Morphisms between abelian differentiable gerbes

Following on from last post I want to talk about the appropriate notion of morphism between the objects I defined. Recall that these are Lie groupoids $X$ with a map to the manifold $M$ satisfying some properties ( $X_0 \to M$ and $X_1 \to X_0\times_MX_0$ are surjective submersions), and then equipped with a little bit of extra structure. We will fix a bundle of abelian Lie groups $\mathcal{A} \to M$ (where this projection map is a surjective submersion). Then the extra structure is that we have a specified isomorphism between the bundle of Lie groups $\Lambda X$ (see the previous post for notation) and the pullback of $\mathcal{A}$ along $X_0\to M$ , such that that the conjugation action of $X$ on $\Lambda X$ factors through the resulting descent data for $\Lambda X$ in the guise of its encoding of an action of the Čech groupoid $\check{C}(X_0)$ on $\mathcal{A}$ . This is the definition of an abelian $\mathcal{A}$ -bundle gerbe.

Now, given two Lie groupoids $X\to \mathrm{disc}(M)$ and $Z\to \mathrm{disc}(M)$ over $M$ (with the surjective submersions as above) we can consider internal functors $X\to Z$ between them, commuting with the projections to $M$ . Such a functor is a map of (generalised bundle) gerbes over $M$ , but we want to consider more specifically maps of abelian $\mathcal{A}$ -bundle gerbes.

The cartoon analogy to keep in mind as to what could happen without further conditions is if you had a pair of principal $G$ -bundles $P \to M$ and $Q\to M$ , and a map $f\colon P\to Q$ over $M$ , but it was only equivariant in a weak way, namely that $f(pg) = f(p)\phi(g)$ where $\phi$ is a group automorphism of $G$ . Abelian $\mathcal{A}$ -bundle gerbes are principal 2-bundles for the 2-group (or bundle of 2-groups) $\mathcal{A} \rightrightarrows 1$ , but the way in which they inherit this action because of the isomorphism $\Lambda X\simeq \mathcal{A}$ , and functors automatically commute with the action of $\Lambda X$ on $X$ . So instead of having to impose a condition analogous to $f(pg) = f(p)\phi(g)$ , this comes for free, in that from a functor (over $M$ ) $f\colon X\to Y$ between abelian $\mathcal{A}$ -bundle gerbes you get an endomorphism of bundles of groups of the pullback of $\mathcal{A}$ along $\pi_X\colon X_0\to M$ . This arises as the composite $\pi_X^* \mathcal{A} \simeq \Lambda X \to f_0^*\Lambda Y \simeq f_0^*\pi_Y^*\mathcal{A} \simeq \pi_X^* \mathcal{A}$ .

So the requirement that we are going to impose is going to firstly ensure this endomorphism of $\pi_X^*\mathcal{A}$ is an automorphism, which we can then ask is actually descendable, so that it arises from an automorphism of $\mathcal{A}$ down on $M$ (in principle, one could ask the endomorphism descends without asking it’s an automorphism, but not today). But, secondly we are going to ask that the automorphism of $\mathcal{A}$ is in fact the identity map. This is the analog of asking the group automorphism $\phi$ above is the identity map on $G$ . However, we need to specify these conditions explicitly in a way that makes sense in the picture of bundle gerbes as internal groupoids.

So, it’s been a long way around to give the intuition (and this blog post has been long delayed, due to work commitments and other things), but the final definition is in fact rather easy. All that we ask is that $\Lambda X \to f_0^*\Lambda Y$ is a pullback, so that the square witnessing the fact $\Lambda X$ is a pullback of $\mathcal{A}$ along $\pi_X$ is the pasting of the similar square for $\Lambda Y$ and the square witnessing the fact $\Lambda X$ is a pullback of $\Lambda Y$ .

One then has to check that from this definition, and the condition that the descent data for $\pi_X^*\mathcal{A}$ “comes from” the adjoint action of $X_1$ on $\Lambda X$ , means that the resulting automorphism of $\pi_X^*\mathcal{A}$ descends to the identity map.

Given the above, one can then go back and examine what happens if you drop the requirement the bundle gerbe is abelian. Certainly the definition given (that $\Lambda X \simeq f_0^*\Lambda Y$ ) still makes sense even when we only ask that we have a “ $\mathcal{G}$ -bundle gerbe”, for any bundle of Lie groups $\mathcal{G}$ on $M$ .

The final point, I think, is that a morphism of abelian $\mathcal{A}$ -bundle gerbes is a functor of Lie groupoids satisfying two conditions: commuting with the functors down to the base, and the pullback condition discussed above. In particular there is no extra data that we need to supply to define a morphism.

