## January 24, 2008

### Classifying Spaces for 2-Groups

#### Posted by John Baez

These days I’ve been working hard finishing off papers. Now you can see this one on the arXiv:

Or, you can just read this summary…

As you probably know if you hang out here, Urs Schreiber and I like ‘higher gauge theory’, which describes what happens when you move a string or higher-dimensional membrane around, just as ordinary gauge theory does for point particles.

I didn’t get into this from string theory. I just like categorifying stuff. So, I got involved in higher gauge theory simply by trying to categorify the ideas of gauge theory.

If you categorify the concept of ‘group’, you get the concept of ‘2-group’: just as a group is a set with multiplication and inverses, a 2-group is a category with multiplication and inverses. In fact, 2-groups have been around for a long time under various names: ‘categorical groups’, ‘gr-categories’, and (in disguise) ‘crossed modules’. Just you can blend groups with topological spaces and get ‘topological groups’ you can do the same with 2-groups. For gauge theory, we want ‘Lie 2-groups’, which are especially nice topological 2-groups.

The story goes on. If you categorify the concept of ‘Lie algebra’, you get the concept of ‘Lie 2-algebra’. If you categorify the concept of ‘bundle’, you get the concept of ‘2-bundle’. And if you categorify the concept of ‘connection’, you get the concept of ‘2-connection’.

Now, if you hand me a topological group $G$, there’s a topological space called $B G$ that serves as a ‘classifying space’ for principal $G$-bundles. In other words, under some mild conditions I won’t worry about here, principal $G$-bundles over a space $M$ are classified by homotopy classes of maps

$f: M \to B G$

So, it’s natural to wonder if the same thing works for topological 2-groups. If $\mathbf{G}$ is a topological 2-group, is there a space $B\mathbf{G}$ such that principal $\mathbf{G}$-2-bundles over a space $M$ are classified by homotopy classes of maps $M \to B\mathbf{G}$?

Yes! And thanks to some beautiful arguments developed by Danny Stevenson, we’re able to give a very simple construction of this space $B\mathbf{G}$.

This fact is not exactly new. In his pioneering work on this subject, back in 2005, Branislav Jurčo asserted that a certain space homotopy equivalent to ours does the job. However, there are some gaps in his proof. This is what got Danny interested in straightening things out.

Later, Nils Baas, Marcel Bökstedt and Tore Kro constructed an even more general sort of classifying space. For any sufficiently nice topological 2-category $C$, they construct a classifying space for $C$-2-bundles. A topological 2-group is just a topological 2-category with one object and with all morphisms and 2-morphisms invertible — and in this special case, their result almost reduces to the fact mentioned above. There are some minor differences — for example, their classifying space matches Jurčo’s rather than ours, and they classify 2-bundles up to ‘concordance’. But it follows from our work that these differences don’t really matter.

A nice thing about having a classifying space is that it lets you define ‘characteristic classes’. For a topological 2-group $\mathbf{G}$, characteristic classes are just elements of the cohomology $H^*(B\mathbf{G})$. Since any $\mathbf{G}$-2-bundle over a space $M$ comes from a map

$f : M \to B \mathbf{G}$

and this gives a map

$f^* : H^*(B\mathbf{G}) \to H^*(M),$

characteristic classes automatically give elements of $H^*(M)$ when you have a principal $\mathbf{G}$-2-bundle over $M$.

We look at an interesting example of this: the ‘string 2-group’. Any simply-connected compact simple Lie group, like $SU(n)$ or $Spin(n)$, automatically gives rise to a continuous 1-parameter family of Lie 2-algebras, and a discrete 1-parameter family of Lie 2-groups. When you set this parameter equal to 1, you get the string 2-group $String(G)$. As the name suggests, this shows up in string theory. Following ideas of Matt Ando and Greg Ginot, we work out the characteristic classes for $String(G)$-2-bundles. The answer is simple: working with rational coefficients at least, $H^*(B String(G))$ is just $H^*(B G)$ mod the ideal generated by the ‘first Pontryagin class’ — the god-given element in $H^4(B G)$.

If you actually read this paper, you’ll see it starts with a big review of known stuff, but then introduces some new technical tricks — three cool lemmas. You’ll learn about Cech cohomology with coefficients in a 2-group — also known as nonabelian cohomology — and you’ll see how the classifying space of a topological 2-group $\mathbf{G}$ is really just the classifying space of a certain topological group $|\mathbf{G}|$: the ‘geometric realization of the nerve of $\mathbf{G}$’. In particular, $|String(G)|$ is a famous thing: it’s the 3-connected cover of $G$, meaning it’s built by ‘unwrapping’ $G$ to make its 3rd homotopy group go away. The universal cover of a Lie group is the 1-connected cover; this is a more intense version of the same idea.

Posted at January 24, 2008 1:31 AM UTC

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### Re: Classifying Spaces for 2-Groups

As a general comment, I would like to better understand the relation between

- the “standard” transformations between ana 2-functors $X \stackrel{\simeq}{\lt\leftarrow} Y \stackrel{g}{\to} \mathbf{B} G$ which you use, and which follows the principle summarized in Morphisms of anafunctors

- and the “concordance” point of view on transformations between anafunctors, which Nils Baas, Tore Kro and M. Bökstedt use in studying classifying spaces of 2-categories.

Notice that concordance is designed exactly such as to deal elegantly with the somewhat subtle issue in transformations of anafunctors: the case where you have a transformation between two of them which are not defined on the same cover $Y$.

In the entry Concordance (pdf) I observe that there should be a nice relation between the two points of view, both being related by the Hom-adjunction in $\omega$-categories (if we restrict attention to strict $\infty$-groups for simplicity and for the moment).

This was supposed to be a contribution to understanding the “almost” in

their result almost reduces to the fact mentioned above.

I haven’t got any feedback on this contribution so far, which probably just means that it’s not relevant. And I didn’t find the time and leisure to persue this further.

But at least, after having thought about it it made me realize that concordance is the right way to go when describing $\infty$-bundles with connection not at the Lie $\infty$-group level, but at the Lie $\infty$-algebra level.

This is now section 6.2 “$L_\infty$-algebra homotopy and concordance” and section 7 “$L_\infty$-Cartan-Ehresmann connections” in arXiv:0801.3480v1.

As discussed there, in this Lie $\infty$-algebraic approach concordances play another important role: they automatically take care of the fact that for a finite transformation between morphisms of $L_\infty$-algebras, you actually do need to “integrate” $L_\infty$-algebra elements over an interval. (Compare the standard formula for the transformation of a Lie-algebra valued connection 1-form: $A' = g A g^{-1} + g d g^{-1}$, which comes from a group element acting on the Lie algebra which we are really interested in.)

But, as we discuss, there is also a way to deal with this using not concordance but “transformations”. See table 5 on p. 32. These higher morphisms of $L_\infty$-algebra reproduce, in particular, the 2-morphisms which John and Alissa describe in HDA VI as is described in the appendix.

So, there must be a very direct, abstract nonsense kind of way to understand the precise relation between concordance and transformation. We should think about it.

By the way, the Lie $\infty$-algebraic Cartan-Ehresmann kind of way to say that String bundles have the same characteristic classes as the underlying $G$-bundles but without the Pontrjagin class corresponds to theorem 3, p. 15.

In the next version we should add a discussion of this in the light of your result here, also relating in more detail to Ginot and Stiénon on Characteristic Classes of 2-Bundles.

Posted by: Urs Schreiber on January 24, 2008 12:45 PM | Permalink | Reply to this

### Re: Classifying Spaces for 2-Groups

Urs wrote:

Yes, Danny had this 3-lemma proof almost worked out at least a year ago. Then I got involved and wanted a ‘cleaner, more conceptual’ proof. We spent the spring of 2007 working on that proof… but it became clear that while certain experts on model categories could carry it out, we weren’t them.

So, when Danny came back to Riverside over Christmas, we decided it was time to switch back to his original strategy. It turned out a few details needed to be straightened out, involving nasty subtleties like the difference between locally trivial fiber bundles and fibrations (and how the former isn’t always the latter), and the difference between locally contractible topological groups, groups where the inclusion $1 \hookrightarrow G$ is a closed cofibration (aka ‘NDR pair’) and groups where the inclusion $1 \hookrightarrow G$ is a strong NDR pair. Technical things I never really wanted to think about! Luckily, in the end Peter May saved us — and in the end, everything worked out perfectly.

All this may sound a bit scary and unpleasant, but it’s actually not, in the end. Milgram, Segal and Steenrod noticed you could build a classfying space $B G$ for a topological group $G$ by taking the geometric realization of the nerve of $G$ (viewed as a topological category with one object). They also got a space $E G$ and a map $E G \to B G$. The question then arises:

When is $E G \to B G$ a fibration?

and the nicest known answer seems to be

When $1 \hookrightarrow G$ is a closed cofibration!

This should be in some textbook somewhere, but we found it in some classic papers by Peter May, in somewhat concealed form.

As a general comment, I would like to better understand the relation between

- the “standard” transformations between ana 2-functors $X \stackrel{\simeq}{\lt\leftarrow} Y \stackrel{g}{\to} \mathbf{B} G$ which you use, and which follows the principle summarized in Morphisms of anafunctors

- and the “concordance” point of view on transformations between anafunctors, which Nils Baas, Tore Kro and M. Böksted use in studying classifying spaces of 2-categories.

Notice that concordance is designed exactly such as to deal elegantly with the somewhat subtle issue in transformations of anafunctors: the case where you have a transformation between two of them which are not defined on the same cover $Y$.

Yes, this subtle issue is also what my ‘cleaner, more conceptual’ proof was supposed to tackle head-on… but in a superficially different way than you suggest, using model categories rather than higher categories:

In the entry Concordance (pdf) I observe that there should be a nice relation between the two points of view, both being related by the Hom-adjunction in $\omega$-categories (if we restrict attention to strict $\infty$-groups for simplicity and for the moment).

In the model category approach, I believe this adjunction should manifest itself as a ‘Quillen equivalence’: a specially nice adjunction between model categories. I’ve gotten an email from a young homotopy theorist who thinks he can work out the technical issues involved, so we’ll see what happens.

I haven’t got any feedback on this contribution so far, which probably just means that it’s not relevant.

It’s probably very relevant! Alas, every time you’ve told me about this, my internal reaction was aargh, Danny and I should just finish writing this paper… I didn’t want to think about yet another ‘cleaner, more conceptual’ strategy for solving a problem, when we had one solution almost ready to write up.

One more thing:

Having criticized it a little, I should add a few words in defense of Danny’s 3-lemma strategy for proving this theorem:

Theorem. Suppose that $\mathbf{G}$ is a well-pointed topological 2-group and $M$ is a paracompact Hausdorff space admitting good covers. Then there is a bijection $\check{H}^1(M,\mathbf{G}) \cong [M,B|\mathbf{G}|]$ where the topological group $|\mathbf{G}|$ is the geometric realization of the nerve of the topological groupoid $\mathbf{G}$.

The main thing I like is Lemma 1, which gives a very practical concrete description of $|\mathbf{G}|$ in terms of the crossed module $(G,H, t,\alpha)$ corresponding to the 2-group $\mathbf{G}$. Namely:

$|\mathbf{G}| \cong (G \ltimes E H)/H$

Segal showed that $E H$ is a topological group. $G$ acts on $H$ via $\alpha$, so it acts on this group $E H$. This lets us define the semidirect product $G \ltimes E H$. There’s a way to see $H$ as a normal subgroup of $G \ltimes E H$, via $h \mapsto (t(h), h^{-1})$. So, we can take the quotient $(G \ltimes E H)/H$, and that turns out to be $|\mathbf{G}|$.

If you remember, Danny was using some very similar tricks already in our paper ‘From Loop Groups to 2-Groups’.

Posted by: John Baez on January 24, 2008 6:03 PM | Permalink | Reply to this

### Re: Classifying Spaces for 2-Groups

John wrote:

The main thing I like is Lemma 1, which gives a very practical cocrete description of $|\mathbf{G}|$ in terms of the crossed module $(G,H,t,\alpha)$ corresponding to the 2-group $\mathbf{G}$. Namely: $|\mathbf{G}| \simeq (G \ltimes E H)/H$

Yes, that’s nice. Back when we first talked about this, I had the impression that a nice way to understand this is by looking at tbe groupoid version of the universal $\mathbf{G}$-2-bundle

$\mathbf{G} \hookrightarrow INN_0(\mathbf{G}) \rightarrow\gt \mathbf{B} \mathbf{G}$

which I described with David Roberts in arXiv:0708.1741v1.

