### Ginot and Stiénon on Characteristic Classes of 2-Bundles

#### Posted by Urs Schreiber

In

G. Ginot & M. Stiénon
*Groupoid extensions, principal 2-group bundles and characteristic classes*

arXiv:0801.1238

the authors regard principal 2-bundles on $X$ for any strict 2-group $G_{(2)} := (H \stackrel{t}{\to} G)$ in terms of their descent data/$G_{(2)}$-cocycles/transport anafunctors $g : X \leftarrow [Y] \rightarrow \mathbf{B} G_{(2)} \,.$

(Here $\mathbf{B}G_{(2)}$ is my notation for the one-object 2-groupoid defined by the 2-group $G_{(2)}$.)

Then they point out two things:

a) they demonstrate that and describe explicitly and in detail how such anafunctors, for $G_{(2)} = AUT(H) := (H \to \mathrm{Aut}(H))$ the automorphism 2-group of an ordinary group $H$, are equivalent to $H$-extensions of groupoids (such $H$-extensions of groupoids are a popular way to think of categorified bundles, as such usually addressed as “(bundle) gerbes” (What is the fiber??).)

b) they define a straightforward generalization of the notion of characteristic classes of principal 1-bundle to principal 2-bundle and prove that in the abelian case these characteristic classes of 2-bundles coincide with the familiar Dixmier-Douady classes known from bundle gerbes.

I very much enjoyed the paper, but I cannot but start talking about it here without mentioning one quarrel I have, not with the technical content, but with one conceptual point of view which the authors mention.

Right in the third paragraph of the introduction, Greg Ginot and and Mathieu Stiénon state that their work differs from other comparable considerations in that they do not consider a local charts and cocycles perspective, but a “global” perspective.

I think this is actually not the case. The conception of a 2-bundle as an
anafunctor (also “Hilsum-Skandalis morphism” or “generalized morphism” and, actually, also “cocycle” and “descent datum”) from the base to the structure 2-group is in fact *precisely* the cocycle perspective. It is precisely
the middle term in the generalized morphism
$g : X \leftarrow [Y] \rightarrow \mathbf{B} G_{(2)}$
which is the “cover” that makes the description local.

**The relation between $AUT(G)$-bundles and $G$-extensions of groupoids.**

I believe their first main theorem, the bijection between 2-bundles over $\Gamma$ with structure 2-group the automorphism 2-group $\mathrm{AUT}(G) := (H \stackrel{\mathrm{Ad}}{\to} Aut(G))$ and $G$-extensions of $\Gamma$ is to be thought of precisely as an instance of groupoid Schreier theory.

This says that extensions of groupoids $\Gamma$ by discrete (all isomorphic object are equal) groupoids $\Gamma_0 \times G$ are classified by pseudofunctors from $\Gamma$ to the automorphism 2-group of $G$.

**Characteristic classes of 2-bundles**

Characteristic classes of ordinary principal $G$-bundles $P \to X$ can neatly be defined as the pullback of the cohomology classes of the classifying space $B G$ along any classifying map $|g| : X \to B G$ $\array{ |g|^* &\to & E G \\ \downarrow && \downarrow \\ X &\stackrel{|g|}{\to}& B G } \,,$ which in turn is best thought of as the image under the nerve realization functor of any anafunctor/cocycle/descent datum $X \stackrel{\simeq}{\leftarrow} Y \times_X Y \stackrel{g}{\to} \mathbf{B}G \,.$

Here again, in my notation $\mathbf{B} G$ denotes the one-object groupoid defined by the group $G$, such that its realization is the classifying space $B G$ of $G$: $|\mathbf{B} G | = B G \,.$

Since there is the standard generalization of the notion of a nerve from 1-categories to 2-categories, there is an obvious definition of the characteristic classes of any 2-bundle (in fact, for any $n$-bundle):

the characteristic classes of a 2-bundle represented by an ana-2-functor/2-cocycle/descent datum $X \stackrel{\simeq}{\leftarrow} Y \times_X Y \stackrel{g}{\to} \mathbf{B}G_{(2)}$ is the pullback of the cohomology classes of the realization of the nerve $B G_{(2)} := |\mathbf{B} G_{(2)}|$ along the image under the nerve functor of the cocycle $g$.

This is the definiton Greg Ginot and Mathieu Stiénon adopt, and which they use to show that the characteristic class of $(U(1) \to 1)$-bundles (they consider more general abelian 2-bundles, too) in this sense is indeed, as one would hope, the well-known Dixmier-Douady class.

