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As you may have noticed, while I do like to talk about principal infinity-bundles in regular intervals, about infinity-gerbes I used to talk much less.
(Here and in the following “gerbe” strictly means the original notion by Giraud, as used also by, say, Breen and Lurie: a 1-connective and 1-truncated object. It does not mean here any of the other meanings of the word that people like to use. These meanings are all equivalent to some extent, but not completely. In particular, “gerbe” here does not tautologically mean “cocycle for an oo-bundle”.)
Why? Because
the classification of oo-gerbes is just a subcase of principal oo-bundles;
their total spaces are less canonically defined,
in applications that I care about the oo-bundles show up much more naturally.
Okay. But nevertheless, for the sake of completeness, now I would like to obtain a full discussion of the relation between nonabelian oo-gerbes and principal oo-bundles. And I think by pointing to the right place in the $\infty$-topos literature, this is easy. The problem has secretly been solved already. We only need to make it manifest.
First to clarify my terminology: with an ambient $\infty$-topos $\mathcal{X}$ fixed, by a nonabelian $n$-gerbe I shall mean a 1-connective and n-truncated object. This is the evident $n$-version of Breen’s notion of 2-gerbe, which is more general (namely nonabelian, as opposed to abelian) than Lurie’s $n$-gerbes, which are moreover required to be n-connective, even.
So this means: a nonabelian $\infty$-gerbe in $\mathcal{X}$ is equivalently simply a connected object . Nothing else. An object whose 0th homotopy sheaf is the terminal sheaf.
This way of saying it indicates why $\infty$-gerbes are relevant on general grounds: by the discussion at looping and delooping, the infinity-group objects in $\mathcal{X}$ are precisely those objects that are connected AND pointed. Hence conversely: $\infty$-groups are equivalently $\infty$-gerbes equipped with a global section. So the notion flows naturally from $\infty$-group theory, simply by dropping one axiom.
In particular, if $G$ is an $\infty$-group and $\mathbf{B}G$ its connected and canonically pointed delooping object, we want to ask for those connected objects that, while not necessarily having a global section, still look like $\mathbf{B}G$ locally. These are the $G$-$\infty$-gerbes $P$: for $U \to *$ any effective epimorphism they admit an equivalence
$P|_U \simeq \mathbf{B} G|_{U} \,.$Extrapolating from the 1-gerbe case, we expect that $G$-$\infty$gerbes are classified by $AUT(G)$-cohomology in $\mathcal{X}$, where $AUT(G)$ is the (internal!) automorphism $\infty$-group… of $\mathbf{B}G$:
$AUT(G) := \underline{Aut}_{\mathcal{X}}(\mathbf{B}G) \,.$We already know that
classified by an $AUT(G)$-cocycle is an $AUT(G)$-principal infinity-bundle;
there is a canonical associated infinity-bundle to any $\underline{Aut}(\mathbf{B}G)$-bundle, of course: the one associated by the canonical action of $\underline{Aut}(\mathbf{B}G)$ on $\mathbf{B}G$.
But such an associated bundle is precisely one whose fibers locally look like $\mathbf{B}G|_U$!
So I guess we want to be claiming
For $G \in \infty Grp(\mathcal{X})$, $G$-$\infty$-gerbes are precisely the $\underline{Aut}(\mathbf{B}G)$-associated infinity-bundles.
Luckily, as you can see behind that last link, the necessary notions and properties of associated $\infty$-bundles in an $\infty$-topos have all already been worked out. By Matthias Wendt.
So just open his article and wherever he says “$F$” for the fiber, take $\mathbf{B}G$. Then notice
his def 3.5 of locally trivial $F$-fibration is then manifestly the definition of $G$-$\infty$-gerbe from above.
his “$B(*,hAut_\bullet(F),*)$” from section 5.3 is then the object $\mathbf{B} AUT(G)$;
his theorem 5.10 is the desired classification result, which identified nonabelian $G$-$\infty$-gerbes precisely with the $AUT(G) := \underline{Aut}(\mathbf{B}G)$-associated $\mathbf{B}G$-$\infty$-bundles.
QED.
Do you see what I mean? Let me know if you think I missing some subtlety that makes this trivial conclusion nontrivial, after all.
The link associated infinity-bundles doesn’t go anywhere. :)
Thanks to naming conventions, I know that it must be at associated infinity-bundle; I’ve now created the redirect.
Yes. I was just being a bit lazy.
Ah, thanks, my fault. The first link to that keyword did work, though.
(By the way, Google also helps in such cases.)
I have added a sketch remark (marked as such) along the above lines to infinity-gerbe. I’ll try to spell this out in more detail a little later.
One thing I need to think about is how to generalize Matthias Wend’s def. 3.5 of local triviality to one that just involves general effective epimorphisms. His definition, as is, applies to the specific effective epimorphism provided by the disjoint union of all objects in the site.