Hm, let me see if I find that old email I once sent. Ah, here it is.

So you considered the exact sequence of topological 2-groups

$1 \to H \to \mathbf{E} \to \mathbf{G} \to 1 \,,$ where $\mathrm{Obj}(\mathbf{E}) = G \ltimes H$ and $\mathrm{Mor}(\mathbf{E}) = (G \ltimes H) \ltimes H \,.$

The statement $|\mathbf{G}| \simeq (G \ltimes E H)/H$ together with $|\mathbf{E}| \simeq G \ltimes E H$ then leads to the sequence of topological 1-groups $1 \to H \to G \ltimes E H \to (G \ltimes E H)/H$ which you use to get an explicit description of $|\mathbf{G}|$.

Back then this reminded me of the following. I am still not sure if it is relevant, but it sure looks interesting to me.

I wrote:

As you know, we point out that for $C$ any 2-groupoid, we get a short exact sequence

$Mor(C) \to T C \to C$ of 2-groupoids with a couple of nice properties. In particular, if

$C = \mathbf{B} \mathbf{G}$

is a 1-object 2-groupoid, we get the exact sequence

$\mathbf{G} \to INN_0(\mathbf{G}) \to \mathbf{B}\mathbf{G}$

and that this “is” the universal $\mathbf{G}$ 2-bundle.

(We don’t try to take nerves and their realizations of this (David might in his thesis, I guess), but instead discuss how it plays the role of the universal 2-bundle in the world of 2-groupoids.)

For $\mathbf{G}$ coming from the crossed module $(H \to G)$ I’d think that the exact sequence which you describe, $H \to \mathbf{E} \to \mathbf{G}$ is obtained from the one above by quotienting out one factor of G.

I am not claiming that I have rigorously proven the following, but it seems to be true:

it seems your sequence is the pushout of our sequence along the canonical morphism

$\mathbf{G} \to \mathbf{B} H$

namely

$\array{ \mathbf{G} &\to& INN_0(\mathbf{G}) &\to& \mathbf{B}\mathbf{G} \\ \downarrow && \downarrow && \downarrow \\ \mathbf{B} H &\to& \mathbf{B} \mathbf{E} &\to& \mathbf{B} \mathbf{G} }$

with “our” sequence on top and “your” sequence in the bottom row, with the left vertical arrow the canonical one, the middle one a pushout and the right one the identity.

As a conistency check, notice that we show in our paper that $INN_0(\mathbf{G})$ is a 3-group coming from the 2-crossed module which is the mapping cone of the identity on the crossed module $(H \to G)$.

This means it is itself given by a complex of group

$H \to G \ltimes H \to G$

which is almost, up to the Peiffer lifting, a crossed module of crossed modules

$(G \ltimes H) \ltimes (G \ltimes H)$

So it’s plausible that dividing out one factor of $G$ here we do get your 2-group

$E$ “=” $(G \ltimes H) \ltimes H$,

that you consider.

If this works as I imagine it does, it would seem to give a nice conceptual and arrow-theoretic way to think the formulas you have.

That’s what I said back then. I haven’t really thought much more about it. Except that back then I really wrote $\Sigma \mathbf{G}$ for the one-object 2-groupoid corresponding to the 2-group $\mathbf{G}$, whereas now I write $\mathbf{B}\mathbf{G}$.

Posted by: Urs Schreiber on January 24, 2008 6:42 PM | Permalink | Reply to this

### Re: Classifying Spaces for 2-Groups

Some points:

1) The notion of 2-group seems a special case of the notion of 2-groupoid and so of globular omega-groupoid. These were shown equivalent to crossed complexes by Philip Higgins and I in 1981 (CTGDC). There is a general question of the appropriate structures for higher dimensional group(oid)s’ as the possibilities proliferate, partly because of the many compact convex sets in R^n for n > 1. All these possibilities and their pros and cons should be discussed. I am against the idea that crossed modules are 2-groups in disguise’, as this is making an assumption. We need to find out which concepts are best in which situations.

2) A simplicial version of the classifying space BC of a crossed complex C was defined and discussed in a paper with Higgins in Proc. Camb. Phil. Soc., 1991. The advantage was a result on homotopy classification of maps of a CW-complex to BC in terms of crossed complex maps \Pi X_* to C. This generalises old results of Eilenberg-Mac Lane, including the local coefficient case. This has been generalised to the equivariant case in work with Golasinski, Porter and Tonks (K-theory, 2001). It is clearly related to nonabelian cohomology.

3) The description of BC was for a crossed complex in the category of sets. If C is a topological crossed complex then its nerve as we defined it is a simplicial space, and so again defines BC. This needs work to exploit its properties.

4) In the abstract case we can successfuly use fibrations of crossed complexes (as coefficient morphisms, leading to families of exact sequences in nonabelian cohomology). Can this be generalised to the topological case?

Posted by: Ronnie Brown on January 24, 2008 10:30 PM | Permalink | Reply to this

### Re: Classifying Spaces for 2-Groups

Two questions:

1) Once we start talking about BG, I wonder why we need inverses? G a monoid works fine, as should a monoidoid (yuch!)

2) In terms of extensions of topological groups
K –> H –> G
it is reasonable to consider H–>G being a principal K bundle.

Then we can forget the group and classify the bundle.
If the bundle is trivial i.e. topologically split we can consider the group structure as usual in terms of a 2-cocyle where now cochains are continuous.

This exists in the literature: Graeme Segal has published one version, another due to D. Johnson I believe remains unpublished.

Posted by: jim stasheff on January 25, 2008 1:20 AM | Permalink | Reply to this

### Re: Classifying Spaces for 2-Groups

Jim wrote:

Once we start talking about $B G$, I wonder why we need inverses? $G$ a monoid works fine, as should a monoidoid (yuch!)

“Yuch”?

I always say “yuck”.

Groupoids are called groupoids, but monoidoids are called categories — and for that, we should all be grateful.

If we define $B G$ as the classifying space of the nerve of the topological category $G$, it’s easy to generalize to the case where $G$ is a topological a category, or a topological 2-category, or even beyond.

As I hinted above, Nils Baas, Marcel Bokstedt and Tore Kro have defined ‘charted $C$-bundles’ for any topological 2-category $C$. When the topological 2-category $C$ and the space $M$ are sufficiently nice, they show the set of ‘concordance classes’ of charted $C$-2-bundles over $M$ is in 1-1 correspondence with the set of homotopy classes

$[M, B C].$

So, there’s no need for inverses! However, you have be a bit careful dealing with bundles (or 2-bundles) where the transition functions don’t have inverses.

The work of Baas Bökstedt and Kro uses Jack Duskin’s idea of the ‘nerve of a 2-category’ — generalized to the topological case. Danny and I only use the ‘nerve of a category’ — generalized to the topological case.

Here’s a charming fact, mentioned in our paper. Say someone walks up and hands you a topological 2-group $\mathbf{G}$.

You can think of it as a topological 2-category with one object, follow Duskin’s prescription to take its nerve $N \mathbf{G}$, and take the geometric realization of that to get a space $|N\mathbf{G}|$.

Or, you can think of it as a topological groupoid that just happens to be equipped with a group structure! If you take the geometric realization of the nerve of this topological groupoid, you get a space we call $|\mathbf{G}|$. Of course, this just happens to have a group structure. So, $|\mathbf{G}|$ is a topological group. So, you can form its classifying space $B|\mathbf{G}|$.

It turns out that these two spaces $|N\mathbf{G}|$ and $B|\mathbf{G}|$ are homotopy equivalent!

There’s even a third way to get your hands on the classifying space of a topological 2-group. All this is discussed in Section 5.2 of our paper.

Posted by: John Baez on January 25, 2008 10:08 PM | Permalink | Reply to this

### Re: Classifying Spaces for 2-Groups

The notion of 2-group seems a special case of the notion of 2-groupoid and so of globular omega-groupoid. These were shown equivalent to crossed complexes by Philip Higgins and I in 1981 (CTGDC) [here I suppose? - urs]. There is a general question of the appropriate structures for higher dimensional group(oid)s’ as the possibilities proliferate, […]. All these possibilities and their pros and cons should be discussed. I am against the idea that crossed modules are 2-groups in disguise’, as this is making an assumption. We need to find out which concepts are best in which situations.

Lately you have kept pointing out to us (or maybe to me, in particular) here that we may not appreciate sufficiently existing results about crossed complexes. I am actually grateful for your persistence in this matter and will try (am trying) to better myself.

But maybe I could add this remark: from my subjective perspective (which is obviously different from your subjective perspective!) it has always been the $n$-group which is the “true” object, and the crossed complex which is “just a way to describe and handle it”.

Possibly the notion of homotopies of crossed complexes is a point in case:

the definition of a homotoy of morphisms of a crossed complex is a list of rather unilluminating (John would probably say: scary) equations.

But it turns out (as it should) that they describe nothing but the obvious 2-commuting naturality tin-can diagrams between morphisms of the corresponding higher groups.

So in one case I have a very immediate construction of the objects of interest, which is manifestly “right”, while on the other hand I have a host of equations whose relevance I can only ascertain by finding that lots of nice facts follow from these formulas.

Personally, I feel this is a good reason for thinking that

crossed modules are 2-groups ‘in disguise’.

I openly admit that this is how I feel and how I have always felt.

But, and here I don’t want to be misunderstood, of course this subjective feeling about concepts in no way justifies being careless about the host of good results about crossed complexes which have been obtained. So I appreciate your comments here.

One last remark: even though at this point (in this entry here) we are exclusively talking about strict 2-groups, maybe one other reason for assigning priority to the concept of an $n$-group is that in general we may want and have to consider weak ones.

Posted by: Urs Schreiber on January 25, 2008 12:39 PM | Permalink | Reply to this

### Re: Classifying Spaces for 2-Groups

Urs, “we can always replace weak 2-groupoids with strict ones without losing any homotopy theoretic information, and this strictification does not alter derived mapping spaces.” From page 2 of this paper by B. Noohi.

Is there some other advantage to “consider weak ones”?

Posted by: Charlie Stromeyer Jr on January 26, 2008 8:39 PM | Permalink | Reply to this

### Re: Classifying Spaces for 2-Groups

and yet, for n=2, bicats are equivalent to 2 cats
but this does not go for n=3

Posted by: jim stasheff on January 27, 2008 7:25 PM | Permalink | Reply to this

### Re: Classifying Spaces for 2-Groups

I agree that there are several equivalent categories of strict objects and that the switching between these is a powerful procedure, which Philip Higgins and I exploit. Where we go furthest away from crossed complexes is to the cubical case, which is even simpler to understand than the globular case.

This is discussed a bit on my page (recently revised)

http://www.bangor.ac.uk/r.brown/nonab-a-t.html

However crossed complexes are useful for computation and for their relation to the widely used chain complexes.

I agree about the weak case, but even here the cubical case may have possibilities and advantages, as shown in work of Richard Steiner.

I am more than happy to discuss these questions directly if you wish.

Posted by: Ronnie Brown on January 27, 2008 3:43 PM | Permalink | Reply to this

### Re: Classifying Spaces for 2-Groups

Ronnie Brown:

I am against the idea that ‘crossed modules are 2-groups in disguise’, as this is making an assumption.

Well, 2-groups are also crossed modules in disguise! The 2-category of crossed modules is equivalent to the 2-category of 2-groups. So, it’s a matter of convenience which we use… and if you look at my paper with Danny, you’ll see we use both viewpoints very heavily.

In fact, we really need four different viewpoints, which we list and explain in Section 3.

A simplicial version of the classifying space $B C$ of a crossed complex $C$ was defined and discussed in a paper with Higgins in Proc. Camb. Phil. Soc., 1991.

Interesting! Thanks, I’ll look at that and add a reference to Section 5.2, where we rapidly summarize a bunch of work on classifying spaces for 2-groups — whoops, I mean crossed modules.

Posted by: John Baez on January 25, 2008 10:23 PM | Permalink | Reply to this

### Re: Classifying Spaces for 2-Groups

I agree with all comments that there are a number of equivalent categories of algebraic objects, and one uses the appropriate one for the context at hand. I am not sure they should be called disguises for each other! The switching between these categories can be enormously powerful, partly because in higher dimensions the equivalences are non trivial; the cubical versus crossed complex switch is exploited in the work with Philip Higgins!

