The n-Category Café

A principal $G$-bundle (possibly with connection) on $X$ is a functor $$\begin{array}{c}{P}_1(X)\\ {\downarrow }^{\mathrm{tra}}\\ G\mathrm{Tor}\end{array}$$ with the special property that it is smoothly locally trivializable, meaning that it can be completed to a square

$$\begin{array}{ccc}{P}_1(Y)& \stackrel{\pi }{\to }& P_1(X)\\ {\downarrow }^{\mathrm{triv}}& {\Downarrow }^t& {\downarrow }^{\mathrm{tra}}\\ \Sigma G& \stackrel{i}{\to }& G\mathrm{Tor}\end{array}$$

such that the descent data obtained from this is smooth.

In order to decouple the discussion from what happens to the connection data, and concentrate attention here to just the $n$-bundles themselves, I now restrict attention to the case where we take all paths to be constant. And I’ll simply write $X$ and $Y$ for the spaces regarded as discrete $n$-categories.

For $$1\to {\Sigma }^{n-1}U(1)\stackrel{t}{\to }\hat{G}\to G\to 1$$ any shifted central extension, we should be able to construct from the $G$-bundle $$\begin{array}{c}X\\ {\downarrow }^{\mathrm{tra}}\\ G\mathrm{Tor}\end{array}$$

canonically the corresponding $n$-bundle obstruction $$\begin{array}{c}X\\ {\downarrow }^{\mathrm{tra}}\\ {\Sigma }^{n}U(1)\mathrm{Tor}\end{array}.$$

The trick to play is now is to keep following what looks like a long exact sequence in homotopy, at the level of $n$-groups:

$${\Sigma }^{n-1}U(1)\to G\hookrightarrow (K\to G)\to {\Sigma }^{n}U(1)\simeq (\hat{G}\to G)$$

using the equivalence $${\Sigma }^{n}U(1)\simeq (\hat{G}\to G)$$ (at the level of smooth $n$-groups realized in terms of anafunctors) where $$(\hat{G}\to G)$$ is the corresponding crossed module For $n=1$ this is literally the obvious crossed module. For higher $n$ it is the analogous crossed $n$-module, not to be discussed here. I’ll focus on $n=1$ now.

Notice that we have canonical inclusions $$G\hookrightarrow (\hat{G}\to G)$$ (the identity on objects) as well as $${\Sigma }^{n}U(1)\hookrightarrow (\hat{G}\to G)$$ (identifying ${\Sigma }^{n}U(1)$ with the fiber over 1).

The construction which I am about to make amounts to accordingly embedding $G$-1-bundles as well as ${\Sigma }^{n}U(1)$-$n$-bundles into the world of $(\hat{G}\to G)$-n-bundles: in that world the given $G$-1-bundle will be equivalent to some ${\Sigma }^{n}U(1)$-n-bundle which is the obstruction to lifting it to a $\hat{G}$-bundle.

What might sound intricate here is actually just the general principle behind a standard fact that people working with bundle gerbes use all the time: many people satisfy all their gerby needs entirely in terms of $\mathrm{PU}(H)$-bundles: these bundles uniquely represent the classes of the lifting gerbes obstructing their lift to $U(H)$-bundles (see for instance these authors). This kind of thinking is useful for various purposes. For instance the associated $K(H)$-bundles of algebras of compact operators may be thought of as the 2-vector bundles associated to the given gerbes: think of each algebra as a placeholder for the 2-vector space given by its category of modules.

Anyway, here is the construction in detail.

It amounts to noticing that there is a canonical injection

$$\begin{array}{c}G\mathrm{Tor}\\ \downarrow \\ (\hat{G}\to G)\mathrm{Tor}\end{array}$$

which formalizes the construction that Bruce distilled out of the discussion in Brylinski’s book:

each $G$-torsor $T$ (my $G$-torsors here are “torsors over a point”(!) since I am following John Baez instead of most of the rest of the world in that respect) is sent to the $(\hat{G}\to G)$-2-torsor that is given by the groupoid of of $\hat{G}$-torsors over $G$.

So an object in that $(\hat{G}\to G)$-torsor is a $\hat{G}$-torsor $\hat{T}$ together with its projection down to a $G$-torsor $$\begin{array}{c}\hat{T}\\ & \searrow \\ & & T\end{array}$$ and a morphism is a $\hat{G}$ torsor morphism $$\hat{T}\stackrel{f}{\to }\hat{T}\prime$$ respecting that projection, such that we get a commuting triangle $$\begin{array}{ccc}\hat{T}& \stackrel{f}{\to }& \hat{T}\prime \\ & \searrow \swarrow & \\ & T\end{array}.$$

One checks that this groupoid has a canonical $(\hat{G}\to G)$ action which makes it a $(\hat{G}\to G)$-2-torsor.