**The Chern-Weil homomorphism**

As, for instance, I tried to emphasize a bit in my slide show, there is an
*integral* and a *differential picture* of all things gauge theoretic
(no real news here, but it deserves to be made explicit in the $n$-categorical context):

$\array{ integral picture &\leftrightarrow& differential picture \\ Lie n-groupoids &\leftrightarrow& Lie n-algebras \\ n-functors between n-groupoids &\leftrightarrow& morphisms of Lie n-algebras }$

The short remark found in example 4.6 on p. 18 of the Ginot-Stiénon paper nicely leads over from the integral picture in which their article is situated, to the differential picture which I was drawing with Jim Stasheff and Hisham Sati in the thread “Lie $\infty$-connections” #.

In a way, passing to the differential picture *and then dualizing* from
$L_\infty$-algebras (= codifferential coalgebras) to differential graded algebras
amounts to nothing more than passing

$from n-groupoids \stackrel{to their}{\to} classifying spaces (realized nerve) \stackrel{to their}{\to} algebras of differential forms. \,.$

There is something at work at the background, which we discussed in On rational homotopy theory and Lie $N$-tegration and in Transgression.

I feel like postponing a more detailed discussion about how to relate Greg Ginot’s and Mathieu Stiénon’s work to the discussion in Lie $\infty$-connection to another installment.

I’ll just end this discussion here with listing without much comment some corresponding ideas, as they appear in the Lie-algebraic approach:

- the middle term of an anafunctor: a surjective submersion $\array{ Y \\ \downarrow^\pi \\ X } \;\;\;\;\; \array{ \Omega^\bullet(Y) \\ \uparrow^{\pi^*} \\ \Omega^\bullet(X) }$

- an $n$-bundle descent datum (“cocycle”) $\array{ \Omega^\bullet_{vert}(Y) &\stackrel{A^*_{vert}}{\leftarrow}& CE(g) \\ \uparrow^{i^*} \\ \Omega^\bullet(Y) \\ \uparrow^{\pi^*} \\ \Omega^\bullet(X) }$

- a connection on that $n$-bundle

$\array{ \Omega^\bullet_{vert}(Y) &\stackrel{A^*_{vert}}{\leftarrow}& CE(g) \\ \uparrow^{i^*} && \uparrow \\ \Omega^\bullet(Y) &\stackrel{(A,F_A)}{\leftarrow}& \mathrm{W}(G) \\ \uparrow^{\pi^*} \\ \Omega^\bullet(X) }$

- the Chern-Weil homomorphism

$\array{ \Omega^\bullet_{vert}(Y) &\stackrel{A^*_{vert}}{\leftarrow}& CE(g) \\ \uparrow^{i^*} && \uparrow \\ \Omega^\bullet(Y) &\stackrel{(A,F_A)}{\leftarrow}& \mathrm{W}(G) \\ \uparrow^{\pi^*} && \uparrow \\ \Omega^\bullet(X) &\stackrel{\{K_i\}}{\leftarrow}& \mathrm{inv}(g) }$

More later.

## Re: Ginot and Stiénon on Characteristic Classes of 2-Bundles

It may be useful to point out that the notion of extensions of groupoids a la Schreier is described in

(with P.J. HIGGINS), “Crossed complexes and non-abelian extensions”,

Category theory proceedings, Gummersbach, 1981, Lecture Notes in Math. 962 (ed. K.H. Kamps et al, Springer, Berlin, 1982), pp. 39-50.and the crossed complex approach to Schreier theory, including computational questions, is also addressed in

(with T. PORTER), “On the Schreier theory of non-abelian extensions: generalisations and computations”,

Proceedings Royal Irish Academy96A (1996) 213-227.which examines extensions of the type of a crossed module, following the lead of Dedecker (CRAS Paris 247 (1958) 1160-1163, and other papers).

The point is to use free crossed resolutions, and the standard such of a group(oid) contains the usual cocycle type formulae in the boundary operator. So a nonabelian cocycle with coefficients in a crossed module becomes just a morphism from the standard free crossed resolution to the crossed module, and equivalence of cocycles is just homotopy. I find this approach easier to understand!

One fun aspect is to use fibrations of the coefficients as in

(with O. MUCUK), “Covering groups of non-connected topological groups revisited”,

Math. Proc. Camb. Phil., 115 (1994) 97-110. This does presume previously established results on the monoidal closed structure and the Quillen model structure on crossed complexes.Soc

Ronnie Brown