The value (for certain purposes) of these different (strict) structures is discussed a bit on my page

http://www.bangor.ac.uk/r.brown/nonab-a-t.html

I am not surprised at people avoiding the cubical version, but this was my basic intuition for conjecturing and proving theorems which are not accessible by other means it seems. It also gave our first (1982) version of the classifying space of a crossed complex (see the above web page).

Urs’ point about the weak theory is a good one. I suspect the cubical theory may eventually show advantages in this respect - see recent papers of Richard Steiner. There is a lot of work to be done here to see if this idea is correct. But I love cubes for the ease of ‘algebraic inverse to subdivision’, and the application of that to local-to-global problems.

Posted by: Ronnie Brown on January 27, 2008 3:27 PM | Permalink | Reply to this

### Re: Classifying Spaces for 2-Groups

“the difference between locally trivial fiber bundles and fibrations (and how the former isn’t always the latter)…”

What! do you then mean by fibration! most defintions dear to alg top types use fibrations as a generalization of bundle = locally trivial e.g. the path fibration is not locally trivial

btw, on your p.4 does principal G-2-bundle mean locally trivial?

Posted by: jim stasheff on January 25, 2008 1:30 AM | Permalink | Reply to this

### Re: Classifying Spaces for 2-Groups

Jim wrote:

Note: This business of e.g. ‘without the Pontryagin class’ follows from a classic Gysin sequence argument, using crucially that the extension is by $S^1$.

Yes — Danny came up with a Gysin sequence computation of the cohomology of $|String(G)|$ (the 3-connected cover of the compact simple Lie group $G$), but he didn’t see how to use this to get the ring structure on the cohomology… so we went ahead with the spectral sequence argument.

All this is a bit beyond my range of expertise, alas. To me a ‘spectral sequence’ is something like the series of ghosts that came to haunt Scrooge on the night before Christmas… by the time we get to the $E_3$ term I break down, repent, and promise to quit being a mathematician. More importantly, I don’t know if you can use the Gysin sequence to figure out the ring structure on cohomology.

Posted by: John Baez on January 25, 2008 5:49 PM | Permalink | Reply to this

### Re: Classifying Spaces for 2-Groups

” I don’t know if you can use the Gysin sequence to figure out the ring structure on cohomology.”

Up to a point which in this case is I bet enough - the connecting morphism in cohomology has a specific form.

When Danny’s done being a witness, let’s ask.

Posted by: jim stasheff on January 25, 2008 6:38 PM | Permalink | Reply to this

### Re: Classifying Spaces for 2-Groups

Yes, you can use the Gysin sequence to figure out the ring structure on cohomology.
One of the maps in the sequence is given by the (wedge or cup) product with the Euler class of the sphere bundle.

You can look it up in Wiki.

Posted by: jim stasheff on January 26, 2008 2:00 PM | Permalink | Reply to this

### Re: Classifying Spaces for 2-Groups

Note: This business of e.g. ‘without the Pontryagin class’ follows from a classic Gysin sequence argument, using crucially that the extension is by S1.

For other groups, life will be much more interesting.

Posted by: jim stasheff on January 25, 2008 1:24 AM | Permalink | Reply to this

### Re: Classifying Spaces for 2-Groups

the Lie $\infty$-algebraic Cartan-Ehresmann kind of way to say that String bundles have the same characteristic classes as the underlying $G$-bundles but without the Pontrjagin class corresponds to theorem 3, p. 15.

There is now a more pronounced statement of this, figure 8 on p. 48.

Posted by: Urs Schreiber on January 25, 2008 3:26 PM | Permalink | Reply to this

### Re: Classifying Spaces for 2-Groups

“just as a group is a set with multiplication and inverses, a 2-group is a category with multiplication and inverses.”

A “set” is a 0-category, and a “category” is a 1-category, so wouldn’t it be more logical to call a group, a “0-group”, and a 2-group a “1-group”?

Also, if you say that a category is a set with morphisms, a 2-category is a category with 2-morphisms between morphisms, a 3-category is a 2-category with 3-morphisms between 2-morphisms, etc, wouldn’t it be more logical to define “2-group” as follows? A category is a set with morphisms between elements. A group is a set with binary operations between elements (with the identity and inverses). Just as a 2-category is a category with 2-morphisms between morphisms, a 2-group would be a group with 2-binary operations between binary operations. In other words, let’s say you had a group with several binary operations, and it was a group under each of those binary operations. Then let’s say you could perform a 2-binary operation on two of those binary operations, and get one of the other binary operations. In fact these these binary operations would themselves form a group, where the various binary operations would be elements of the group, which would then be a group under the 2-binary operation.

Jeffery winkler

Posted by: Jeffery Winkler on February 4, 2008 7:08 PM | Permalink | Reply to this

### Re: Classifying Spaces for 2-Groups

Jeffrey wrote:

A “set” is a 0-category, and a “category” is a 1-category, so wouldn’t it be more logical to call a group, a “0-group”, and a 2-group a “1-group”?

Perhaps! I often think about these things.

Right now, the custom is to call the categorified version of a familiar thingie a “2-thingie”. For example, a categorified category is a 2-category, a categorified group is a 2-group, and so on. This system has one big advantage, which is that you can take all your favorite theorems, stick “2-” in front of lots of words, and hope that some similar result is true.

Of course if we stuck to this system in a truly stubborn way, we might call a category a “2-set”, and a functor a “2-function”.

I’m not eager to force the terminology to be “logical” too soon, because we don’t really understand things very well yet.

For example, it turns out that $n$-categories make sense not only for $n = 0$, but also for $n = -1$ and $-2$ — see the section on ‘The Power of Negative Thinking’ here. Lots of things might more sense if our numbering system took this into account. But, I think it’s too early to optimize things.

It’s hard to get notation right the first time, and bad old notation has a way of sticking around. For example, in astronomy, Population I stars are older than Population II stars, and oxygen counts as a “metal”. The crazy system for determining the “magnitude” of a star dates back to Ptolemy, and shows no sign of going away.

This is why we need civilization to collapse now and then.

Posted by: John Baez on February 4, 2008 7:44 PM | Permalink | Reply to this

### Re: Classifying Spaces for 2-Groups

wouldn’t it be more logical to call a group, a “0-group”

I’d think not, because a group is the abstract diagrams specifying a group internalized in a 1-category.

Whenever a concept is internalized in an $n$-category, we call the result an $n$-thing.

But sometimes people give 2-things new atomic names. Then things can become a little confused.

So happened with “stacks” and “gerbes” and “branes”. A “stack” as well as a “gerbe” is really a 2-sheaf. Hence “2-stacks” and “2-gerbes” are really 3-sheaves.

Of course being consistent with being inconsistent reduced the confusion. So for instance 1-branes couple to 1-gerbes with connection. Which is okay, since 1-branes are really 2-particles.

So, “sheaves” and “particles” have been around before “gerbes” and “branes” were named, and if people had payed more attention (or rather: had cared) they would have called the latter 2-this and 2-that.

Posted by: Urs Schreiber on February 4, 2008 9:50 PM | Permalink | Reply to this

### Re: Classifying Spaces for 2-Groups

This is why we need civilization to collapse now and then.

Ah - stimulated annealing in large scale.

Posted by: Urs Schreiber on February 4, 2008 9:53 PM | Permalink | Reply to this

### Re: Classifying Spaces for 2-Groups

stimulated annealing

Was that a pun, a mistake, or has it become a recognised term? I have heard it once or twice before.

I see Google gives 1260 hits for your version, and 610000 for simulated annealing.

Posted by: David Corfield on February 5, 2008 9:14 AM | Permalink | Reply to this

### Re: Classifying Spaces for 2-Groups

I asked John the same question when he told me in Vienna that intellectual progress requires forgetting your mistakes and that this is like in stimulated annealing.

Does seem to make sense, conceptually: you want to catalyze (hence stimulate) that procees.

But who am I to talk about single letter differences between words.

Posted by: Urs Schreiber on February 5, 2008 11:46 AM | Permalink | Reply to this

### Re: Classifying Spaces for 2-Groups

Doesn’t the ‘annealing’ part already give us the sense of causal agency? ‘Simulated’ is drawing attention to the fact there are no real metals being heated – it’s just an analogy, albeit a very good one.

Posted by: David Corfield on February 5, 2008 4:08 PM | Permalink | Reply to this

### Re: Classifying Spaces for 2-Groups

Urs wrote:

I asked John the same question when he told me in Vienna that intellectual progress requires forgetting your mistakes and that this is like in stimulated annealing.

Really? I must have been tired or joking or something. The usual term is ‘simulated annealing’. I don’t remember ever talking about ‘stimulated annealing’.

If you look in Wikipedia, you’ll see it begins by describing simulated annealing as a ‘generic probabilistic meta-algorithm for the global optimization problem’. Impressive — but this article needs a bit of editing so ordinary people like me can understand it.

Basic idea: if you get stuck on what you think is the best thing to do, you’ll need to get pushed around a bit to find something even better.

Posted by: John Baez on February 5, 2008 6:42 PM | Permalink | Reply to this

### Re: Classifying Spaces for 2-Groups

Really?

Oh, dear. After I sent that message I was getting afraid that this would happen.

Sorry, never mind, it’s probably me misremembering things. And in any case, it doesn’t really matter.

Time for me to forget that particular mistake of mine…

Posted by: Urs Schreiber on February 5, 2008 7:51 PM | Permalink | Reply to this

### Re: Classifying Spaces for 2-Groups

“The crazy system for determining the “magnitude” of a star dates back to Ptolemy, and shows no sign of going away.”

I think it was Hipparchus who devised the system of star magnitudes still in use today.

Posted by: Jeffery Winkler on February 6, 2008 5:22 PM | Permalink | Reply to this

### Re: Classifying Spaces for 2-Groups

Dear Professors Baez and Stevenson,

I have only just started to read your paper above and so I only have so far just one comment concerning the last sentence in paragraph 2 of page 2:

Non-abelian cohomology can also be defined via a generalization of abelian gerbes (in terms of 2-categories). This is explained in this paper about a tower of n-gerbes by A. Tsemo.

Posted by: Charlie Stromeyer Jr on January 25, 2008 2:45 PM | Permalink | Reply to this

### Re: Classifying Spaces for 2-Groups

Non-abelian cohomology can also be defined via a generalization of abelian gerbes (in terms of 2-categories). This is explained in this paper about a tower of n-gerbes by A. Tsemo.

I haven’t absorbed that article yet. But I know the following way to characterize nonabelian cohomology by collections of higher abelian cohomologies:

Given a classifying space $|\mathbf{G}|$ for some $n$-group $\mathbf{G}$, we can consider for each ordinary degree $k$ cohomology class of $|\mathbf{G}|$ the $(k-2)$-gerbe = $(k-1)$-line bundle classified by it.

Hence, given any classifying map $X \to |\mathbf{G}|$ we can not only pull back the universal (in general non-abelian) $\mathbf{G}$-bundle to $X$ and get a class in nonabelian cohomology, but we can also pull back all these higher abelian line bundles. Their classes know all about the characteristic classes of the nonabelian $\mathbf{G}$-bundle. That characterizes the nonabelian cocycle to a great extent (though in general not entirely).

This is proposition 35 on p. 64 in the paragraph Line $n$-bundles on classifying spaces here.

I can’t tell yet if that is at all related to what the paper you pointed to is addressing, but the term “tower of $n$-gerbes” vaguely reminded me of this.

Posted by: Urs Schreiber on January 25, 2008 3:40 PM | Permalink | Reply to this

### Re: Classifying Spaces for 2-Groups

Is there a notion of Eilenberg-MacLane spaces for 2-groups? So $K(G, n)$ has homotopy only in dimensions $n$ and $n + 1$ given by the 2-group $G$, and is unique in some suitable sense.

I feel sure I’ve asked this before.

Posted by: David Corfield on January 25, 2008 2:51 PM | Permalink | Reply to this

### Re: Classifying Spaces for 2-Groups

Sure
Never seen it said that way but
t:H –> G
determines the k-invariant K(H,n+1)–>K(G,n+1)

Posted by: jim stasheff on January 25, 2008 6:45 PM | Permalink | Reply to this

### Re: Classifying Spaces for 2-Groups

David wrote:

Is there a notion of Eilenberg-MacLane spaces for 2-groups? So $K(\mathbf{G},n)$ has homotopy only in dimensions $n$ and $n+1$ given by the 2-group $\mathbf{G}$, and is unique in some suitable sense.

This works, but you have to be a bit careful. You’ll like this!

First let’s warm up a bit:

$K(G,1)$ makes sense for any group $G$.

