The n-Category Café

Urs wrote:

I am glad this article is finally seeing the day of light. I remember that Danny in particular had started thinking about this long ago.

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’.

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.

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.

“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

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.

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

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.

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.

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.

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. ;-)

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.

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?

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

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.

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

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.)