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January 11, 2008

Ginot and Stiénon on Characteristic Classes of 2-Bundles

Posted by Urs Schreiber


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

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

(Here BG (2)\mathbf{B}G_{(2)} is my notation for the one-object 2-groupoid defined by the 2-group G (2)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):=(HAut(H))G_{(2)} = AUT(H) := (H \to \mathrm{Aut}(H)) the automorphism 2-group of an ordinary group HH, are equivalent to HH-extensions of groupoids (such HH-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[Y]BG (2) g : X \leftarrow [Y] \rightarrow \mathbf{B} G_{(2)} which is the “cover” that makes the description local.

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

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

This says that extensions of groupoids Γ\Gamma by discrete (all isomorphic object are equal) groupoids Γ 0×G\Gamma_0 \times G are classified by pseudofunctors from Γ\Gamma to the automorphism 2-group of GG.

Characteristic classes of 2-bundles

Characteristic classes of ordinary principal GG-bundles PXP \to X can neatly be defined as the pullback of the cohomology classes of the classifying space BGB G along any classifying map |g|:XBG|g| : X \to B G |g| * EG X |g| BG, \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 XY× XYgBG. X \stackrel{\simeq}{\leftarrow} Y \times_X Y \stackrel{g}{\to} \mathbf{B}G \,.

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

the characteristic classes of a 2-bundle represented by an ana-2-functor/2-cocycle/descent datum XY× XYgBG (2) 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 BG (2):=|BG (2)| B G_{(2)} := |\mathbf{B} G_{(2)}| along the image under the nerve functor of the cocycle gg.

This is the definiton Greg Ginot and Mathieu Stiénon adopt, and which they use to show that the characteristic class of (U(1)1)(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 nn-categorical context):

integralpicture differentialpicture Liengroupoids Lienalgebras nfunctorsbetweenngroupoids morphismsofLienalgebras \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 L_\infty-algebras (= codifferential coalgebras) to differential graded algebras amounts to nothing more than passing

fromngroupoidstotheirclassifyingspaces(realizednerve)totheiralgebrasofdifferentialforms.. 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 NN-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 Y π XΩ (Y) π * Ω (X) \array{ Y \\ \downarrow^\pi \\ X } \;\;\;\;\; \array{ \Omega^\bullet(Y) \\ \uparrow^{\pi^*} \\ \Omega^\bullet(X) }

- an nn-bundle descent datum (“cocycle”) Ω vert (Y) A vert * CE(g) i * Ω (Y) π * Ω (X) \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 nn-bundle

Ω vert (Y) A vert * CE(g) i * Ω (Y) (A,F A) W(G) π * Ω (X) \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

Ω vert (Y) A vert * CE(g) i * Ω (Y) (A,F A) W(G) π * Ω (X) {K i} inv(g) \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.

Posted at January 11, 2008 11:09 PM UTC

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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 Academy 96A (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.

Ronnie Brown

Posted by: Ronnie Brown on January 13, 2008 10:23 PM | Permalink | Reply to this

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

Thanks a lot for this comment.

I realize that I need to learn more about the general theory of crossed complexes (for higher nn and for their groupoid version).

When you write:

The point is to use free crossed resolutions

is this referring to the crossed complex analog of a surjective equivalence π:Y>X \pi : Y \stackrel{\simeq}{\to}\gt X of (nn-)groupoids?

If so, I believe I understand the paragraph you write then as stating the crossed complex analog of what John (following Makkai) calls an ana(-nn-)functor (and others call by other names), namely a span X<YG, X \lt\stackrel{\simeq}{\leftarrow} Y \to G\,,

where GG is here supposed to denote the thing that the cocycle takes values in.

I find this approach easier to understand!

I guess I should once and for all grok the general statement, which presumeably establishes some isomorphism between crossed complexes and higher groupoids. I am very familiar with this for 2-groupoids and partly familiar for 3-groupoids. Then I can guess how it continues, but am not sure what precisely is known.

I understand that a step in the right direction for me would be to read more of your papers…

Posted by: Urs Schreiber on January 14, 2008 6:00 PM | Permalink | Reply to this

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