$K(G,2)$, $K(G,3)$ etcetera make sense when $G$ is abelian.

Backing up, $K(G,0)$ makes sense whenever $G$ is a mere set!

So: set, group, abelian group, abelian group… what we’re seeing here is the first column of the periodic table!

A $k$-tuply groupal $n$-groupoid is an $(n+k)$-groupoid with only one $j$-morphism for $j$ less than $k$. The periodic table shows the pattern:

...............................................
k-tuply groupal n-groupoids

n = 0       n = 1          n = 2

k = 0        sets        groupoids      2-groupoids

k = 1        groups      2-groups       3-groups

k = 2        abelian     braided        braided
groups      2-groups       3-groups

k = 3        "     "     symmetric      sylleptic
2-groups       3-groups

k = 4        "     "     "      "       symmetric
3-groups

k = 5        "     "     "      "       "      "
....................................................


According to the homotopy hypothesis, $k$-tuply groupal $n$-groupoids are secretly the same as spaces with only $\pi_k, \dots, \pi_{n+k}$ nontrivial.

Now you’re asking about the second column. Let $\mathbf{G}$ be a 2-group. According to the homotopy hypothesis, a 2-group is secretly a space with only $\pi_1$ and $\pi_2$ nontrivial. You’re calling this space $K(\mathbf{G},2)$.

What about $K(\mathbf{G},3)$? According to the periodic table, this should make sense when $\mathbf{G}$ is a braided 2-group. And, it should be a space with only $\pi_2$ and $\pi_3$ nontrivial.

Indeed, that’s right: braided 2-groups classify spaces with only $\pi_2$ and $\pi_3$ nontrivial. A proof can be found lurking rather deeply here.

But here’s the interesting subtlety, which you will surely enjoy. For a group to be abelian is a mere property. But, for a 2-group to be braided is an extra structure.

So, you can’t get $K(\mathbf{G},3)$ unless you equip your 2-group $\mathbf{G}$ with a braiding.

I leave the case of $K(\mathbf{G},4)$ as an exercise.

It takes work to prove some of these things, but the pattern is simple and beautiful! It could be taught in high school.

Posted by: John Baez on January 25, 2008 7:23 PM | Permalink | Reply to this

### Re: Classifying Spaces for 2-Groups

So say we have a symmetric 2-group, $\mathbf{G}$. This lets us define, you say, Eilenberg-MacLane spaces $K(\mathbf{G}, n)$ for all $n$.

Now surely we’re going to want to start mapping into these spaces from other spaces. So we might think about $[X, K(\mathbf{G}, n)]$ and expect this to be the $n$th cohomology of $X$ of some kind, linked to the spectrum $E(n) = K(\mathbf{G}, n)$.

I wonder if $[X, K(\mathbf{G}, n)]$ was ambitious enough – mere homotopy classes of maps. Is this not to decategorify from a groupoid of such mappings?

Posted by: David Corfield on January 26, 2008 10:06 AM | Permalink | Reply to this

### Re: Classifying Spaces for 2-Groups

I wonder if $[X, K(\mathbf{G}, n)]$ was ambitious enough…

And further down the page you’re asking something similar.

Posted by: David Corfield on January 26, 2008 2:50 PM | Permalink | Reply to this

### Re: Classifying Spaces for 2-Groups

David wrote:

So say we have a symmetric 2-group, $\mathbf{G}$.

It’s also good to know how topologists would say this, so we can tap into their knowledge. Instead of saying ‘symmetric 2-group’, they’d say ‘homotopy 2-type which happens to be an infinite loop space’, or ‘spectrum with only $\pi_1$ and $\pi_2$ nontrivial’.

(Remember: the words ‘symmetric’, ‘infinite loop space’ and ‘spectrum’ are all ways of saying that we’ve entered the ‘stable’ range of the periodic table.)

This lets us define, you say, Eilenberg-MacLane spaces $K(\mathbf{G},n)$ for all n.

Right. Of course no topologist will understand you when you say this.

But, if you tell them “For each $n \ge 2$ I’ve got a space $K(\mathbf{G},n)$ with only $\pi_{n-1}$ and $\pi_n$ nontrivial, and these spaces are related by looping: $\Omega K(\mathbf{G},n) = K(\mathbf{G},n-1)$”, they’ll say something like “Oh! You mean you’ve got an infinite loop space with only $\pi_1$ and $\pi_2$ nontrivial! Why didn’t you just say so?”

And, they’ll proceed to tell you that these things are classified by a pair of abelian groups $\pi_1$, $\pi_2$ and a certain extra piece of data linking the two, studied extensively by Whitehead, Eilenberg and Mac Lane in the 1950s. This extra piece of data is really the associator and braiding in our symmetric 2-group, as Joyal and Street proved in 1986.

Now surely we’re going to want to start mapping into these spaces from other spaces. So we might think about $[X,K(\mathbf{G},n)]$ and expect this to be the $n$th cohomology of $X$ of some kind, linked to the spectrum $E(n)=K(\mathbf{G},n)$.

Oh, good! You figured it all out already!

Homotopy theorists regard Eilenberg–Mac Lane spectra (coming from abelian groups, mind you!) as the simplest of spectra. These ones coming from symmetric 2-groups would also be regarded as pathetically simple. The really juicy ones have nontrivial homotopy groups going all the way up… and all the way down, too: that’s how spectra go beyond infinite loop spaces.

Posted by: John Baez on January 27, 2008 4:05 AM | Permalink | Reply to this

### Re: Classifying Spaces for 2-Groups

Homotopy theorists regard Eilenberg–Mac Lane spectra (coming from abelian groups, mind you!) as the simplest of spectra. These ones coming from symmetric 2-groups would also be regarded as pathetically simple.

Pathetically simple though they may be, you yourself have expressed interest in finding out geometric descriptions of the spaces of the very simplest of spectra – $K(\mathbb{Z}, n)$. For example, here you are in TWF 149 wanting to know $K(\mathbb{Z}, 3)$:

Basically, the point is that the integers, the group U(1), and infinite-dimensional complex projective space are all really important in quantum theory. This is perhaps more obvious for the latter two spaces - the integers are so basic that it’s hard to see what’s so “quantum-mechanical” about them. However, since each of these spaces is just the loop space of the next, they’re all part of tightly linked sequence… and I want to know what comes next!

Now, are there some nice simple 2-groups which would yield geometrically interesting $K(\mathbf{G}, n)$’s?

Might there even be cases where you find something as pleasant as $CP^{\infty} = K(\mathbb{Z}, 2)$, with its universal line bundle?

Or is it that all you’ll get is a simple-ish composition of spaces from each of the two levels?

Perhaps a nice case to consider would be that weak 2-group we were talking about.

Posted by: David Corfield on January 27, 2008 10:55 AM | Permalink | Reply to this

### Re: Classifying Spaces for 2-Groups

Right. Of course no topologist will understand you when you say this.

>:O

So, since evidently our definitions differ.. what’s a topologist?

Posted by: John Armstrong on January 27, 2008 5:34 PM | Permalink | Reply to this

### Re: Classifying Spaces for 2-Groups

I meant someone who could help me on questions about $K(\mathbf{G},n)$’s with $\mathbf{G}$ a 2-group. For me, part of the fun of learning homotopy theory has always been taking questions about $n$-groupoids, translating them into homotopy theory lingo, asking experts these questions, and translating the answers back. Someday this may not be necessary, thanks to people like you… and everyone at this café. But right now it seems to be.

Posted by: John Baez on January 27, 2008 6:38 PM | Permalink | Reply to this

### Re: Classifying Spaces for 2-Groups

David wrote:

Pathetically simple though they may be, you yourself have expressed interest in finding out geometric descriptions of the spaces of the very simplest of spectra – $K(\mathbb{Z},n)$.

Indeed, I’m only interested in the pathetically simple aspects of mathematics: these alone are sufficient for a lifetime. And, while Eilenberg–Mac Lane spaces are considered dull from a homotopy-theoretic perspective, their homology groups can already be quite interesting, and so can their geometry.

So, yes, I’d like very much to understand $K(\mathbf{G},n)$’s for the simplest, most beautiful 2-groups.

Or is it that all you’ll get is a simple-ish composition of spaces from each of the two levels?

Don’t forget the theory of Postnikov towers: if we have a space $E$ with only $\pi_1, \dots \pi_n$ nonzero, it’s a ‘twisted product’ of $K(\pi_n,n)$ and some space $B$ that is just like $E$ except that its $\pi_n$ has been killed.

By ‘twisted product’, I really mean ‘bundle’ So, $E$ is the total space of a bundle with base space $B$ and fiber $K(\pi_n,n)$’. Topologists would write it this way:

$K(\pi_n,n) \hookrightarrow E \to B$

$E$ could just be the product of the base and the fiber… or the fiber could ‘twist around’ as we travel around the base space.

So, the answer to your question is yes if a fiber bundle counts as a ‘simple-ish composition’ of the base space and the fiber. Fiber bundles can certainly be understood, but the ‘twisting’ phenomenon can make them not so simple. People have developed a lot of technology to understand this stuff, though.

Here’s an easy example: let $\mathbf{G}$ be the 2-group with $\pi_1 = \mathbb{Z}/2$, $\pi_2 = \mathbb{Z}$, the nontrivial action of $\mathbb{Z}/2$ on $\mathbb{Z}$ (there’s only one nontrivial option here), and a trivial associator $a :(\mathbb{Z}/2)^3 \to \mathbb{Z}$. What’s $K(\mathbf{G},2)$?

In other words: what space has $\mathbf{G}$ as its fundamental 2-group?

Maybe you can guess. It’s the total space of a bundle with fiber ??? and base space ???. But, the nontrivial action of $\pi_1$ on $\pi_2$ means its a nontrivial bundle. Can you see how it’s twisted?

I’m afraid this 2-group $\mathbf{G}$ can’t be made into a braided or symmetric 2-group, thanks to that nontrivial action. So, we’d need another example before we march on and understand $K(\mathbf{G},n)$ for higher $n$.

We’ve looked at other examples of 2-groups, the same sorts of questions are interesting for all of these.

Posted by: John Baez on January 27, 2008 6:07 PM | Permalink | Reply to this

### Re: Classifying Spaces for 2-Groups

Hmm, $\mathbf{G}$ a 2-group with $\pi_1 = \mathbb{Z}/2$ and $\pi_2 = \mathbb{Z}$. Does this suggest $RP^{\infty}$ as base and $CP^{\infty}$ as fibre?

You could almost see something flag-like going on, but I’m not so sure about mixing the real and the complex.

Posted by: David Corfield on January 28, 2008 9:30 AM | Permalink | Reply to this

### Re: Classifying Spaces for 2-Groups

Oh well, let’s live dangerously and plump for the infinite flag manifold, the direct limit of lines within 3-dimensional subspaces within $R^n$.

Posted by: David Corfield on January 28, 2008 11:15 AM | Permalink | Reply to this

### Re: Classifying Spaces for 2-Groups

David almost wrote:

Hmm, $\mathbf{G}$ a 2-group with $\pi_1 = \mathbb{Z}/2$ and $\pi_2 = \mathbb{Z}$. Does this suggest that if we’re looking for $K(\mathbf{G},2)$, we need the total space of a bundle with $RP^{\infty}$ as base and $CP^{\infty}$ as fibre?

Right! And it doesn’t just suggest it — it demands it, at least up to homotopy equivalence.

So, the only question is: which bundle?

We’re trying to use the interesting action of $\mathbb{Z}/2$ on $\mathbb{Z}$ to build this bundle — the action where the nontrivial element of $\mathbb{Z}/2$ acts by ‘flipping’ $\mathbb{Z}$:

$n \mapsto -n$

I hope you can visualize $\mathbb{R}P^\infty$ and the noncontractible loop in it, which corresponds to the nontrivial element of $\mathbb{Z}/2$.

(Can’t visualize $\mathbb{R}P^\infty$? Well, just visualize $\mathbb{R}P^2$ and let the higher dimensions fend for themselves.)

So, here’s how we get that action of $\mathbb{Z}/2$ on $\mathbb{Z}$ into the game: we build a bundle of $\mathbb{C}P^\infty$’s over $\mathbb{R}P^\infty$ so that when we carry our $\mathbb{C}P^\infty$ all the way around that noncontractible loop, it comes back ‘flipped’ somehow.

But how?

