Classes of bundles
Examples and Applications
(∞,1)-category of (∞,1)-sheaves
Extra stuff, structure and property
locally n-connected (n,1)-topos
locally ∞-connected (∞,1)-topos, ∞-connected (∞,1)-topos
structures in a cohesive (∞,1)-topos
Special and general types
In full generality, we have the following definition of gerbe .
The first condition says that a gerbe is an object in the (2,1)-topos inside . This means that for any (∞,1)-site of definition for , a gerbe is a (2,1)-sheaf on , : a stack on .
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 to the terminal object of is an effective epimorphism);
the 0th categorical homotopy group is isomorphic to the terminal object as objects in the sheaf topos . Here is the sheafification of the presheaf of connected components of the groupoids that 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 of a topological manifold or smooth manifold or a topological stack or differentiable stack . One then speaks of a gerbe over .
More precisely, we may associate to any Top or Diff the corresponding big site and form the (2,1)-topos . In terms of this a gerbe is given by a collection of groupoids assigned to patches of , satisfying certain conditions.
Equivalent to this is the over-(2,1)-topos , where is the big (2,1)-topos over (and denotes its (2,1)-Yoneda embedding).
Since this 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” akin to a principal bundle. Indeed, while not the same concept, these -gerbes are equivalent to -principal 2-bundles, for the automorphism 2-group of .
The following definition characterizes gerbes that are locally of the form of remark 1.
Let be a group object. A gerbe is a -gerbe if there exists an effective epimorphism and an equivalence
where and .
Equivalence of -gerbes to -2-bundles
Let be any ambient (∞,1)-topos.
Let be a group object (a 0-truncated ∞-group).
for the core of the full sub-(∞,1)-category on -gerbes in .
for the 2-group object called the automorphism 2-group of .
-gerbes in are classified by first -nonabelian cohomology
In the general perspective of (∞,1)-topos theory this appears as (JardineLuo, theorem 23).
Since nonabelian cohomology with coefficients in also classified -principal 2-bundles it follows that also
Notice that under this equivalence a -gerbe is not identified with the total space object of the corresponding -principal 2-bundle. The latter differs by an -factor. Where a -gerbe is locally equivalent to
an -principal 2-bundle is locally equivalent to
Instead, under the above equivalence a gerbe is identified with the associated ∞-bundle with fibers that is associated via the canonical action of on .
For , the automorphism 2-group has a canonical morphism to its 0-truncation, the ordinary outer automorphism group object of :
Therefore every -cocycle has an underlying -cocycle (an -principal bundle):
By prop. 1 this an assignment of -cohomology classes to -gerbes:
For one says that is its band.
Sometimes in applications one considers not just the restriction from all gerbes to -gerbes for some , but further to -banded -gerbes for some .
The groupoid of -banded gerbes is the -twisted -cohomology of (where is the center of ): it is the homotopy pullback
More details on gerbes is at the following sub-entries:
The definition of gerbe goes back to (see also nonabelian cohomology)
- J. Giraud, Cohomologie non abélienne , Springer (1971)
A discussion from the point of view of (∞,1)-topos theory is in
- Rick Jardine, Z. Luo, Higher order principal bundles , K-theory (2004) (web)
The definition for -gerbes as -truncated and -connected objects (see ∞-gerbe) is in