(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 full generality, we have the following definition of gerbe .
Given an (∞,1)-topos $\mathcal{X}$, a gerbe in $\mathcal{X}$ is an object $\mathcal{G} \in \mathcal{X}$ that is
The first condition says that a gerbe is an object in the (2,1)-topos $\tau_{\leq 1 } \mathcal{X} \hookrightarrow \mathcal{X}$ inside $\mathcal{X}$. This means that for $C$ any (∞,1)-site of definition for $\mathcal{X}$, a gerbe is a (2,1)-sheaf on $C$, $\mathcal{G} \in Sh_{(2,1)}(C)$: a stack on $C$.
The second condition says that a gerbe is a stack that locally looks like the delooping of a sheaf of groups. More precisely, it says that
the morphism $\mathcal{G} \to *$ to the terminal object of $\mathcal{X}$ is an effective epimorphism);
the 0th categorical homotopy group $\pi_0 \mathcal{G}$ is isomorphic to the terminal object $*$ as objects in the sheaf topos $\tau_{\leq 0} \mathcal{X} = Sh_{(1,1)}(C)$. Here $\pi_0 \mathcal{G}$ is the sheafification of the presheaf of connected components of the groupoids that $\mathcal{G} : C^{op} \to Grpd \hookrightarrow \infty Grpd$ assigns to each object in the site.
Traditionally this is phrased before sheafification as saying that a gerbe is a stack that is locally non-empty and locally connected . This is the traditional definition, due to Giraud.
Also traditionally gerbes are considered in the little (2,1)-toposes $\tau_{\leq 1} \mathcal{X}$ of a topological manifold or smooth manifold $X$ or a topological stack or differentiable stack $X$. One then speaks of a gerbe over $X$ .
More precisely, we may associate to any $X \in C :=$ Top or $X \in C :=$ Diff the corresponding big site $C/X$ and form the (2,1)-topos $\tau_{leq} \mathcal{X} := Sh_{(2,1)}(C/X)$. In terms of this a gerbe is given by a collection of groupoids assigned to patches of $X$, satisfying certain conditions.
Equivalent to this is the over-(2,1)-topos $\tau_{\leq 1} \mathcal{H}/j(X)$, where $\tau_{\leq 1}\mathcal{H} := Sh_{(2,1)}(C)$ is the big (2,1)-topos over $C$ (and $j$ denotes its (2,1)-Yoneda embedding).
Since this $\mathcal{H}$ is a cohesive (∞,1)-topos we may think of its objects a general continuous ∞-groupoids or smooth ∞-groupoids. In large parts of the literature coming after Giraud gerbes, or related structures equivalent to them, are described this way in terms of topological groupoids and Lie groupoids. This perspective is associated with the notion of a bundle gerbe .
We discuss gerbes that have a “strucure group” $G$ akin to a principal bundle. Indeed, while not the same concept, these $G$-gerbes are equivalent to $AUT(G)$-principal 2-bundles, for $AUT(G)$ the automorphism 2-group of $G$.
The definition 1 of gerbe is almost verbatim that of Eilenberg-MacLane object in degree 1. The only difference is that the latter is required to have not only the homotopy sheaf $\pi_0 = *$, but even have a “global section” in the form of a morphism $* \to P$.
First consider this locally. A gerbe (as any 1-connected object) necessarily has local sections:
for
any topos point, the stalk functor $x^*$, being an inverse image is left exact and hence preserves homotopy sheaves and terminal objects. It follows that the 0th homotopy sheaf is trivial
as are all the degree-$p$ homotopy sheaves for $p \gt 1$. Therefore $x^* P$ is a groupoid with a single object: the delooping groupoid of a group $G_x$:
More generally, by the discussion at looping and delooping we have in an equivalence of (∞,1)-categories
between the ∞-group objects in the ambient (∞,1)-topos $\mathcal{X}$ and the pointed connected objects.
It follows that for a gerbe $P$ that admits a global section $* \to P$ the above relation holds not only stalk-wise, but globally: it is the delooping of its own first sheaf of homotopy groups
The following definition characterizes gerbes that are locally of the form of remark 1.