By abstract nonsense, we know that any automorphism of a group $A$ acts as a symmetry of $K(A,n)$. So, the ‘flipping’ automorphism of $\mathbb{Z}$:

$n \mapsto -n$

must give a symmetry of $K(\mathbb{Z},2) = \mathbb{C}P^\infty$.

But, what does this actually look like?

I’ll leave this as a fun puzzle for anyone who wants to tackle it.

(Hint: it might be easier to tackle the case of $K(\mathbb{Z},1)$ first. What does the automorphism $n \mapsto -n$ do to $K(\mathbb{Z},1)$?)

Oh well, let’s live dangerously and plump for the infinite flag manifold, the direct limit of lines within 3-dimensional subspaces within $\mathbb{R}^n$.

Hmm! Interesting guess. I don’t know if it’s right, though I see why you made it. The interesting thing is that you’re seeking a nicer sort of answer to this puzzle than any I actually know. I just know how to describe this space as the total space of a bundle. You’re seeking a more ‘synthetic’ description of this total space.

Now that you’ve raised the stakes in this way, I feel sure there must be a good answer of the sort you’re dreaming of!

But, I don’t know how the answer goes.

A point in this space $K(\mathbf{G},2)$ should be a real line in $\mathbb{R}^\infty$ together with a complex line in some copy of $\mathbb{C}^\infty$… where this copy of $\mathbb{C}^\infty$ depends on the real line we’ve chosen in a topologically twisted way.

Since there are only two actions of $\mathbb{Z}/2$ on $\mathbb{Z}$ — the trivial one and the one we want — I think we have a good shot at getting this ‘topological twisting’ correct as long as we do anything nontrivial.

Posted by: John Baez on January 30, 2008 11:59 PM | Permalink | Reply to this

### Re: Classifying Spaces for 2-Groups

My guess has the right homotopy groups, yes? Perhaps using this for help.

And isn’t there a twist going on as you rotate a line round 180 degrees onto itself? Doesn’t it change the orientation of any orthogonal plane?

Posted by: David Corfield on January 31, 2008 11:11 AM | Permalink | Reply to this

### Re: Classifying Spaces for 2-Groups

Indeed, isn’t orientation what it’s all about? that is, orientation of the CP.

If we are after a bundle (= twisted product po russki), considr what the classifying map MUST be.

btw, what’s the simplest example of a topological twist you’ve encountered?

Posted by: jim stasheff on January 31, 2008 1:40 PM | Permalink | Reply to this

### Re: Classifying Spaces for 2-Groups

I still can’t see the bigger picture. Any infinite flag of $m$-dimensional spaces in $n$-dimensional spaces in $\mathbb{R}^{\infty}$ will give a fibring with base space the infinite $m$-Grassmannian (if you see what I mean - the limit of Grassmannians $G_m(R^k)$ as $k \to \infty$) and fibre the infinite $(n - m)$-Grassmannian.

But I’m not sure how twisting goes, and whether the higher homotopy always gets killed off.

Hmm, why do we generally stop with one space fibred over another? What’s wrong with three spaces, such as the infinite flag of $m$-dimensional spaces in $n$-dimensional spaces in $p$-dimensional spaces in $\mathbb{R}^{\infty}$?

Is it that these are just compositions of fibings?

Posted by: David Corfield on February 1, 2008 2:17 PM | Permalink | Reply to this

### Re: Classifying Spaces for 2-Groups

David wrote:

I still can’t see the bigger picture.

I still don’t know if your original guess was right. But I can advise you on the general rules of the game.

If you’re seeking a connected space with a given $\pi_1$ and $\pi_2$, and vanishing higher homotopy groups, you need to specify those groups together with an action of $\pi_1$ and $\pi_2$ and a 3-cocycle

$a: \pi_1^3 \to \pi_2$

Of course, this is precisely the data needed to specify a 2-group.

Then there will be a unique space fitting this description — at least, up to weak homotopy equivalence. This space will be a bundle with fiber $K(\pi_2,2)$ and base space $K(\pi_1,1)$. The action of $\pi_1$ on $\pi_2$ describes how the fiber ‘twists around’ as we move it around any noncontractible loop in the base. The 3-cocycle describes something subtler. By carrying the fiber around the base, our space gives a kind of ‘action’ of $\pi_1$ on $K(\pi_2,2)$. But, this action is associative only up to homotopy! The 3-cocycle describes this homotopy: in category language, it’s the ‘associator’.

Luckily we’re doing an example where the ‘associator’ is trivial.

But I’m not sure how twisting goes, and whether the higher homotopy always gets killed off.

I’m not sure what you want to know about twisting — but for starters, just picture a Möbius strip! (I may have just given away the answer to Jim Stasheff’s puzzle.)

As for whether the higher homotopy gets killed off, here’s the deal: if you have a bundle whose base space and fiber both have vanishing homotopy groups above the $n$th, the same is true for the total space!

Moreoever, if you have a bundle with fiber $K(\pi_2,2)$ and base space $K(\pi_1,1)$, the total space will automatically be some space with $\pi_1$, this $\pi_2$, and vanishing higher homotopy groups. To say which space, we just need to specify the action of $\pi_1$ on $\pi_2$, and the 3-cocycle $a: \pi_1^3 \to \pi_2$.

All this stuff is part of the theory of ‘Postnikov towers’.

Hmm, why do we generally stop with one space fibred over another?

It depends what you’re doing! The theory of Postnikov towers says to get any connected space (up to weak homotopy equivalence), you first take a bundle of $K(\pi_2,2)$’s over a $K(\pi_1,1)$. Then you take the total space of that bundle. If your space has vanishing higher homotopy groups, you’re done.

If not, you look at a bundle of $K(\pi_3,3)$’s over what you’ve got so far, and take the total space of that. And so on. At each stage we need to specify an action of $\pi_1$ on our new $\pi_n$, and also a certain cocycle called a ‘Postnikov $k$-invariant’.

This is a Postnikov tower!

I thought you knew all this stuff. If you do, maybe you’re asking some other, deeper questions which I’m failing to catch. If you don’t, I’ve failed in my mission to explain the deep inner meaning of cohomology and Postnikov towers.

But, I shouldn’t complain. I used to think Postnikov Towers was a 1980’s-era Soviet sitcom about life in a government-run apartment complex.

Posted by: John Baez on February 1, 2008 7:00 PM | Permalink | Reply to this

### Re: Classifying Spaces for 2-Groups

There’s probably something of the kind of thing Rumsfeld forget to mention going on – the unknown knowns – along with the three he did refer to.

I’m aware if you press me that given a space, say the 2-sphere, you can build up a Postnikov tower, with $K(\mathbb{Z}, 2)$ as base, and $K(\mathbb{Z}, 3)$ as fibre. Then over that total space a bundle with fibre $K(\mathbb{Z}_2, 4)$, and so on, putting in all that extra Postnikovian stuff.

I guess I could then see that homotopically you’ve sorted out the 2-sphere. But given just that tower you wouldn’t know it was as special as such a simple thing as a 2-sphere. Is there any way you could know you could do as well as that from the tower?

Just as from the simple two-tiered tower we were considering, we could rest content with some twisted product of $CP^{\infty}$ and $RP^{\infty}$, but might hope for something lovelier, perhaps flag-ish. Is there anything to be said in general about how lovely we might hope the total space could be?

Posted by: David Corfield on February 2, 2008 12:19 PM | Permalink | Reply to this

### Re: Classifying Spaces for 2-Groups

Would surprise me. If I give you a simplicial set which happens to be K(Z,3),
how would you determine that?

Posted by: jim stasheff on February 2, 2008 12:46 PM | Permalink | Reply to this

### Re: Classifying Spaces for 2-Groups

David wrote:

I’m aware if you press me that given a space, say the 2-sphere, you can build up a Postnikov tower, with $K(\mathbb{Z}, 2)$ as base, and $K(\mathbb{Z}, 3)$ as fibre. Then over that total space a bundle with fibre $K(\mathbb{Z}_2, 4)$, and so on, putting in all that extra Postnikovian stuff.

I guess I could then see that homotopically you’ve sorted out the 2-sphere. But given just that tower you wouldn’t know it was as special as such a simple thing as a 2-sphere.

Right. Luckily, nobody is likely to “give you” such a tower anytime soon, since people don’t really know the homotopy groups of the 2-sphere, much less all the Postnikov data.

By the way — once you blogged about somebody’s presentation of the homotopy groups of $S^2$, or maybe $S^3$. I can’t find that anymore! And, I’ve learned some other similar things, lately. Can you remember the guy’s name, or anything?

Is there any way you could know you could do as well as that from the tower?

That sounds tough to me.

The point is, there are two ways to describe and understand spaces, and it’s very hard to translate between these descriptions:

• The first way is to build the space by gluing together cells — a CW complex. This makes it easy to compute the cohomology groups (maps into $K(\mathbb{Z},n)$’s) but hard to compute the homotopy groups (maps out of $S^n$’s). A typical space that’s easy to understand this way is $S^n$.
• The second way is to build the space by fibrations with $K(\mathbb{Z},n)$’s as fibers — a Postnikov tower. This makes it hard to compute the cohomology groups, but easy to compute the homotopy groups. A typical space that’s easy to understand this way is $K(\mathbb{Z},n)$.

You could say that the tension between these perspectives is what makes homotopy theory endlessly complicated and fascinating.

You could also say that spectra are an attempt to steer a compromise. For example, the homotopy groups of the sphere spectrum are easier to compute than the homotopy groups of $S^2$, while its cohomology groups are still very simple.

But, while the homotopy groups of the sphere spectrum are ‘easier to compute’, they’re still not all known.

There could even be some kind of ‘Heisenberg uncertainty principle’ putting limitations on how well we can compute both cohomology groups and homotopy groups, but I don’t think anyone has formalized anything quite like that.

Just as from the simple two-tiered tower we were considering, we could rest content with some twisted product of $\mathbb{C}P^{\infty}$ and $\mathbb{R}P^{\infty}$, but might hope for something lovelier, perhaps flag-ish.

I’m sure there is something lovelier — and you may have guessed it already! The action of $\mathbb{Z}/2$ on $\mathbb{Z}$, and their relation to $\mathbb{R}$ and $\mathbb{C}$ respectively, are so beautiful and simple that there should be a very poignant description of this particular homotopy 2-type. I just need more time to think about this (someday).

Is there anything to be said in general about how lovely we might hope the total space could be?

I’d be very happy to understand more examples, for starters.

Posted by: John Baez on February 4, 2008 10:02 PM | Permalink | Reply to this

### Re: Classifying Spaces for 2-Groups

…once you blogged about somebody’s presentation of the homotopy groups of $S^2$, or maybe $S^3$

That would be Wu’s theorem.

Posted by: David Corfield on February 5, 2008 9:08 AM | Permalink | Reply to this

### Re: Classifying Spaces for 2-Groups

For only slightly greater precision, that’s Jie Wu.

Posted by: jim stasheff on February 5, 2008 1:00 PM | Permalink | Reply to this

### Re: Classifying Spaces for 2-Groups

Great, thanks! Wu’s theorem gives a tantalizing description of all the homotopy groups of $S^3$. I just learned from somebody a superficially quite different — but even more tantalizing — description of all the homotopy groups of $S^2$. Clearly it’s time to write about these in This Week’s Finds!

Posted by: John Baez on February 5, 2008 11:51 PM | Permalink | Reply to this

### Re: Classifying Spaces for 2-Groups

It seems to me that if one takes the program of categorification seriously, one should categorify the notion of the classifying space as well. The collection of principal $G$-bundles over a fixed manifold $M$ ($G$ a 1-group) is a category and the homotopy classes of maps $M$ to $BG$ classify the equivalence classes of objects of that category. I’m guessing that if $G$ is a 2-group, then the principal $G$-bundles over $M$ form a 2-category with all 2-arrows being isomorphism. So one should probably look for a classifying “space” so that maps into it would recover more than just equivalence classes of objects of the 2-cateogory of $G$-bundles over $M$.

Posted by: Eugene Lerman on January 25, 2008 3:48 PM | Permalink | Reply to this

### Re: Classifying Spaces for 2-Groups

It seems to me that if one takes the program of categorification seriously, one should categorify the notion of the classifying space as well.

As long as we are talking about $n$-groupoids as opposed to general $n$-categories (in whatever guise) we have in fact that spaces already correspond to $\infty$-groupoids.

spaces $\leftrightarrow$ $\infty$-groupoids

(for instance John’s The Homotopy Hypothesis for an overview).

