The n-Category Café

Ordinary bundles from bundles of groupoids obtained from their transition function

You all know how it works for ordinary bundles. But one of my points will be that instead of being thought of as a groupoid, the things we see when constructing bundle gerbes are actually best thought of as two groupoids. In order to make that point, I will first emphasize how the ordinary construction for bundles already uses 1-groupoids.

So suppose you have a principal $G$-bundle $P \to X$ which you have locally trivialized over a cover $\pi : Y \to X$ $$t : \pi^* P \stackrel{\simeq}{\to} Y \times G$$ to obtain a transition function $$g : Y^{[2]} \to G \,.$$

Over each point $y \in Y$ of the cover we have identitfied the fiber $P_x$ with $(y,G)$. Different points $y'$ covering the same point $x$ have the same typical fiber sitting over them, $(y',G)$, which will be identified with the former one by the action of the transition function $g(y,y')$.

This situation is modeled by a groupoid over each point $x$: it’s objects are the elements $h$ in the typical fibers corresponding to $y$

$$(y,h)$$

and there is a unique morphism from $(y,h)$ to that $(y',h')$ which is related by the action of $g(y,y')$:

$$(y,h) \stackrel{g(y,y')}{\to} (y',h' = g(y,y') h) \,.$$

Let’s call this groupoid $$\tilde P_x := \left\{ (y,h) \stackrel{g(y,y')}{\to} (y',h' = g(y,y') h) | y \in \pi^{-1}(x)\,,h \in G \right\} \,.$$

It’s a big fat groupoid which encodes all the information of our trivialization. But actually, this big fat groupoid is equivalent to a mere set:

to start with, it is non-canonically isomorphic to the set underlying $G$ itself $$\tilde P_X \simeq G \,.$$

This equivalence is established by picking any (and that’s the non-canonical choice) element $y_0$ in $\pi^{-1}(x)$ and using the cocycle to identify any morphism in $\tilde P_x$ with a morphism starting and ending at $y$. This equivalence thus involves naturality squares of the form

$$\array{ (y,h) &\stackrel{g(y,y')}{\to} &(y',h' ) \\ \;\;\downarrow^{g(y,y_0)} && \;\;\downarrow^{g(y',y_0)} \\ (y_0,h_1) &\stackrel{\mathrm{Id}}{\to} &(y',h_2 ) } \,,$$

where you can figure out $h_1$ and $h_2$ from the general rules. They are not important. What is important is that the squares of this kind always commute, precisely due to the cocycle property of the transition function $g$ (i.e. the fact that $g(y_1,y_2) g(y_2,y_3) = g(y_1,y_3)$).

This establishes the (non-canonical) equivalence of our groupoid with the typical fiber $G$ of the $G$-bundle that we started with.

But that typical fiber $G$ is itself non-canonically equivalent to the real fiber, $P_x$. In fact, if we remember the trivialization $t$ from above, then we find that our groupoid is indeed canonically equivalent to this fiber $$\tilde P_x \simeq P_x \,.$$

So, and that’s where I am getting to the point of the discussion with the bundle gerbes: we could say that the bundle of groupoids $$\tilde P \to X$$ which we have constructed from the transition function is the total space of the bundle which we started with.

But actually, that would be slightly imprecise. Rather, we would first want to apply this equivalence to reduce all the 1-groupoid fibers to mere sets.

My point is: a similar argument applies to bundle gerbes: while it is posible to regard the bundle gerbe itself as a 2-bundle, it is better thought of as just the puffed-up thing analogous to $\tilde P$ which we obtain from the transition function of a 2-bundle, and to address as a true 2-bundle only the result of operating with some quotienting operation on our bundle gerbe.

Before closing this first part, I’ll reformulate the above in the language of universal bundles using groupoids, as mentioned at the end of this

Writing $Y^{[2]}$ for the groupoid associated to $Y \to X$, writing $\Sigma G$ for the one-object groupoid obtained from the group $G$ (it is crucial here to distinguish $G$ from $\Sigma G$, otherwise we’ll get lost) and $G//G = \mathrm{INN}(G)$ for the action groupoid of $G$ acting on itself by left action, we have:

the transition function itself is a functor

$$Y^{[2]} \stackrel{g}{\to} \Sigma G \,.$$

Along this functor we pull back the universal $G$-bundle in its groupoid incarnation $$\array{ && \mathrm{INN}(G) \\ && \downarrow \\ Y^{[2]} &\stackrel{g}{\to}& \Sigma G }$$ to obtain the bundle of groupoids $\tilde P$ $$\array{ \tilde P &\to& G // G \\ \downarrow && \downarrow \\ Y^{[2]} &\stackrel{g}{\to}& \Sigma G } \,.$$ The orginal $G$-bundle we started with is reobtained as the pushout $$\array{ \tilde P &\stackrel{t}{\to}& Y \times G \\ \downarrow^s && \downarrow \\ Y \times G &\to& P } \,.$$