What does this tell us in the case of a vanilla (abelian) $U(1)$ -bundle gerbe, as originally defined by Michael Murray? (Though note that in the paper, everything’s done with $\mathbb{C}^\times$ -bundles instead) He defined a notion of morphism of bundle gerbes $X\to Y$ , which amounts to a functor over $M$ such that $X_1 \to Y_1$ is a map of $U(1)$ -bundles covering the map $X_0\times_M X_0 \to Y_0\times_M Y_0$ . If I take $\mathcal{A}= M\times U(1)$ , then as last time, a bundle gerbe on $M$ is a Lie groupoid $X\to M$ over $M$ equipped with a trivialisation $\Lambda X\simeq X_0\times U(1)$ as a bundle of groups, satisfying conditions. Given two such, say $X$ and $Y$ , a morphism between them in my sense is a functor over $M$ such that the restriction of $f_1$ to $\Lambda X$ respects the trivialisations. This condition then ensures that the full $f_1$ is a map on the total spaces of $U(1)$ -bundles.that is $U(1)$ -equivariant, so we have a morphism of bundle gerbes in Murray’s sense. Now assume we have a pair of abelian $U(1)$ -bundle gerbes $X,Y$ , and a morphism between them in Murray’s sense. Then since $f_1\colon X_1 \to Y_1$ is a map of $U(1)$ -bundles covering $X_0\times_M X_0 \to Y_0\times_M Y_0$ , the resulting square is a pullback, and then the restriction of this square along the diagonal maps $X_0\to X_0\times_MX_0$ and $Y_0\times_MY_0$ (to get the map $\Lambda X\to \Lambda Y$ over $X_0\to Y_0$ ) is still a pullback, and then we have recovered my definition of morphism. This I have recovered the traditional definition up to equivalence.

The natural continuation of the above discussion is to examine what we should do with natural transformations (or rather, natural isomorphisms, as they all are). Of course, the data of a natural isomorphism $a\colon f\Rightarrow g\colon X\to Y$ over $M$ is a smooth function $X_0\to Y_1$ satisfying the usual naturality condition. To compare to the literature, we need to go to section 3.4 of Danny Stevenson’s PhD thesis (which builds a 2-category of bundle gerbes and morphisms à la Murray, in the lead up to Proposition 3.8). The definition of 2-arrow is slightly nontrivial and on inspection of Lemma 3.3 one can see that the definition of abelian bundle gerbe I have given is being implicitly relied on, where descent data is given by conjugation in the groupoid, to descend a certain bundle down to $M$ . The data of a 2-arrow $a\colon f\Rightarrow g\colon X\to Y$ in Stevenson’s sense gives a smooth map $X_0 \to Y_1$ that, I think by the construction in Lemma 3,3, is in fact the data of a natural isomorphism (implicit in the equation at the bottom of page numbered 27). The question is: what else do we get? Or, another tactic is to see if an arbitrary natural isomophism $f\Rightarrow g$ is enough to define a 2-arrow in Stevenson’s sense. Given the component map $a\colon X_0\to Y_1$ , it gives a section $s_a$ of Danny’s bundle $\hat{D}\to X_0$ (which is the pullback of $Y_1\to Y_0\times_M Y_0$ by $(f_0,g_0)\colon X_0\to Y_0\times_MY_0$ ). What we need to check is that $s_a$ descends to a section of the descended bundle $D_{\bar{f},\bar{g}}$ . However, we know that the descent data for $\hat{D}$ that is constructed in Lemma 3.3 is by conjugation, and the naturality condition on $a$ in fact tells us that $s_a$ is compatible with this descent data, and so descends to a section as needed.

So in fact we can define a 2-arrow between morphisms (in my sense) to just be a natural isomorphism satisfying no additional conditions. And so we can define a 2-category of bundle gerbes as a locally full—and not full—sub-2-category of the 2-category of Lie groupoids over $M$ . My discussion in the past two blog posts (this and the previous one) is much longer than the bare definition requires; this is all to motivate and explain. The actual definition could be at most a single paragraph long—cf the more than two pages including two lemmas with proofs needed to define what turns out to be an isomorphic 2-category in Danny’s thesis, in addition to the theory of bundle gerbes that requires.

It turns out this post is longer than I thought it would be, but that’s ok. The next step is to move on from this naive 2-category of bundle gerbes, because the above really isn’t the one that people want. But that will fall out from an existing construction in the literature. I’m not sure, though, if I should first treat the analog of the above definition stack where we add connections to things, before moving on to the “real” definitions for both.

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