### 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:

- John Baez and Danny Stevenson, The classifying space of a topological 2-group.

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.

## Re: Classifying Spaces for 2-Groups

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.

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

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

finitetransformation 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 agroupelement 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

concordanceandtransformation. 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.