that are stalks at the origin of the n-disk.

Hmm, I thought that up here you were saying that Smooth∞Grpd has precisely $\mathbb{N}$-many points. Now are you saying there’s another one too?

Posted by: Mike Shulman on April 25, 2012 7:47 PM | Permalink | Reply to this

### Re: Principal ∞-Bundles

Hmm, I thought that up here you were saying that Smooth∞Grpd has precisely $\mathbb{N}$-many points.

True, I made it sound this way and probably believed this. But I had never tried to formally prove that these are all. So far all I really cared about was that these $\mathbb{N}$-many exist and are enough.

Thanks for the pointer to the Elephant! I’ll look into it. Good that we started talking about this issue.

Posted by: Urs Schreiber on April 25, 2012 8:42 PM | Permalink | Reply to this

### Re: Principal ∞-Bundles

Continuing my question from the nForum, given

Claim. Since $Sh_{\infty}(SmthMfd)$ is “cohesive”, there is a notion of differential refinement of the above discussion, yielding connections on $\infty$-bundles,

ought there to be a parallel claim for ‘holomorphic refinement’ if only we knew what holomorphic cohesiveness was?

By the way, on p. 366 of ‘Differential cohomology in a cohesive $\infty$-topos’ there’s a typo

The left colum displays…

Posted by: David Corfield on April 24, 2012 9:26 AM | Permalink | Reply to this

### Re: Principal ∞-Bundles

I think I have a clearer way of putting this question I keep bugging you with.

I might think to investigate a smooth manifold by taking it as a special kind of object of the cohesive $(\infty, 1)$-topos $Top$. Or I might decide it’s better to look it as an object in a more closely tailored setting where smoothness prevails, i.e., in $Smooth \infty Grpd$.

Now if I’m dealing with, say, a complex or symplectic manifold, should I rest happy with taking it to be a special object of $Smooth \infty Grpd$, or should I seek out a better fitting setting, somewhere where holomorphic or symplectic cohesion prevails?

In the symplectic case you adopt the former stategy.

Posted by: David Corfield on April 24, 2012 11:11 AM | Permalink | Reply to this

### Re: Principal ∞-Bundles

Hi David,

I see. So when I replied to your question on the $n$Forum, I was not speaking in the context of cohesion, but just of sheaf cohomology in general. So I guess we were talking past each other to some extent.

Your general question here, is a very good one, I think, in the sense that it is very interesting, hence also hard. So I may not have a really good answer. But I can try to indicate better what I know and what I don’t know.

First, it is absolutely true that a single traditional object may naturally present very different objects in various toposes, reflecting very different aspects of the former object. For instance taking a smooth manifold to present an object in $\mathrm{ETop}\infty Grpd \hookrightarrow \Smooth \infty Grpd$ instead of directly in $Smooth \infty Grpd$ means forgetting its smooth structure, regarding it just as a topological space, and then “freely” adding a diffeological structure again (which is a weird thing to do from the point of traditional differential geometry, but it makes sense).

Then concerning the question of complex and analytic geometry: as you know, I don’t know if these geometries are cohesive. For the analytic case it might be true if interpreted suitably, but I still don’t know. Accordingly, I have no intuition really what the difference will be – or would be – between regarding an object as a plain object in an analytic cohesive $\infty$-topos or as an object with extra structure in some other $\infty$-topos. Sorry, I realize that this is a huge frustrating gap from the point of view of getting a general overview idea of cohesion, but for the moment I can’t help it.

On the other hand, concerning your question about symplectic structure: such structure is very naturally and usefully regarded as a special form of differential cohomological structure in $Smooth \infty Grpd$. So while I don’t know if there might be a useful “symplectic cohesive $\infty$-topos”, I first of all doubt it, and secondly it doesn’t bother me at all that I don’t know it, because I see no indication that I need to know it, with instead the theory in $Smooth \infty Grpd$ with extra differential cohomological structure working so very well.

