structures in a cohesive (∞,1)-topos
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 be an (∞,1)-topos.
For an extended natural number,
In particular we have then the following.
An -gerbe in is a connected object.
The notion of morphisms between EM -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 -gerbe” the condition that has a “global section”, a global element , then it becomes the definition of an Eilenberg-MacLane object in degree . In other words, EM -gerbes are just like Eilenberg-MacLane objects but without necessarily a global section.
Therefore an -gerbe is much like the delooping of an -group, only that a global section may be missing.
But notice that a section always exists locally: for
any topos point and an -gerbe, the stalk ∞Grpd is connected, because the inverse image 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 --gerbe for a fixed . This is done below.
This is called the band of and that is banded by .
In the case that or that we have a “general” -gerbe the notion of band is refined by nonabelian cohomology-information. See [below].
Let be an (∞,1)-topos.
For an ∞-group, a --gerbe is
Let be an abelian group object and fix , .
Recall the notion of -banded -gerbes from def. 3.
There is a canonical bijection
This appears as HTT, cor. 188.8.131.52.
The morphisms of EM -gerbes in HTT, p. 576-577 are more restrictive than the morphisms of these objects regarded as general -gerbes.
By the discussion below, if EM -gerbes are regarded as special -gerbes, then their classification is by a nonabelian cohomology-extension of the above result.
We discuss partial generalizations of the above result to nonabelian -gerbes
Comparing with the discussion at associated ∞-bundle one finds that def. 4 of --gerbes defines “locally trivial -fibrations”. By the main theorem there, these are classified by cohomology in with coefficients in the internal automorphism ∞-group
(Compare to the analogous discussion in the special case of gerbes.)
For an -group (with ) write
Call this the outer automorphism infinity-group of . By definition there is a canonical morphism
By the above classification, this induces a morphism
from --gerbes to nonabelian cohomology in degree 1 with coefficients in . For the pair
We call the center of the infinity-group.
Notice that if itself is higher connected then is accordingly cohomology in higher degree.
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 on either of the s is the canonical one given by ). This is seen as follows: every invertible 2-functor comes from a group auotmorphism of , of which there are . A pseudonatural transformation necessarily goes from any such to itself, and there are of them Similarly, any modification of these necessarily is an endo, and there are also of them.
Therefore -2-gerbes are classified by the nonabelian cohomology
This has a subgroup coming from the canonical inclusion
on the trivial automorphism of . The classification of EM-2-gerbes in HTT gives only this subgroup
The notion of “EM” -gerbes in an -topos appears in section 7.2.2 of
The general notion (but without a concept of ambient -topos made explicit) for (see 2-gerbe) appears for instance in
Lawrence Breen, On the classification of 2-gerbes and 2-stacks , Astérisque 225 (1994).
The general notion for arbitrary in an -topos context is discussed in section 2.3 of
Discussion of the classification of --gerbes and more general fiber bundles in 1-localic -toposes is in