Now, moreover this injection fits into this square

$$\begin{array}{ccc}\Sigma G& \hookrightarrow & G\mathrm{Tor}\\ \downarrow & {\Downarrow }^{\simeq }& \downarrow \\ \Sigma (\hat{G}\to G)& \hookrightarrow & (\hat{G}\to G)\mathrm{Tor}\end{array},$$

where the transformation filling it is non-canonically given, essentially, by choosing for each $\hat{G}$-torsor a trivialization. Since there is no smoothness condition or anything involved at this point, this thing exists (otherwise, if we want this to be smooth in some sense or other, we can apply the standard machinery like using anafunctors et al – but the point of the $n$-transport formalism which I am using here being that I don’t need to do that here, if I don’t want to, since I don’t need a smooth structure on the global codomain of my $n$-transport, just on its descent data).

So, we can now send any $G$-bundle

$$\begin{array}{c}X\\ {\downarrow }^{\mathrm{tra}}\\ G\mathrm{Tor}\end{array}$$

canonically to the corresponding $(\hat{G}\to G)\simeq {\Sigma }^{n}U(1)$-$n$-bundle

$$\begin{array}{c}X\\ {\downarrow }^{\mathrm{tra}}\\ G\mathrm{Tor}\\ \downarrow \\ (\hat{G}\to G)\mathrm{Tor}\end{array}.$$

For that simple construction to be admissable in the world of $n$-transport, we need to check that this composite still has smooth local $(\hat{G}\to G)$-trivializations. But it does, we simply paste any smooth local trivialization of the given $G$-bundle

$$\begin{array}{ccc}Y& \stackrel{\pi }{\to }& X\\ {\downarrow }^{\mathrm{triv}}& {\Downarrow }^t& {\downarrow }^{\mathrm{tra}}\\ \Sigma G& \stackrel{i}{\to }& G\mathrm{Tor}\end{array}$$

with the square which we just discussed, to obtain

$$\begin{array}{ccc}Y& \stackrel{\pi }{\to }& X\\ {\downarrow }^{\mathrm{triv}}& {\Downarrow }^t& {\downarrow }^{\mathrm{tra}}\\ \Sigma G& \stackrel{i}{\to }& G\mathrm{Tor}\\ \downarrow & {\Downarrow }^{\simeq }& \downarrow \\ \Sigma (\hat{G}\to G)& \hookrightarrow & (\hat{G}\to G)\mathrm{Tor}\end{array},$$

which is indeed a smooth local trivialization of our $(\hat{G}\to G)$-$n$-bundle.

This is the global construction of the obstructing $n$-bundle that Bruce was asking for in the world of $n$-transport.

But it might be noteworthy that this abstract nonsense reduces in terms of local data to the following simple statement:

we check that the $n$-cocycle of the $(\hat{G}\to G)$-$n$-bundle obtained as above is simply nothing but the original $G$-1-cocycle of the $G$-bundle we started with, but now regarded as a $(\hat{G}\to G)$-$n$-cocycle under the canonical inclusion $$G\hookrightarrow (\hat{G}\to G).$$

Hence the “real work” happens not before we take the $(\hat{G}\to G)$-bundle constructed as above and realize it concretely as an equivalent ${\Sigma }^{n}U(1)$-$n$-bundle, using the other injection $${\Sigma }^{n}U(1)\hookrightarrow (\hat{G}\to G)$$ already mentioned before.

Doing this computation amounts to finding a pseudonatural transformation tin can diagram (a prism for $n=1$) connecting the $G$-cocycle triangles $$\begin{array}{ccc}& & •\\ & {}^{g_{\mathrm{ij}}}\nearrow & {\Downarrow }^{=}& {\searrow }^{g_{\mathrm{jk}}}\\ •& & \stackrel{g_{\mathrm{ik}}}{\to }& & •\end{array}$$

with the ${\Sigma }^{n}U(1)$-triangles (thought and drawn for $n=1$ now) $$\begin{array}{ccc}& & •\\ & {}^{\mathrm{Id}}\nearrow & {\Downarrow }^{f_{\mathrm{ijk}}}& {\searrow }^{\mathrm{Id}}\\ •& & \stackrel{\mathrm{Id}}{\to }& & •\end{array}.$$

But finding and choosing these prisms is precisely the operation encoded in the composition

$$Y^{[2]}\stackrel{g}{\to }\Sigma G\stackrel{\simeq }{\to }({\Sigma }^{n-1}U(1)\to \hat{G})\to {\Sigma }^{n}U(1)$$

which expresses, as mentioned at the beginning, the obstruction directly as an operation on local data.