That’s what I can say for the moment. I wish I had a more conceptual answer than these pragmatic attitudes. But maybe we’ll get there in the future.

Posted by: Urs Schreiber on April 25, 2012 9:40 AM | Permalink | Reply to this

### Re: Principal ∞-Bundles

Just a couple of quick remarks.

I thought you’d say that about the symplectic case, but then while rooting around generalized complex geometry wondered if the unification of the complex and symplectic cases suggested parallel treatment.

In Lurie’s Structured Spaces he treats the complex analytic case in section 4.4, using Stein manifolds in a pregeometry. So maybe if you’re right here

There ought to be a good way to connect the big and the little perspective on higher/derived geometry,

there’s a way through.

Which raised one, final I promise, thought. If in higher geometry you treat the big and small topos approaches, are there big and small approaches in the dual higher algebra?

Posted by: David Corfield on April 25, 2012 10:05 AM | Permalink | Reply to this

### Re: Principal ∞-Bundles

The complex analytic case is treated at greater length in sections 11 and 12 of DAGIX.

Posted by: David Corfield on April 25, 2012 3:24 PM | Permalink | Reply to this

### Re: Principal ∞-Bundles

David,

you point to

In Lurie’s Structured Spaces he treats the complex analytic case in section 4.4, using Stein manifolds in a pregeometry.

and to

The complex analytic case is treated at greater length in sections 11 and 12 of DAGIX.

Therefore I should maybe amplify that it is clear that higher symplectic/analytic geometry is naturally considered in the $\infty$-topos over a category of symplectic/analytic spaces. This is not the problem that I am running into. The problem that I still can’t see the solution to is whether there is a way to tweak these sites such that the resulting $\infty$-topos is cohesive.

Of course this is only a problem if you really feel the need for a cohesive $\infty$-topos, and for many applications you may not. I realize that you may have gotten the impression that I said that there is some problem with considering things like cohomology of symplectic/analytic spaces in terms of higher toposes. But this is absolutely not so. Of course many things are straightforward at least to define, if maybe not to study.

But all this is not directly related to the question whether there is a cohesive context for analytic geometry. If there is, it just means that it will imply various nice extra structure and properties on the structures that you have by default in higher analytic geometry.

Posted by: Urs Schreiber on April 26, 2012 1:57 PM | Permalink | Reply to this

### Re: Principal ∞-Bundles

So when you suggest

There ought to be a good way to connect the big and the little perspective on higher/derived geometry,

this presumably won’t involve any kind of recipe for passing from the little to the big.

But won’t there be signs in certain cases of geometry that there really must be a cohesive site about, e.g., when the accompanying structures are found?

Posted by: David Corfield on April 26, 2012 2:43 PM | Permalink | Reply to this

### Re: Principal ∞-Bundles

while rooting around generalized complex geometry wondered if the unification of the complex and symplectic cases suggested parallel treatment.

This is an aspect interesting in another way: we can describe generalized (complex) geometry entirely in terms of reductions of structure groups, without even introducing differential form structure explicitly (this is one of the items in the table of twists).

As far as I am aware, there is not in the literature a conceptually clean way to combine the 2-forms that arise as components of such “generalized vielbein fields” that constitute such a choice of reduction of structure group with the curvature 2-form of a gerbe. This is always imposed by hand.

This is, to my mind, a major open question in the conceptual understanding in the wider context of exceptional generalized geometry, controlling backgrounds of higher dimensional supergravity theories:

here the generalized vielbein fields appearing in a plain reduction of structure group contain higher forms, and these clearly want to be identified with the local connection forms of the 11d supergravity $C$-field circle 3-bundle and its magnetic dual. As far as I can see, this is done “by hand” in the literature and for restricted cases. There needs to be a conceptual understanding of this though.

I think this will be a major insight. (If somebody seeing me say this has this insight already, please let me know. Also, I can expand on details, if desired).