The fact that ordinary spaces don’t know about $\infty$-categories which are not $\infty$-groupoids is due to the fact that every path in a space may be traveresed in both directions. To get rid of that in-built invertibility of everything in the world of spaces, people are trying to develop directed homotopy theory.

So as long as we are talking spaces, no further categorification is to be expected.

But, to some degree it is a prejudice that we always need everything to be a space. The best argument for it is that homotopy theorists know so very much about spaces. But passing to spaces also makes things more difficult.

The issue that John and Danny addressed here (as did Baas-Bökstedt-Kro) only becomes difficult because spaces are so much more “smoothed out” than other things.

For instance, if we stay in the world of categories, the classification of $G$-bundles becomes a triviality:

$G$-bundles on a space $X$ are classified by equivalence classes of anafunctors

$X \stackrel{\simeq}{\leftarrow} Y \stackrel{g}{\to} \mathbf{B} G \,,$

where $\mathbf{B} G$ just denotes the category with a single object and $G$ worth of morphisms.

This is just the familiar statement that $G$-bundles are given by $G$-cocycles. It is only after hitting the above statement with the nerve realization functor:

$|\cdot| : \infty-Grpd \to spaces$

that things become more involved.

Namely it is clear that every transformation between $\infty$-functors between $\infty$-categories leads to a homotopy between the corresponding continuous maps between the corresponding topological spaces. But, since there is so much more room to “wiggle around” in a space, the converse of this statement becomes difficult to show.

That’s where all these extra conditions then come in that John mentiones, cofibrations, NDR pairs etc.

But if we just stay at the level of $\infty$-groupoids without passing to spaces the situation is clear:

for instance 2-bundles are classified by ana2-functors with values in the 2-groupoid $\mathbf{B} \mathbf{G}$.

Notice that here you see how the classifiying object does raise in dimension, too. That one does not see this for spaces is just due to the fact that spaces already are $\infty$-structures.

That’s why so much of $n$-category theory is like playing hare and hedgehog: the topologists have always already been there before.

Posted by: Urs Schreiber on January 25, 2008 4:51 PM | Permalink | Reply to this

### Re: Classifying Spaces for 2-Groups

Except unlike the hedgehogs, the topologists aren’t pretending in this case.

Posted by: David Corfield on January 25, 2008 5:07 PM | Permalink | Reply to this

### Re: Classifying Spaces for 2-Groups

every path in a space may be traversed in both directions.

but that’s invertibility only up to homotopy - at least `thin homotopy’
and BOmega X doesn’t care about that
Omega = based loop space

Posted by: jim stasheff on January 25, 2008 6:43 PM | Permalink | Reply to this

### Re: Classifying Spaces for 2-Groups

Urs, I agree that if $G$ is a 2-group, then there is a 2-stack $BG$ and for a manifold $X$ the 2-category of $G$ bundles over $X$ is equivalent to the 2-category $Hom (X, BG)$ (morphisms of stacks). But I don’t understand your explanation of why one would want $BG$ is a topological space.

Posted by: Eugene Lerman on January 25, 2008 7:07 PM | Permalink | Reply to this

### Re: Classifying Spaces for 2-Groups

As I understand them (if at all), stacks are a way of enlarging the cat if your construction e.g. BG doesn’t stay in the cat
but if X is a manifold, a lesser enlargement would be to the cat of top spaces
whee the result i known to hold

in the old old days, BG ws approximated by manifolds

Posted by: jim stasheff on January 25, 2008 9:54 PM | Permalink | Reply to this

### Re: Classifying Spaces for 2-Groups

Urs, I agree that if G is a 2-group, then there is a 2-stack

By the way: I agree with that, too, but didn’t actually allude to it in my comment, nor do I think it is a very helpful point of view for what we were discussing: all we need to talk about is the small, nice and fine 2-groupoid with a single object and $\mathbf{G}$-worth of morphisms and 2-morphisms. And its nerve and the realization of that.

But I don’t understand your explanation of why one would want BG is a topological space.

The answer was: because it is sufficient.

categorify the notion of the classifying space as well

and, while I think I see where you are coming from with this suggestion, tried to indicate that this is not necessary/good: a space (at least in the role it plays here) is already an $\infty$-structure. Classifying spaces are the image of structure $n$-groups under the nerve realization functor, which forms something like an equivalence or so between $\infty$-groupoids and spaces. So when talking about classifying spaces, we have already, in a way, moved into an $\infty$-context (model category context, really).

That’s what I tried to say.

That’s why $n$-bundles have classifying spaces, not classifying $n$-spaces.

Sorry if I misunderstood you.

Posted by: Urs Schreiber on January 26, 2008 11:23 AM | Permalink | Reply to this

### Re: Classifying Spaces for 2-Groups

One reason why one might want a classifying space is that for a compact manifold $X$ (or a finite CW-complex), we can classify $\mathbf{G}$-bundles by a the realisation of a skeleton of the nerve of $\Sigma\mathbf{G}$, which is a space constructed by a finite process from $\mathbf{G}$. If $\mathbf{G}$ itself is made from finite CW complexes, we have a finite complex classifying bundles over $X$. This is what Jim was saying much more succinctly about approximating $BG$ by manifolds.

Posted by: David Roberts on January 27, 2008 3:06 PM | Permalink | Reply to this

### Re: Classifying Spaces for 2-Groups

Eugene almost wrote:

So one should probably look for a classifying “space” so that maps into it would recover more than just equivalence classes of objects of the 2-category of $\mathbf{G}$-2-bundles over $M$.

Hi!

A map from $M$ into the classifying space of the topological 2-group $\mathbf{G}$ gives more than a mere “equivalence class” of $\mathbf{G}$-2-bundles over $M$. It gives a specific one! There’s a universal $\mathbf{G}$-2-bundle over the classifying space, and we can pull this back via our map and get a specific $\mathbf{G}$-2-bundle over $M$.

However, in our paper we don’t talk about that. We just show that a homotopy class of maps from $M$ to the classifying space gives an equivalence class of $\mathbf{G}$-2-bundles over $M$.

It’s because we take homotopy classes on one side that we must take equivalence classes on the other. Evil begets evil.

It would be nice to prove something like this: the 2-groupoid of $\mathbf{G}$-2-bundles over $M$ is equivalent to the 2-groupoid of:

• maps from $M$ to the classifying space of $\mathbf{G}$;
• homotopies between such maps;
• homotopy classes of homotopies between such maps.

That would imply our result, but it would be much better.

Has anyone ever proved a result like this better one for the case of plain old $G$-bundles? Namely, something like this:

Conjecture. For any well-pointed topological group $G$ and paracompact Hausdorff space $M$, the groupoid of principal $\mathbf{G}$-bundles over $M$ is equivalent to the groupoid of:

• maps from $M$ to the classifying space of $G$;
• homotopy classes of homotopies between such maps.

Usually people state the decategorified version: a bijection between equivalence classes of $G$-bundles and homotopy classes of maps.

Posted by: John Baez on January 25, 2008 7:38 PM | Permalink | Reply to this

### Re: Classifying Spaces for 2-Groups

John stated the

Conjecture. For any well-pointed topological group $G$ and paracompact Hausdorff space $M$, the groupoid of principal $G$-bundles over $M$ is equivalent to the groupoid of

• maps from $M$ to the classifying space of $G$
• homotopy classes of homotopies between such maps

I started enjoying thinking about all questions of this sort from the kind of Lie algebraic approach that I keep going on about #.

In that setup, this conjecture is true by definition, essentially, at an algebraic level (and hence ignoring, probably non-rational phenomena, alas). Which is not much more than saying that, similarly, it is true trivially an a groupoid level (i.e. anafunctors between groupoids instead of maps of spaces).

Like there is geometric realization of nerves of groupoids, there is the functor from $L_\infty$-algebras over their CE-algebras to generalized smooth spaces.

And indeed, that does send an $L_\infty$-algebra to (a smooth version of) the classifying space of the corresponding $\infty$-group, as indicated here.

Now, I don’t really know, but it could be that the Lie oo-algebraic “realization functor” which sends L-oo-algebras over their CE-algebras to smooth spaces $Hom(--,\Omega^\bullet(--)) : DGCAs \to smooth spaces$ is better behaved than the nerve realization functor in that it preserves more of the structure of its domain (because, for instance, everything is smooth as opposed to just continuous). If so, it might have a better chance of carrying over the conjecture you state from the world of L_oo algebras (where it is true essentially by definition) to the wold of spaces (or generalized smooth spaces at least).

(I am hesitant, fearing to get on his nerves with all my questions, but I would tend to ask for Todd’s help at this point… :-)

Posted by: Urs Schreiber on January 26, 2008 11:45 AM | Permalink | Reply to this

### Re: Classifying Spaces for 2-Groups

Urs, what is known about the nerves of n-groupoids? For example, the nerve of every 2-groupoid is a Kan complex.

Posted by: Charlie Stromeyer Jr on January 26, 2008 1:02 PM | Permalink | Reply to this

### Re: Classifying Spaces for 2-Groups

One of the most useful definitions of a weak $\infty$-groupoid is that it is a Kan complex. I advocate this approach in a paper with James Dolan (see Section 3). The point is that it captures the essence of the notion, and takes advantage of simplicial technology that’s been under development for many decades. In this approach, $n$-groupoid is then a Kan complex satisfying certain conditions which we explain.

This approach seems widely accepted by those in the know. What’s nice is that it makes weak $\infty$-groupoids a special case of Joyal’s ‘quasicategories’ — also known as $(\infty,1)$-categories because, morally speaking, all $j$-morphisms with $j \gt 1$ are weakly invertible. These in turn are a special case of Street’s simplicial weak $\infty$-categories.

In any decent approach to weak $n$-categories one should be able to define weak $(\infty,n)$-categories for any $n$: these are weak $\infty$-categories where all $j$-morphisms above degree $n$ are weakly invertible. Then $\infty$-groupoids are $(\infty,0)$-categories, while $\infty$-categories are $(\infty,\infty)$-categories. I’m not sure anyone has defined weak $(\infty,n)$-categories, but in the simplicial framework it should not be hard, now that Street has done the hard work.

If you want to use a globular approach to weak $\infty$-groupoids, one of your first jobs should be to construct a nerve for such a thing which is a Kan complex. I forget if Batanin has done this in his approach.

Posted by: John Baez on January 27, 2008 1:49 AM | Permalink | Reply to this

### Re: Classifying Spaces for 2-Groups

Thanks for the reference to your paper with James Dolan which I am just starting to read. Do you or anyone else already happen to know if the weak infinity-groupoids that Baez and Dolan consider are like the weak L-groupoids which Ezra Getzler equates with an enriched Kan complex on page 7 of this paper ? Duskin called these weak L-groupoids “L-dimensional hypergroupoids”.

Posted by: Charlie Stromeyer Jr on January 27, 2008 3:36 PM | Permalink | Reply to this

### Re: Classifying Spaces for 2-Groups

More specifically, on the top of page 8 of the Getzler paper his n-groupoids are Kan complexes with the extra condition that every a-simplex is thin if a > n.

Is this not the same as saying that there is a cutoff on the dimension of j-morphisms as on page 16 of the Baez-Dolan paper?

Posted by: Charlie Stromeyer Jr on January 27, 2008 10:18 PM | Permalink | Reply to this

### Re: Classifying Spaces for 2-Groups

Charlie wrote:

Do you or anyone else already happen to know if the weak infinity-groupoids that Baez and Dolan consider are like the weak $\ell$-groupoids which Ezra Getzler equates with an enriched Kan complex on page 7 of this paper?

I just glanced at this, but it seems that Getzler’s weak $\ell$-groupoids are the same as our weak $n$-groupoids… when $\ell = n$!

The intuition is certainly the same in both cases: a Kan complex is an $n$-groupoid if it has no holes of dimension $\gt n$.

Posted by: John Baez on January 28, 2008 12:56 AM | Permalink | Reply to this

### Re: Classifying Spaces for 2-Groups

Wouldn’t quasi-cats do as well or better?

Posted by: jim stasheff on January 27, 2008 7:19 PM | Permalink | Reply to this

### Re: Classifying Spaces for 2-Groups

Jim wrote:

Wouldn’t quasi-cats do as well or better?

Do what as well or better?

A quasicategory is a simplicial set with horn-filling properties that make it act like an $(\infty,1)$-category: that is, a weak $\infty$-category where all $j$-morphisms with $j \gt 1$ are invertible.

A Kan complex is a simplicial set with horn-filling properties that make it act like an $\infty$-groupoid: that is, a weak $\infty$-category where all $j$-morphisms with $j \gt 0$ are invertible.

