(2,1)-quasitopos?
structures in a cohesive (∞,1)-topos
group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
In the language of (∞,1)-topos theory the ordinary definition of gerbe has the following simple re-formulation:
In a given (2,1)-topos, a gerbe is an object which is
There are various evident generalizations of this, where one allows the degree in either of the two clauses to vary. In the literature one finds higher gerbes defined in either way, and so there are more general and more specific definitions.
Let $\mathcal{X}$ be an (∞,1)-topos.
For $n \in \overline{\mathbb{N}}$ an extended natural number,
a $n$-gerbe is an object $\mathcal{G} \in \mathbf{G}$ that is
an EM $n$-gerbe is an an $n$-gerbe that is
The definition of “EM $n$-gerbes” appears as HTT, def. 7.2.2.20. For $n = 2$ the definition of “general $n$-gerbe” (called a nonabelian 2-gerbe at 2-gerbe) appears for instance in (Breen).
In particular we have then the following.
An $\infty$-gerbe in $\mathcal{X}$ is a connected object.
Write
for the core of the full-sub-(∞,1)-category on the $\infty$-gerbes.
The notion of morphisms between EM $n$-gerbes in HTT, pages 576, 577 is more restrictive than this. This is the reason why the classification found there is a restriction of the general classification. This is discussed below.
If one adds to the definition of “EM $n$-gerbe” the condition that $\mathcal{G}$ has a “global section”, a global element $* \to \mathcal{G}$, then it becomes the definition of an Eilenberg-MacLane object in degree $n$. In other words, EM $n$-gerbes are just like Eilenberg-MacLane objects but without necessarily a global section.
By the main theorem at looping and delooping the connected and pointed objects in an (∞,1)-topos $\mathcal{X}$ are equivalently (the deloopings of) ∞-group objects: we have an equivalence of (∞,1)-categories
Therefore an $\infty$-gerbe is much like the delooping of an $\infty$-group, only that a global section may be missing.
But notice that a section always exists locally: for
any topos point and $P \in \mathcal{X}$ an $\infty$-gerbe, the stalk $x^* P \in$ ∞Grpd is connected, because the inverse image $x^*$ preserves the finite limits that define categorical homotopy groups in an (∞,1)-topos and preserves the terminal object $*$:
Therefore it is natural to consider the notion of an $G$-$\infty$-gerbe for a fixed $G \in \infty Grp(\mathcal{X})$. This is done below.
A “restricted” $n$-gerbe $E$ has, by definition, a single non-trivial homotopy sheaf?. For $n \geq 2$ this is an sheaf of abelian groups $A$ in the underlying topos
This $A$ is called the band of $E$ and that $E$ is banded by $A$.
In the case that $n = 1$ or that we have a “general” $n$-gerbe the notion of band is refined by nonabelian cohomology-information. See [below].
Let $\mathcal{X}$ be an (∞,1)-topos.
For $G \in \infty Grp(\mathcal{X})$ an ∞-group, a $G$-$\infty$-gerbe $P$ is
an $\infty$-gerbe, def. 2;
such that there exists an effective epimorphism $U \to *$ in $\mathcal{X}$ and an equivalence
Let $A \in Grp(\mathcal{X}) \subset \infty Grpd(\mathcal{X})$ be an abelian group object and fix $n \in \mathbb{N}$, $n \geq 2$.
Recall the notion of $A$-banded $n$-gerbes from def. 3.
Write
for the cohomology in $\mathcal{X}$ of the terminal object with coefficients in $A$ in degree $n+1$.
There is a canonical bijection
This appears as HTT, cor. 7.2.2.27.
The morphisms of EM $n$-gerbes in HTT, p. 576-577 are more restrictive than the morphisms of these objects regarded as general $n$-gerbes.
By the discussion below, if EM $n$-gerbes are regarded as special $n$-gerbes, then their classification is by a nonabelian cohomology-extension of the above result.