Have to rush off now. I’ll get back to you on your other points later. Please don’t stop mentioning these questions, as long as you have them! I am happy to have discussion of this stuff.

Posted by: Urs Schreiber on April 25, 2012 12:38 PM | Permalink | Reply to this

### Re: Principal ∞-Bundles

there’s a typo

Thanks! Fixed now. Though that whole section needs more editorial attention. I’ll come to that in a little while…

Posted by: Urs Schreiber on April 26, 2012 1:49 PM | Permalink | Reply to this
Read the post Principal ∞-Bundles -- general theory and presentations
Weblog: The n-Category Café
Excerpt: Notes on the theory of principal and twisted bundles in higher geometry.
Tracked: June 25, 2012 8:08 PM

### Re: Principal ∞-Bundles

This is not the right place for my query, but it’s the best the n-cat search could do:
has there been a comment or discussion anywhere on the blog about:
Blomgren and Chacholski - On the classification of fibrations
which addresses the title from a pov many of you guys might like

Posted by: jim stasheff on June 28, 2012 7:04 PM | Permalink | Reply to this

### Re: Principal ∞-Bundles

Hi Jim

were you thinking of this?

Posted by: David Roberts on June 29, 2012 1:16 AM | Permalink | Reply to this

### Re: Principal ∞-Bundles

Hmm, no that’s not it. Only Chacholski, not Blomgren. But that is the only other mention of Chacholski on the n-category cafe.

Posted by: David Roberts on June 29, 2012 1:18 AM | Permalink | Reply to this

### Re: Principal ∞-Bundles

has there been a comment or discussion anywhere on the blog about:

Blomgren and Chacholski - On the classification of fibrations

Thanks for the pointer!

While I had not seen that particular article yet, we have had various discussions here about this topic: all the discussions about univalence for instance are about this result and its generalizations. You might like Ieke Moerdijk’s short and sweet version of the proof of the classification of fibrations here.

Also if you look at the talk handout slides that are announced in the above entry you see that on p. 4 a generalization of this result is stated. The details I had posted just the other day, at Principal ∞-Bundles – general theory and presentations.

It is noteworthy that all of this follows as a fairly straightforward consequence from what in modern language is a central result of $\infty$-topos theory: the existence of “object classifiers”. I once added a little discussion that makes explicit how your classical result and the formulations of it by May, and now Blomgren-Chacholski follow from this here on the $n$Lab.

Which brings me to a question I have for you: can you help me organize the history of who is to be attributed for what? It seems to me that the classification of fibrations is being re-proven various times, and it is not always entirely clear to me to which extent the authors are actively aware of their predecessors.

An attempt on my part to list the relevant literature is in the reference section at associated infinity-bundle here.

You would do me a favour if you could have a look at that list and tell me what you think, which items and/or comments should be added. Thanks!

Posted by: Urs Schreiber on June 29, 2012 11:25 AM | Permalink | Reply to this

### Re: Principal ∞-Bundles

will look and ponder
may take a while

BUT I was hoping for comments on the extreme generality of the B and G approach for which I
find the title a bit misleading/understated

Posted by: jim stasheff on June 29, 2012 1:21 PM | Permalink | Reply to this

### Re: Principal ∞-Bundles

BUT I was hoping for comments on the extreme generality of the B and G approach

Maybe I am bit confused about this. I will need to read the article in more detail when I have more time. Could you say which extreme generality specifically you mean here?

I mostly looked at the introduction. Probably that’s a reason for missing something. But Theorem A there I would have said I know already, as I indicated above, and Theorem B seems to be a fairly direct consequence of Dwyer-Kan 84, 2.3 and 2.4.

I am certainly missing something. I will now read the article in more detail. But please let me know your perspective on it.

Posted by: Urs Schreiber on June 29, 2012 2:01 PM | Permalink | Reply to this

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