So, every Kan complex is automatically a quasicategory, but not vice versa. Neither is ‘better’; they serve different roles.

Posted by: John Baez on January 28, 2008 12:50 AM | Permalink | Reply to this

### Re: Classifying Spaces for 2-Groups

Obviously I don’t understand the question for which Kan complexes are the answer and quasi-categories are not.

Posted by: jim stasheff on January 28, 2008 1:28 PM | Permalink | Reply to this

### Re: Classifying Spaces for 2-Groups

Alas, I don’t know what question you’re talking about. So, I was just trying to make a general remark: Kan complexes are a model of $\infty$-groupoids, quasicategories aren’t.

Posted by: John Baez on January 30, 2008 11:32 PM | Permalink | Reply to this

### Re: Classifying Spaces for 2-Groups

There’s a universal G-2-bundle over the classifying space,

but it’s UNIversal only up to the appropriate equivalence

as for

maps from M to the classifying space of G;

homotopies between such maps;

homotopy classes of homotopies between such maps.

reminds me of the problem that H-spaces and H-maps do NOT form a category

Posted by: jim stasheff on January 26, 2008 1:47 PM | Permalink | Reply to this

### Re: Classifying Spaces for 2-Groups

It’s because we take homotopy classes on one side that we must take equivalence classes on the other. Evil begets evil.

Classically it was the other way around. Contemplate (cf. Steenrod’s book) what it would mean to classify distinct bundles when they are equivalent.

Posted by: jim stasheff on January 26, 2008 1:57 PM | Permalink | Reply to this

### Re: Classifying Spaces for 2-Groups

Jim wrote:

John wrote:

the difference between locally trivial fiber bundles and fibrations (and how the former isn’t always the latter)…

What!

Yeah, I was surprised too — in fact, I’m still a bit shocked. However, I can’t find any proof in the literature that an arbitrary locally trivial fiber bundle $E \to B$ is a Hurewicz fibration or even a Serre fibration.

The best I can find is that $E \to B$ is a Hurewicz fibration (and thus a Serre fibration) when $E \to B$ is a numerable locally trivial fibration. This is automatic when $B$ is paracompact Hausdorff.

(Apparently there’s something about this in the 1981 edition of Spanier, around Theorem 2.7.13. However, I don’t have that book at hand right now.)

Just out of morbid curiosity, I’d love to see a counterexample when $B$ is not paracompact Hausdorff.

What! do you then mean by fibration!

As you know, we’ve been discussing this with Danny over on email. Right now it seems like we need Hurewicz fibrations. But, I was hoping we could get away with Serre fibrations, because I’d like to be the very model of a modern major model category theorist, and that’s what they use.

btw, on your p.4 does principal $\mathbf{G}$-2-bundle mean locally trivial?

Yes. As you can see, we’re describing it by means of local trivializations.

Posted by: John Baez on January 25, 2008 10:56 PM | Permalink | Reply to this

### Re: Classifying Spaces for 2-Groups

Fair enough. But do you really want to consider more general spaces? Cf. people who think manifolds ar 2nd countable Housdorff.

As for being the
very model of a modern major model category theorist ( a reference I appreciate), the alternative is to appeal to John Wellington Wells. ;-)

Posted by: jim stasheff on January 26, 2008 1:40 PM | Permalink | Reply to this

### Re: Classifying Spaces for 2-Groups

I’ll say again that crossed complexes are equivalent to globular omega-groupoids, and both give a kind of linear model of homotopy types.

A classifying space for a crossed complex was defined (cubically) in my paper

“Non-abelian cohomology and the homotopy classification of maps”, in {\em Homotopie alg'ebrique et algebre locale, Conf. Marseille-Luminy} 1982, ed. J.-M. Lemaire et J.-C. Thomas, Ast'erisques 113-114 (1984), 167-172.

and a homotopy classification theorem generalising that of Eilenberg-Mac Lane proved.

There is a nice discussion of 3-types, with good references, in

On Algebraic Models for Homotopy 3-Types
Z. Arvasi and E. Ulualan
Journal of Homotopy and Related Structures, Vol. 1(2006), No. 1, pp. 1-27

Loday’s cat^n-groups also model weak, pointed homotopy (n+1)-types (a very good theorem this) and also can be calculated in some instances. Simona Paoli has related these to other models.

Posted by: Ronnie Brown on January 26, 2008 9:48 PM | Permalink | Reply to this

### Re: Classifying Spaces for 2-Groups

Something which has caught my attention is a couple of results which require open covers to be numerable.

The first is the classification of numerable principal bundles by Milnor’s construction of the classifying space (unlike Segal’s construction it requires slightly less assumption, and boils down to forming a homotopy colimit correctly).

The second is Segal’s that for a numerable cover $U \to X$, the canonical map $BU \to X$ is a homotopy equivalence.

These aren’t entirely independent, but some of my own work also seems to require numerable covers for a particular result about topological groupoids to work.

It is easy to check that numerable covers form a Grothendieck topology, so we can say things like “$BG$ classifies $G$-bundles on the site $\mathbf{Top}_n$” where $\mathbf{Top}_n$ is the site with this topology. Is there any reason why general open covers are more desirable than numerable covers?

Posted by: David Roberts on January 27, 2008 3:06 PM | Permalink | Reply to this

### Re: Classifying Spaces for 2-Groups

I suspect it’s an historical accident -pre-numeracy.

Posted by: jim stasheff on January 27, 2008 7:22 PM | Permalink | Reply to this

### Re: Classifying Spaces for 2-Groups

David R. wrote:

Something which has caught my attention is a couple of results which require open covers to be numerable.

Right! This is the sort of technicality that Danny and I were busy fighting with last week. I still need to understand it better.

For those not in the know, an open cover is called numerable if it admits a partition of unity.

In a recent post I mentioned one reason numerable covers show up in homotopy theory. Namely: a locally trivial fiber bundle $E \to B$ is a Hurewicz fibration if the bundle is numerable — that is, trivial when restricted to open sets forming a numerable covering of $B$. I’ve never see a proof that doesn’t include this ‘numerable’ assumption. But, I don’t know a counterexample!

Fibrations are very important in homotopy theory, so it’s sort of useless having a locally trivial fiber bundle unless you know it’s a fibration.

A closely related issue is one you mention: if $G$ is a topological group and $B G$ is its classifying space constructed a la Milnor, and $M$ is any topological space, $[M, B G]$ classifies numerable principal $G$-bundles over $M$.

And then there’s this other fact you mention: given an open covering $U$ of a space $M$, Segal builds a simplicial space now called the Cech nerve of $U$, and he shows the geometric realization of this is homotopy equivalent to $M$ if the covering is numerable. Here it’s easy to see how partitions of unity are used to build the homotopy equivalence.

Here’s a big question:

Could we drop this ‘numerable’ condition if throughout the paragraphs above I replaced ‘Hurewicz fibration’ by ‘Serre fibration’, and replaced ‘homotopy equivalence’ by ‘weak homotopy equivalence’, and replaced $[M, B G]$ by the homset in the homotopy category $Ho(Top)$?

If so, model category theory would have succeeded in rescuing us from this ‘numerable’ baloney… because model category theory suggests making all these replacements!

Posted by: John Baez on January 27, 2008 8:40 PM | Permalink | Reply to this

### Re: Classifying Spaces for 2-Groups

given an open covering $U$ of a space $M$, Segal builds a simplicial space now called the Cech nerve of $U$, and he shows the geometric realization of this is homotopy equivalent to $M$ if the covering is numerable.

It is true that we can remove the assumption that $U$ is numerable and get a weak homotopy equivalence (Dugger and Isaksen, “Toplogical hypercovers and $\mathbf{A}^1$-realizations”), and it doesn’t seem too hard to show that a general locally trivial $G$-bundle is a Serre fibration, but the last point about homsets in Ho(Top) requires some thought.

Posted by: David Roberts on January 28, 2008 3:39 AM | Permalink | Reply to this

### Re: Classifying Spaces for 2-Groups

The only thing I know about Ho(Top) is that there is an equivalence between Ho(2Gp) and the full subcategory of Ho(Top*) consisting of connected pointed homotopy 2-types.

Can someone please explain why JB wants to consider the homset (i.e. the locally small category) in Ho(Top)? Thanks.

Posted by: Charlie Stromeyer Jr on January 28, 2008 1:57 PM | Permalink | Reply to this

### Re: Classifying Spaces for 2-Groups

JOHN B:

Could we drop this numerable condition if throughout the paragraphs above I replaced Hurewicz fibration by Serre fibration, and replaced homotopy equivalence by weak homotopy equivalence, and replaced [M,BG] by the homset in the homotopy category Ho(Top)?

JIM S: Almost certainly since Serre fibrations can be shown to have the homotopy lifting property for CW complexes, which is really the model theorists’ retreat.

JOHN B:

If so, model category theory would have succeeded in rescuing us from this ‘numerable’ baloney… because model category theory suggests making all these replacements!

JIM S: Notice you are settling for weak homotopy type and that might could lose some of the structure you like. To say ΩBG is weakly equivalent to G is indeed weak unless you assume CW and then indeed: voila! Of course, then you could settle for quasifibrations.

JOHN B:

More generally, I’m wondering if the problem is that David Roberts and Danny Stevenson and I are learning homotopy theory from old papers written before model categories came in and regimented all the notions involved.

JIM S: Regimentation indeed!

Posted by: jim stasheff on January 28, 2008 1:46 PM | Permalink | Reply to this

### Re: Classifying Spaces for 2-Groups

More generally, I’m wondering if the problem is that David Roberts and Danny Stevenson and I are learning homotopy theory from old papers written before model categories came in and regimented all the notions involved.

Posted by: John Baez on January 27, 2008 8:50 PM | Permalink | Reply to this

### Re: Classifying Spaces for 2-Groups

Hatcher’s book Algebraic Topology has a proof that a locally trivial fibre bundle is a Serre fibration (Proposition 4.48).

Posted by: David Roberts on January 28, 2008 4:25 AM | Permalink | Reply to this

### Re: Classifying Spaces for 2-Groups

David Roberts wrote:

It is true that we can remove the assumption that $U$ is numerable and get a weak homotopy equivalence (Dugger and Isaksen, “Topological hypercovers and $\mathbf{A}^1$-realizations”.

Excellent! Thanks! Score one for model categories!

Hatcher’s book Algebraic Topology has a proof that a locally trivial fibre bundle is a Serre fibration (Proposition 4.48).

Excellent! Score two!

As you noted earlier, this one seemed plausible since Serre fibrations really involve lifting over cubes, which are as paracompact and Hausdorff as you’d possibly want. But, I wasn’t sure.

So, it’s only the business about $[M, B G]$ classifying principal $G$-bundles over $M$ if $M$ is paracompact Hausdorff that hasn’t been smoothed down with the help of model categories.

It’s possible we need to not only 1) replace $[M, B G]$ by the homset in $Ho(Top)$, but 2) replace the concept of ‘principal $G$-bundle’ by a more model-category-friendly concept, something like ‘principal $G$-fibration’.

Posted by: John Baez on January 30, 2008 7:12 PM | Permalink | Reply to this

### Re: Classifying Spaces for 2-Groups

The only thing I know of in this direction, sort of, (the Ho(Top) homset, not the G-fibrations) is Beke’s “Simplicial Torsors”, which you can find in TAC.

Posted by: David Roberts on January 31, 2008 1:36 AM | Permalink | Reply to this

### Re: Classifying Spaces for 2-Groups

Sorry to revive an old post. The tangent bundle to the long line is not trivial (Morrow, 1969) because otherwise, the long line would be a riemannian manifold, hence a metric space, hence paracompact. However, for $G=\R_+^*$, $BG$ is contractible.

Posted by: Antoine Chambert-Loir on February 19, 2014 4:52 AM | Permalink | Reply to this

### Re: Classifying Spaces for 2-Groups

Don’t be sorry — thanks, that’s a very interesting fact!

Posted by: John Baez on February 19, 2014 12:00 PM | Permalink | Reply to this

### Re: Classifying Spaces for 2-Groups

Eugene almost wrote:

It seems to me that if one takes the program of categorification seriously, one should categorify the notion of the classifying space as well. The collection of principal $G$-bundles over a fixed manifold $M$ ($G$ a 1-group) is a category and the homotopy classes of maps $M$ to $B G$ classify the equivalence classes of objects of that category. I’m guessing that if $\mathbf{G}$ is a 2-group, then the principal $\mathbf{G}$-2-bundles over $M$ form a 2-category with all 2-arrows being isomorphisms. So one should probably look for a classifying “space” so that maps into it would recover more than just equivalence classes of objects of the 2-category of $\mathbf{G}$-2-bundles over $M$.

