The String Coffee Table

Hey,

Regarding how String(n)-bundles are “the same” as a Str_G-2-bundle. Maybe someone can explain a simpler version to me: Is it the case that circle-bundles are “the same” as 2-bundles for the 2-group with no 1-arrows and Z 2-arrows? What does it mean?

D.

The proof given in 5.6 of math.DG/0510078 uses the fact that, given a crossed module bundle gerbe bundle $E \to Y^{[2]}\to X$ classified by $Y^{[2]} \to |K|$, we can always choose a gauge such that pulling back to open patches of $X$ by local sections $U_{ij} \overset{s_{ij}}{\to} Y^{[2]}$ we get 1-cocylces $g_{ij}g_{jk} = g_{ik}$ of a $|K|$ bundle on the nose, i.e. without any 2-coboundary appearing.

First I did not see how this is true. Now, I think, i do (benefiting from discussion with various people, which I’ll be glad to name – unless they do not want to be associated with what I am saying here :-).

I’ll first indicate the diagrammtic reasoning, then I give the more pedestrian version.

In the following, “1-cocycle” refers to a Čech cocycle of an ordinary bundle (i.e. $g_{ij}$ satisfying $g_{ij}g_{jk} = g_{ik}$), while “2-cocycle” refers to the analogous thing for gerbes.

1) Let us choose $Y \to M$ to be a good covering $Y = \{U_i\}_i$ by open sets. One checks that the 2-morphism $t$ on p. 16 of the script given above are stritly invertible in the present situation. According to the general construction (def. 4 in these notes) this means that we may always choose the 2-morphism $\phi$ on the same page to be in fact the identity.

But this is equivalent to saying that the Čech cocycles representing the transition bundles satisfy (under their product, wich is composition of arrows on p. 16) the 1-cocycle on the nose. Taking realized nerves everywhere should hence give us a choice of representing function $U_{ij} \overset{g_{ij}}{\to} |K|$ of the transition bundle which, too, satisfies the 1-cocycle.

Conversely, by reverse-engineering this we find that any cocycle $U_{ij} \overset{g_{ij}}{\to} |K|$ gives rise to cocycle data for $K$-crossed module transition bundles that satisfy the diagram on p. 16 with $\phi = \Id$.

Moreover, one quickly sees that a different choice of trivializations $t$ for the 2-bundle corresponds precisely to a 1-coboundary for the $|K|$-bundle cocycle $g_{ik}$. (That’s easy in diagrams, but harder in words.)

The important point here is the strict invertibility of $t$. Had we chosen not to work at the level of Čech data, but at the level of bitorsor bundles (as indicated here), we’d deal with diagrams of precisely the same form, but now nontrivial 2-isomorphisms coming from an adjunction appear in the triangles in def 4, making it non-obvious how to obtain a strict 1-cocycle on the transition bundles.

2) Spelled out, this means the following.

We start with a 2-cocycle representing our $K$-2-bundle. We choose another good cover $Y$. Over each open patch of this cover the above 2-cocycle is cobordant to the trivial 2-cocycle. This coboundary is the $t$ from above.

Hence on double intersections we get a coboundary $\bar t_i \circ t_j$ from the trivial 2-cocycle to itself. One checks that such a 2-coboundary of the trivial 2-cocycle is the same as a 1-cocycle, hence represents a bundle (the transition bundle).

Moreover, the 2-coboundary $t$ has a strict inverse (with respect to the product induced by the product of $K$!). It hence follows that we get 1-cocycle for transition bundles which, under the above product, satisfy themselves a 1-cocycle, strictly.

By passing to classfying functions this 1-cocycle of $K$ crossed module bundles becomes a 1-cocycle of $|K|$-valued functions.