Let $G \in Grp(\mathcal{X})$ be a group object. A gerbe $P \in \mathcal{X}$ is a $G$-gerbe if there exists an effective epimorphism $U \to *$ and an equivalence
where $P|_U := P \times U$ and $G|_U := G \times U$.
In a typical application one considers gerbes over some topological space $X$. In that case
$\mathcal{X} = Sh_{(\infty,1)}(Op(X))$ is the (∞,1)-category of (∞,1)-sheaves on the category of open subsets of $X$;
the terminal object of $\mathcal{X}$ is the space $X$, regarded as an object in its own $(\infty,1)$-topos, hence we can write $X := * \in \mathcal{X}$;
a group object $G \in \mathcal{X}$ is sheaf of groups on $X$;
an effective epimorphism $U \to *$, hence $U \to *$ is obtained from any open cover $\{U_i \to X\}$ by setting $U := \coprod_i U_i$;
with such a choice of effective epimorphism, $G|_U = \coprod_i G|_{U_i}$ is simply the restriction of the sheaf of groups $G$ to each open subset that is a member of the cover;
$\mathbf{B}G_{U} \in \mathcal{X}/U$ is the stack of $G_{U}$-principal bundles on $U$.
Let $\mathcal{X}$ be any ambient (∞,1)-topos.
Let $G \in Grp(\mathcal{X}) \subset \infty Grpd(\mathcal{X})$ be a group object (a 0-truncated ∞-group).
Write
for the core of the full sub-(∞,1)-category on $G$-gerbes in $\mathcal{X}$.
Write
for the 2-group object called the automorphism 2-group of $G$.
$G$-gerbes in $\mathcal{X}$ are classified by first $AUT(G)$-nonabelian cohomology
In the general perspective of (∞,1)-topos theory this appears as (JardineLuo, theorem 23).
Since nonabelian cohomology with coefficients in $AUT(G)$ also classified $AUT(G)$-principal 2-bundles it follows that also
Notice that under this equivalence a $G$-gerbe is not identified with the total space object of the corresponding $AUT(G)$-principal 2-bundle. The latter differs by an $Aut(H)$-factor. Where a $G$-gerbe is locally equivalent to
an $AUT(G)$-principal 2-bundle is locally equivalent to
Instead, under the above equivalence a gerbe is identified with the associated ∞-bundle with fibers $\mathbf{B}G$ that is associated via the canonical action of $AUT(G) = Aut(\mathbf{B}G)$ on $\mathbf{B}G$.
For $G \in Grp(\mathcal{G})$, the automorphism 2-group $AUT(G)$ has a canonical morphism to its 0-truncation, the ordinary outer automorphism group object of $G$:
Therefore every $AUT(G)$-cocycle has an underlying $Out(G)$-cocycle (an $Out(G)$-principal bundle):
By prop. 1 this an assignment of $Out(G)$-cohomology classes to $G$-gerbes:
For $P \in G Gerbe$ one says that $Band(P)$ is its band.
Sometimes in applications one considers not just the restriction from all gerbes to $G$-gerbes for some $G$, but further to $K$-banded $G$-gerbes for some $K \in H_{\mathcal{X}}^1(X,Out(G))$.
The groupoid $G Gerbe_K(X)$ of $K$-banded gerbes is the $K$-twisted $\mathbf{B}^2 Z(G)$-cohomology of $X$ (where $Z(G)$ is the center of $G$): it is the homotopy pullback
More details on gerbes is at the following sub-entries:
principal 2-bundle / gerbe / bundle gerbe
The definition of gerbe goes back to (see also nonabelian cohomology)
Introductions include
Lawrence Breen, Notes on 1- and 2-gerbes in John Baez, Peter May (eds.) Towards Higher Categories (arXiv:math/0611317).
Ieke Moerdijk, Introduction to the language of stacks and gerbes (arXiv:math/0212266)
Lawrence Breen, Notes on 1- and 2-gerbes in John Baez, Peter May (eds.) Towards Higher Categories (arXiv:math/0611317).
A discussion from the point of view of (∞,1)-topos theory is in
The definition for $n$-gerbes as $n$-truncated and $n$-connected objects (see ∞-gerbe) is in