Abelian differentiable gerbes – recap

Bundle gerbes are a class of objects that are my bread and butter, but there are many different ways of thinking about them. In my paper with Raymond Vozzo we wrote a different take that starts from the point of view of plain Lie groupoids (which we allow to be infinite-dimensional) and slowly builds in the conditions so that we end up with something equivalent to Murray’s original definition.

I want to summarise our approach to bundle gerbes, because I want to do a series of posts building up to a new construction of the bicategory of bundle gerbes on a given manifold. As a first step, we need to define bundle gerbes, but also define morphisms between them as a certain class of internal functors. Actually, it’s going to develop a theory of a slightly more general object, which won’t impact the results. Ultimately this is because I need to fix The Paper where my coauthor and I used a somewhat naive approach to gauge transformations of curvings on a bundle gerbe, and have an idea I will slowly develop in public, since my time for pure research is now more constrained. And I miss the early days of the n-Category Café where half-developed research was posted all the time for public discussion.

I will take as given the definition of Lie groupoid, and will use notation like $X_1 \rightrightarrows X_0$ for a Lie groupoid $X$ . The first thing we need is to assume that there is a functor (all of them will be assumed to be smooth, that is, internal functors in the category of manifolds) from the Lie groupoid $X$ to the identity-arrows-only groupoid $\mathrm{disc}(M)$ associated to a smooth manifold $M$ . Further, we assume that $X_0 \to M$ is a surjective submersion. In the world of gerbes qua stacks, this corresponds to the condition that the presheaf of groupoids is locally non-empty. The other condition in the definition of a gerbe (the presheaf of groupods is locally connected) corresponds to the condition on $X$ that $(s,t)\colon X_1 \to X_0\times_M X_0$ is a surjective submersion.

With these two conditions on $X\to \mathrm{disc}(M)$ , we have a presentation on a differentiable gerbe, with no other considerations. However, we are interested in pushing towards a structure much closer to bundle gerbes. In particular, there is no assumption yet that $(s,t)$ is a principal bundle, nor that its structure group is abelian etc. We will get there in stages, and want to pay attention to what is structure, and what is a property on structure. The functor to $\mathrm{disc}(M)$ is clearly additional structure, but in the end we are interested in thinking about the bicategory of bundle gerbes on $M$ , so every Lie groupoid we are looking at will be equipped with such a thing. Then the requirements that $X_0\to M$ and $(s,t)$ are surjective submersions are properties.

We then note that the automorphism groups of objects of the Lie groupoid act on arrows out of said objects (or into said objects) but we need to do this in a global way. So we define the bundle of groups $\Lambda X \to X_0$ that is the pullback of $(s,t)$ along the unit map $X_0\to X_1$ (this is a surjective submersion onto $X_0$ , where there is a fibrewise group structure encoded by $\Lambda X \times_{X_0} \Lambda X \to \Lambda X$ etc. Now this bundle of groups can be made to act on the left and on the right of $X_1 \to X_0\times_M X_0$ , by pulling back $\Lambda X$ along the two projections $\mathrm{pr}_i\colon X_0\times_M X_0 \to X_0$ . We will call the two bundles of groups on $X_0\times_M X_0$ that act on the left and right $\Lambda X_L$ and $\Lambda X_R$ respectively. These actions come for free, given the data so far.

To get something close to a bundle gerbe, but allowing for different actions on the left and the right, we actually want $\Lambda X$ to descend to a bundle of groups $\mathcal{A}\to M$ . This can be ensured by asking that there is a groupoid action of the Čech groupoid $\check{C}(X_0) = X_0\times_M X_0 \rightrightarrows X_0$ on $\Lambda X$ (as a group object), via the anchor map $\Lambda X\to X_0$ . This is additional structure, and is equivalent to asking for descent data for $\Lambda X\to X_0$ . So then we have a given isomorphism $\Lambda X_L\simeq \Lambda X_R$ of bundles of groups, and both of these are uniquely and compatibly identified with the pullback of $\mathcal{A}\to M$ along $X_0\times_M X_0 \to M$ . And so we can think of $X_1\to X_0\times_M X_0$ as a bibundle for this single bundle of groups. At this point, we have an $\mathcal{A}$ -gerbe, which is also close to an example of what some colleagues called a bundle gerbe with non-constant structure group bundle.

The second-to-last condition we want to place on the groupoid is to make sure this bibundle structure is in fact just a one-sided principal bundle (which amounts to the left and right actions agreeing). We first note that there is always a groupoid action of $X$ on $\Lambda X$ (via the same anchor map), which is by conjugation. Pointwise, if we are given an automorphism $a\colon x\to x \in \Lambda X$ and an arrow $f\colon x \to y\in X_1$ , then we get an automorphism $faf^{-1}\colon y\to y$ . Asking that the left and right bundle-of-groups actions agree is equivalent to asking that the conjugation action of $X$ factors through the identity-on-objects functor $X\to \check{C}(X_0)$ to give the Čech groupoid action in the previous paragraph. This is an additional property on the groupoid.

Lastly, we can impose the condition that the bundle of groups $\mathcal{A}\to M$ should be a bundle of abelian Lie groups (another condition), as the notation might have hinted.

At this point, we have abelian bundle gerbes, with the caveat that instead of the map $X_1 \to X_0\times_M X_0$ be a principal $A$ -bundle, for a fixed Lie group $A$ , we have a bundle of abelian Lie groups on $M$ , and a principal bundle for (a pullback of) it. One final (possibly optional) condition is asking that $\mathcal{A}\simeq M\times A$ is the trivial bundle of groups. And this, in the special case that $A=U(1)$ is the traditional structure that goes by the name of bundle gerbe (modulo the fact sometimes the associated line bundles are sometimes used instead of principal bundles). In this special case, we can ask not for a trivialisation of the descended bundle onto $M$ , but instead ask for a trivialisation of $\Lambda X$ , and then ask that the resulting trivialisations of $\Lambda X_L$ and $\Lambda X_R$ are compatible with the isomorphism $\Lambda X_L\simeq \Lambda X_R$ . This means we don’t have to work with any data that lives down on $M$ . The trivialisation of $\Lambda X$ is additional structure but the compatibility with the descent data is a property.

Next time, I will have a shorter post, and talk about morphisms of bundle gerbes, in the original, overly strict sense, and how they work in this more elaborate setup.

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kamilic says:

April 23, 2026 at 4:30 pm

Thanks for your sharing.

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