Hm, I am looking at Wendt’s article in a bit more detail now.
For the present purpose it’s sufficient to start on p. 29.
I think in theorem 5.10 there is a secret assumption: that the base space object $X$ is 0-truncated. Otherwise the ordinary (1-categorical) Cech nerve of $\coprod_i U_i \to X$ (which seems to be being used) is not necessarily locally equivalent to $X$.
With that assumption the argument on p. 31-32 reduces to alluding to the usual cocycle-Yoga.
For the moment I have some further notes from page 105 on in my cohesive document.
added to infinity-gerbe a paragraph on nonabelian “banded” $\infty$-gerbes.
added to infinity-gerbe Lurie’s classification result for abelian $n$-gerbes. And slightly polished the following paragraphs on classification of nonabelian $n$-gerbes.
One has to be careful with comparing the general classification with the classification result in section 7.2.2 of HTT: the latter is more restrictive in two ways:
not only does it restrict attention to $n$gerbes that are what I started calling now EM $n$-gerbes (since they are globalizations of Eilenberg-MacLane objects) ,
but also it restricts the morphisms between them more than one does elsewhere in the literature – see page 576-577 on HTT.
In particular the second diagram on p. 577 shows that nontrivial automorphisms of the band don’t show up in this definition.
This is why for the case $n = 2$ Lurie’s classification contradict’s Breen’s :
in the general context $U(1)$-banded 2-gerbes are classified by nonabelian cohomology with coefficients in the 3-group coming from the crossed complex $[U(1) \to 1 \to \mathbb{Z}_2]$. But for Lurie’s more restricted notion of morphism the classification is by cohomology with coefficients in $[U(1) \to 1 \to 1]$ hence degree-3 cohomology with coefficients in $U(1)$. The nontrivial automorphisms in $\mathbb{Z}_2 = Aut(U(1))$ don’t appear.
I have added some remarks along these lines to infinity-gerbe now.
Urs, why $U(1) \to 1 \to \mathbb{Z}/2$? Why not $U(1) \to \mathbb{Z}/2 \to 1$? (Playing devil’s advocate here) Technically a 2-gerbe has to be banded by a 2-group, so one can’t say $U(1)$-banded 2-gerbe without some ambiguity. How do you want to think of $U(1)$ as a 2-group. Then we can talk about $AUT$ of this 2-group.
Hi David,
the 2-gerbes that I am talking about are either Breen-style 2-gerbes and then $G$-2-gerbes for the 2-group
$G = \mathbf{B}U(1) = [U(1) \to 1]$or they are Lurie-style 2-gerbes, and then “banded by $U(1)$”.
Their classification as Breen-style $G$-2-gerbes is by $AUT(G)$-cohomology. And
$AUT(G) = \underline{Aut}(\mathbf{B} G) = \underline{Aut}(\mathbf{B}^2 U(1))$is (let’s see, maybe above I missed one copy of $U(1)$)
$\cdots = [U(1) \stackrel{0}{\to} U(1) \stackrel{0}{\to} \mathbb{Z}_2]$as far as I can see: invertible 2-functors $\mathbf{B}^2 U(1) \to \mathbf{B}^2 U(1)$ need to come from automorphisms of $U(1)$, of which there are $\mathbb{Z}_2$. There is $U(1)$-worth of pseudonatural transformations from any such to itself. And finally there are $U(1)$ worth of modifications of any of these to itself.
So Breen-style $U(1)$-2-gerbes are classified by $H^1(X, [U(1) \stackrel{0}{\to} U(1) \to \mathbb{Z}_2])$. And not by $H^1(X, [U(1) \to 1 \to 1]) = H^3(X, U(1))$ as are Lurie-style $U(1)$-2-gerbes.
We can look at the same discussion for $U(1)$-1-gerbes, where it is easier. The reason why I went to degree 2 is because Lurie in his account excludes 1-gerbes.
So let’s consider this again for $U(1)$-1-gerbes. Giraud-style $G$-1-gerbes are classified by $AUT(G)$-cohomology. For $G = U(1)$ we have $AUT(U(1)) = \underline{Aut}(\mathbf{B}U(1)) = [U(1) \to \mathbb{Z}_2]$. So Giraud’s $U(1)$-gerbes are not equivalent to $U(1)$-bundle gerbes. Rather, they are equivalent to $U(1)$-“Jandl gerbes”.
For clarification: of course the Breen-style $\mathbf{B}U(1)$-2-gerbes with trivial “band” in the sense of the associated class in $H^1(X,Out(\mathbf{B}U(1)))$, these are indeed classified by $H^3(X, U(1))$.
Ah, good. That makes sense. I just couldn’t figure out where you got the crossed complex from, and now it works.
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