2-bundles from bundles of 2-groupoids obtained from their transition 2-function

For our line 2-bundle the transition function is now a pseudofunctor

$$Y^{[2]} \to \Sigma (\Sigma U(1))$$ mapping a triangle $$\array{ && y' \\ & \nearrow && \searrow \\ y &&\to&& y'' }$$

to a filled triangle

$$\array{ && \bullet \\ & \nearrow &\;\,\;\downarrow^{g(y,y',y'')}& \searrow \\ \bullet &&\to&& \bullet } \,.$$

Again we form the pullback along $$\array{ && \mathrm{INN}_0(\Sigma U(1)) \\ && \downarrow \\ Y^{[2]} &\stackrel{g}{\to}& \Sigma \Sigma U(1) }$$

to obtain $\tilde P$. This is now a bundle of 2-groupoids!

A typical 2-morphism here looks like

$$\array{ && y' \\ & \multiscripts{^h}{\nearrow}{}\; &\;\,\;\Downarrow^{g(y,y',y'')}& \;\searrow^{h'} \\ y &&\stackrel{h''}{\to}&& y'' } \,,$$ where $$h'' = g(y,y',y'')h h' \,,$$ for all $y$, $y'$ and $y''$ and all all $h$ and $h'$.

To see the bundle gerbe here consider this as the “Hitchin-Chatterjee” version of bundle gerbes, where we have assumed the transition line bundle $$\array{ L \\ \downarrow \\ Y^{[2]} }$$ to be trivial. Alternatively, replace the group elements on the edges by elements of $U(1)$-torsors or by elements of complex 1-dimensional vector spaces, if you like. Then $g(\cdots)$ is the component map of the bundle gerbe multiplication mormphism $$\mu_g : \pi_1^*(L) \otimes \pi_2^*(L) \to \pi_3^* L \,.$$ Notice that there are three different ways to interpret this associative product:

a) as defining a central extension of the 1-groupoid $Y^{[2]}$ (that’s the point of view found in much of the literature)

b) as a $U(1)\mathrm{Tor}$ or $1d\mathrm{Vect}$-enriched groupoid (that’s pretty obvious and will probably strike few people as being really different from a))

c) as a two-groupoid where horizontal composition is just the tensor product, and where the “multiplication” is only given by the 2-morphisms.

Recall how we already had a 1-groupoid $\tilde P$ for the 1-bundle case. From this point of view we clearly want to be thinking of $\tilde P$ now here as a 2-groupoid.

But the point is: like our 1-groupoid before was equivalent to the typical fiber (a set, aka 0-groupoid) our 2-groupoid here is equivalent to a 1-groupoid, namely to $$\Sigma U(1) \,.$$

And that’s good! Because that is indeed the typical fiber we expect for a line 2-bundle.

To see the equivalence, use the cocycle condition, in complete analogy to what we did before:

we want to build a commuting triangular cylinder whose top triangle is

$$\array{ && y' \\ & \multiscripts{^h}{\nearrow}{}\; &\;\,\;\Downarrow^{g(y,y',y'')}& \;\searrow^{h'} \\ y &&\stackrel{h''}{\to}&& y'' }$$

and whose bottom triangle is of the form

$$\array{ && y_0 \\ & \multiscripts{^{h_1}}{\nearrow}{}\; &\;\,\;\Downarrow^\mathrm{Id}& \;\searrow^{h_2} \\ y_0 &&\stackrel{h_3}{\to}&& y_0 } \,.$$

That’s a simple exercise. I am too tired to try to fake drawing the diagrams here. The point is: it can be done naturally and smoothly, simply by filling everything with transitions in the only obvious way.

So, this way we find that our bundle gerbe, which we should really think of as giving rise, directly, to a bundle $\tilde P$ of 2-groupoids, becomes equivalent to the bundle of 1-groupoids which we expect, namely a bundle of groupoids each of which are equivalent to $\Sigma U(1)$.

Bottom line slogan

A bundle gerbe “is” a 2-transition function. From any $n$-transition function we obtain, by a pullback operation, a bundle of $(n+1)$-groupoids. The bundle of $n$-groupoids which we identify as the total space of the $n$-bundle associated with our transition function is obtained by performing a certain pushout on that.

Hm.

Rereading this, I am worried that everybody will think how completely pointless this is. But I wanted to write it down somewhere. I think it is both important for putting bundle gerbes into perspective and, more importantly, it helps understand the concept of principal 2-bundles and their construction from transition functions, the way Toby Bartels describes it in his thesis (check the proof of theorem 22!) or the way I think it can equivalently be reformulated in terms of pullback of the universal tangent groupoid sequence $\mathrm{INN}_0(G_{(2)}) \to \Sigma G_{(n)}$.

By the way, Toby tells me that the most up-to-date version of his thesis is provided here.