We discuss partial generalizations of the above result to nonabelian $\infty$-gerbes
Comparing with the discussion at associated ∞-bundle one finds that def. 4 of $G$-$\infty$-gerbes defines “locally trivial $\mathbf{B}G$-fibrations”. By the main theorem there, these are classified by cohomology in $\mathcal{X}$ with coefficients in the internal automorphism ∞-group
At least for $\mathcal{X}$ a 1-localic (∞,1)-topos we have a canonical bijection
This follows as a special case of the result by (Wendt) discussed at associated infinity-bundle.
The right hand side classifies als $AUT(G)$-principal ∞-bundles and this equivalence identifies $G$-$\infty$-gerbes as the canonical $AUT(G)$-associated ∞-bundles.
(Compare to the analogous discussion in the special case of gerbes.)
For $G \in \infty Grpd(\mathcal{X})$ an $n$-group (with $n \geq 1$) write
Call this the outer automorphism infinity-group of $G$. By definition there is a canonical morphism
By the above classification, this induces a morphism
from $G$-$n$-gerbes to nonabelian cohomology in degree 1 with coefficients in $Out(G)$. For $E \in G Gerbe$ the pair
is called the band of $E$. For $[K] \in H^1_{\mathcal{X}}(X, Out(G))$ the (n+1)-groupoid $G Gerbe_K$ of $K$-banded $G$-$n$-gerbes is the homotopy pullback
Write $\mathbf{B}^2 Z(G)$ for the homotopy fiber of $\mathbf{B}AUT(G) \to \mathbf{B}Out(G)$, producing a fiber sequence
We call $Z(G)$ the center of the infinity-group.
In terms of this the above $G Gerbe_K$ is the cocycle $(n+1)$-groupoid of the K-twisted Z(G)-cohomology in degree 2:
Notice that if $Z(G)$ itself is higher connected then $H^2_K(X, Z(G))$ is accordingly cohomology in higher degree.
For $X$ a topological space, let $\mathcal{X} = Sh_{(\infty,1)}(X)$ be its (∞,1)-category of (∞,1)-sheaves. Write
for the sheaf of circle group-valued functions. And $\mathbf{B}U(1)$ for its delooping
Then
where on the right we display the crossed complex corresponding to this 3-group (the morphisms are all constant on the unit element, the action of $\mathbb{Z}_2$ on either of the $U(1)$s is the canonical one given by $\mathb{Z}_2 \simeq Aut(U(1))$ ). This is seen as follows: every invertible 2-functor $F : \mathbf{B}^2 U(1) \to \mathbf{B}^2 U(1)$ comes from a group auotmorphism of $U(1)$, of which there are $\mathbb{Z}_2$. A pseudonatural transformation necessarily goes from any such $F$ to itself, and there are $U(1)$ of them Similarly, any modification of these necessarily is an endo, and there are also $U(1)$ of them.
Therefore $\mathbf{B}U(1)$-2-gerbes are classified by the nonabelian cohomology
This has a subgroup coming from the canonical inclusion
on the trivial automorphism of $U(1)$. The classification of $U(1)$ EM-2-gerbes in HTT gives only this subgroup
In the terminology of orientifold/Jandl gerbes the more general objects on the right are the “Jandl 2-gerbes” or “orientifold 2-gerbes”.
principal ∞-bundle / associated ∞-bundle / $\infty$-gerbe.
Notice that the notion of bundle gerbe and bundle 2-gerbe etc. is not unrelated, but is a priori a rather different notion.
The notion of “EM” $\infty$-gerbes in an $(\infty,1)$-topos appears in section 7.2.2 of
The general notion (but without a concept of ambient $(3,1)$-topos made explicit) for $n = 2$ (see 2-gerbe) appears for instance in
Lawrence Breen, On the classification of 2-gerbes and 2-stacks , Astérisque 225 (1994).
Lawrence Breen, Notes on 1- and 2-gerbes in John Baez, Peter May (eds.) Towards Higher Categories (arXiv:math/0611317).
The general notion for arbitrary $n$ in an $(\infty,1)$-topos context is discussed in section 2.3 of
Discussion of the classification of $G$-$\infty$-gerbes and more general fiber bundles in 1-localic $(\infty,1)$-toposes is in