I haven’t said yet how much I sympathize with this question!

If $\mathbf{G}$ is a 2-group without a topology, it’s a special sort of 2-groupoid, which is a special sort of simplicial set, or homotopy type… and this homotopy type is just the classifying space of $\mathbf{G}$. So, in this case, the classifying space of a 2-group is really just a space — or more precisely, a homotopy type.

Furthermore, as I suggested a while back, this space should already be enough to reconstruct the 2-groupoid of $\mathbf{G}$-2-bundles over a fixed base space. This hasn’t been proved… but it’s gotta be true.

However, if $\mathbf{G}$ is a topological 2-group, there is a lot more information in $\mathbf{G}$ than in its classifying space… unless we deliberately use some trick — as people do — to eliminate this extra information!

You see, in this case, $\mathbf{G}$ is a special sort of topological 2-groupoid, which is a special sort of simplicial space. It’s destructive to turn this into a mere space. But this is the crucial trick used by Baas, Bökstedt, Kro and also Jurč… and what Danny and I do is equivalent.

In other words: a topological 2-group has $n$-categorical stuff going on in ‘two directions’, while its classifying space only has $n$-categorical stuff going on in ‘one direction’.

Here’s another way to see what I mean by that. As I mentioned, a topological 2-group gives a simplicial space. This gives a ‘simplicial simplicial set’, which people usually call a ‘bisimplicial set’: a functor from $\Delta \times \Delta$ to $Set$.

There’s a Quillen model structure on the category of bisimplicial sets that makes it Quillen equivalent to the category of simplicial sets. But the weak equivalences in this model structure are very coarse: we’re squashing two kinds of ‘simplicialness’ down to one!

This is yet another way of thinking about the ‘crucial trick’.

This trick loses information. So, maybe you want to avoid it — and if so, I sympathize.

Posted by: John Baez on January 27, 2008 8:14 PM | Permalink | Reply to this

### Re: Classifying Spaces for 2-Groups

John wrote:

In other words: a topological 2-group has n-categorical stuff going on in directions, while its classifying space only has n-categorical stuff going on in one direction.

But that already happens for n=1: The classifying space BG has combined the topology and algebra of G into the topology of BG, but they can be recoverd *up to equivalence*:

ΩBG ~ G

Posted by: jim stasheff on January 28, 2008 1:49 PM | Permalink | Reply to this

### Re: Classifying Spaces for 2-Groups

John wrote:

In other words: a topological 2-group has $n$-categorical stuff going on in directions, while its classifying space only has $n$-categorical stuff going on in one direction.

Jim wrote:

But that already happens for $n=1$: The classifying space $B G$ has combined the topology and algebra of $G$ into the topology of $B G$.

I see what you mean, but there’s actually a big difference.

Very roughly, the homotopy hypothesis tells us that a topological space is secretly the same as an $\infty$-groupoid. There are various well-known ways to make this rough statement completely precise — but I don’t want to digress, so let’s just accept this as a fact for now.

Given this, what about a topological group? ‘Topological group’ is just another name for ‘group in the world of topological spaces’. By the homotopy hypothesis, this is secretly the same as a ‘group in the world of $\infty$-groupoids’.

Now, any group in the world of $\infty$-groupoids can be reinterpreted as an $\infty$-groupoid with one object. It’s a simple level-shifting trick.

So: a topological group can be harmlessly reinterpreted as an $\infty$-groupoid with a certain special property.

This isn’t true for a topological 2-group! A topological 2-group is a 2-group in the category of spaces. By the homotopy hypothesis, this is secretly a ‘2-group in the category of $\infty$-groupoids’. But at this point we are stuck — this isn’t just an $\infty$-groupoid with an extra property anymore.

If we wanted the same level-shifting trick to work, we could consider a certain special class of topological 2-groups, namely those where the space of objects has the discrete topology. A thing like this is again secretly the same as an $\infty$-groupoid with one object.

The same is true for topological $n$-groups as long as the topology fails to be discrete only ‘on top’ — on the space of $n$-morphisms.

But in general, a topological $n$-group is a much richer thing. A topological $n$-group is secretly the same as an $n$-group in the world of $\infty$-groupoids. This is a special sort of $\infty$-groupoid in the world of $\infty$-groupoids!

Now, a $\infty$-groupoid in the world of $\infty$-groupoids should be called a double $\infty$-groupoid, since it has $(j,k)$-morphisms for each pair of natural numbers $j,k$. You may have heard of ‘double groupoids’ — these are the simplest truly interesting examples. Another interesting class of examples are the ‘bicomplexes’ — more precisely, $\mathbb{N} \times \mathbb{N}$-graded chain complexes of abelian groups.

Indeed, you should think of a double $\infty$-groupoid as a nonabelian, weaken version of a bicomplex. And the same issue shows up in both situations! Just as you can turn a bicomplex into a chain complex in various ways, you can turn a double $\infty$-groupoid into an $\infty$-groupoid in various ways. However, you always lose information.

(By the way: I sketched how to take any group in the world of $\infty$-groupoids, say $G$, and turn it into an $\infty$-groupoid with one object, which Urs would call $\mathbf{B}G$.

It’s harder to go backwards — to take an $\infty$-groupoid with one object and get a group in the world of $\infty$-groupoids. The reason is that when we say ‘group’, we assume the associative and other laws hold strictly. In an $\infty$-groupoid with one object, these laws only hold weakly.

However, a guy named Stasheff figured out the key idea to get around this problem — a thing called the associahedron. Using this and subsequent work, one can show that any $\infty$-groupoid with one object is equivalent to a group in the world of $\infty$-groupoids.)

Posted by: John Baez on January 28, 2008 10:09 PM | Permalink | Reply to this

### Re: Classifying Spaces for 2-Groups

So to summarize this:

- a (topological) space is already an $\infty$-structure.

Hence $n$-bundles with structure $n$-group have classifying spaces (not $n$-spaces) for all $n$

… unless the structure $n$-group is itself internal to spaces, in which case it is something like an $n + \infty$-structure itself…

- hence $n$-bundles for topological $n$-groups can be thought of as having doubly-infinite structures as classifying objects…

… which is a little scary. ;-)

Not just because it sounds twice as intimidating as “infinity” does, but also we are used to carelessly think of most of the stuff around as as being made of spaces, all my Lie groups, all my little naive Euclidean target spaces, etc. Am I to think of all of that as being an $\infty$-structure already, all the time?

Might we not have to distinguish between the use of a topological group as a group internal to the mere category of topological spaces, as opposed to one internal to the model category of topological spaces?

I know that its sounds like splitting hairs. But given my $SU(3) \times SU(2) \times U(1)$-bundle here. Do we really want to think of it as an $\infty$-structure rather than as a 1-structure?

Posted by: Urs Schreiber on January 28, 2008 10:59 PM | Permalink | Reply to this

### Re: Classifying Spaces for 2-Groups

Urs almost wrote:

I know that it sounds like splitting hairs.

A Zen koan for academics: what is the sound of one hair splitting?

But given a $SU(3)\times SU(2) \times U(1)$-bundle here…. do we really want to think of it as an $\infty$-structure rather than as a 1-structure?

Excellent question!!!

Here’s a somewhat related question: given a bundle with the string group as gauge group… do we really want to think of it as a 2-bundle with the string 2-group as gauge 2-group? Isomorphism classes of the former are in 1-1 correspondence with equivalence classes of the latter. They both have the same classifying space! So, in switching to the 2-bundle perspective, we’re really just taking the groupoid of bundles and seeing that it comes from decategorifying a 2-groupoid of 2-bundles. Is this switch of perspective important?

In this case, I bet you’d say the answer is yes.

Of course in this case, we’re not just taking a topological group and reinterpreting it as a topological 2-group with a discrete space of objects. Is that the big difference?

Maybe.

But still, the result of this paper, saying that the classifying space of a topological 2-group $\mathbb{G}$ is the same as the classifying space of the topological group $|\mathbb{G}|$, shows that we can very often reinterpret bundles as 2-bundles. Our proof, especially of Lemma 2, makes it clear how this works. But is this something we should always seek to do?

Posted by: John Baez on January 29, 2008 7:01 AM | Permalink | Reply to this

### Re: Classifying Spaces for 2-Groups

The example with the String-group is a good one. Here is why I am inclined to regard String-bundles as 2-bundles, but not as “$2+\infty$“-bundles, in applications:

when I actually use String-bundles to do something, I am doing this:

- I map curves into base spaces of String bundles and lift these curves to the total space by using the 1-structure of the String bundle;

- then I check how I may deform these curves along surfaces, and for that I use the 2-structure of the String-bundle

- what I am not interested in here is in how I can deform the lifts of these surfaces using the potentially available “$\infty$“-structure of the String-bundle.

The last bit is of course best exhibited for ordinary bundles: the reason that I do not want to regard my $SU(3)\times SU(2) \times U(1)$-bundle as an $\infty$-structure means that this step would divide out all (or most of) the information that the bundle itself was considered for in the first place: for doing Yang-Mills theory, it is important that I have this Lie gauge bundle, and not any old space merely homotopy equivalent to it.

Hm, probably what I am trying to say can be said better, but this is as I far as I get right now.

Posted by: Urs Schreiber on January 29, 2008 10:42 AM | Permalink | Reply to this

### Re: Classifying Spaces for 2-Groups

URS wrote:
for doing Yang-Mills theory, it is important that I have this Lie gauge bundle, and not any old space merely homotopy equivalent to it.

JIM writes:
Not so fast! Depends on what you want to do. Certainly some computations will be easier (maybe doable only) if you can write things in terms of a finite number of coordinates, but you might figure out/envison that computation in terms of something inf dim or conceptual
cf. a vector is NOT an n-tuple of numbers
and
Rainich in his book on relativity:
the last thing you want to do is to write it in coordinates

Posted by: jim stasheff on January 29, 2008 2:36 PM | Permalink | Reply to this

### Re: Classifying Spaces for 2-Groups

JIM writes: Not so fast! Depends on what you want to do.

Yes, I suppose so. I was mainly thinking about bundles with connection.

A point about the strict version of the String 2-group which, despite all advertisement, we possibly did not advertize enough is that it is indeed Lie, i.e. smooth.

If one tries to think of connections on String bundles it is, as I don’t have to tell you, crucial that we have this smooth structure around. It’s the crucial advantage.

Forgetting this and regarding everything only up to homotopy (hence making the $\infty$-structure on spaces manifest) divides out all the interesting information.

But the same argument applies for more down-to-earth examples, too. If we do Yang-Mills theory we really want bundles with connection. As soon as we do that, we want to be sure not to work up to homotopy. The entire information in a connection would be lost otherwise.

But I think even more generally, we often do indeed not want to refard many spaces around us objects of an $\infty$-structure.

One notable exception are in fact classfying spaces, though.

On the other hand, while saying all this one part of myself is whispering: “Urs, you are just fighting the step into a larger universe, where all things dear and well-known to you know will suddenly look strange and scary.”

Posted by: Urs Schreiber on January 29, 2008 5:17 PM | Permalink | Reply to this

### Re: Classifying Spaces for 2-Groups

Urs’ conscience wrote:

“Urs, you are just fighting the step into a larger universe, where all things dear and well-known to you know will suddenly look strange and scary.”

Of course, what’s well-known to you is already strange and scary to most. But, we all have our own resistance to new stuff. If we didn’t, we couldn’t function.

Since you often seem to like $\infty$-groupoids better than topological spaces (so do I!), you might at least spend a couple hours imagining a world where $SU(3) \times \SU(2) \times \mathrm{U}(1)$ is really an $\infty$-group, and what bundles with that $\infty$-group as gauge group would be like, and their connections too.

I haven’t tried!

Posted by: John Baez on January 31, 2008 4:02 AM | Permalink | Reply to this
Read the post Differential Forms and Smooth Spaces
Weblog: The n-Category Café
Excerpt: On turning differential graded-commutative algebras into smooth spaces, and interpreting these as classifying spaces.
Tracked: January 30, 2008 10:16 AM
Read the post Twisted Differential Nonabelian Cohomology
Weblog: The n-Category Café
Excerpt: Work on theory and applications of twisted nonabelian differential cohomology.
Tracked: October 30, 2008 7:46 PM

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