infinity-gerbe in nLab

Contents

Contents

Idea

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

1. 1-connective;

2. 1-truncated.

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.

Definition

General

The definition of “EM nn-gerbes” appears as HTT, def. 7.2.2.20. For n=2n = 2 the definition of “general nn-gerbe” (called a nonabelian 2-gerbe at 2-gerbe) appears for instance in (Breen).

In particular we have then the following.

In the case that n=1n = 1 or that we have a “general” nn-gerbe the notion of band is refined by nonabelian cohomology-information. See [below].

GG-∞\infty-Gerbes

Let 𝒳\mathcal{X} be an (∞,1)-topos.

Properties

Classification of EM nn-gerbes.

Let A∈Grp(𝒳)⊂∞Grpd(𝒳)A \in Grp(\mathcal{X}) \subset \infty Grpd(\mathcal{X}) be an abelian group object and fix n∈ℕn \in \mathbb{N}, n≥2n \geq 2.

Recall the notion of AA-banded nn-gerbes from def. .

Write

H 𝒳 n+1(X,A):=π 0𝒳(*,B n+1A) H^{n+1}_{\mathcal{X}}(X, A) := \pi_0 \mathcal{X}(*, \mathbf{B}^{n+1}A)

for the cohomology in 𝒳\mathcal{X} of the terminal object with coefficients in AA in degree n+1n+1.

This appears as HTT, cor. 7.2.2.27.

Classification of GG-∞\infty-gerbes

We discuss partial generalizations of the above result to nonabelian ∞\infty-gerbes

Comparing with the discussion at associated ∞-bundle one finds that def. of GG-∞\infty-gerbes defines “locally trivial BG\mathbf{B}G-fibrations”. By the main theorem there, these are classified by cohomology in 𝒳\mathcal{X} with coefficients in the internal automorphism ∞-group

AUT(G):=Aut̲(BG) AUT(G) := \underline{Aut}(\mathbf{B}G)

This follows as a special case of the result by (Wendt) discussed at associated infinity-bundle.

(Compare to the analogous discussion in the special case of gerbes.)

Nonabelian banded ∞\infty-gerbes

For G∈∞Grpd(𝒳)G \in \infty Grpd(\mathcal{X}) an nn-group (with n≥1n \geq 1) write

Out(G):=τ n−1AUT(G)=τ n−1Aut̲(BG). Out(G) := \tau_{n-1} AUT(G) = \tau_{n-1}\underline{Aut}(\mathbf{B}G) \,.

Call this the outer automorphism infinity-group of GG. By definition there is a canonical morphism

BAUT(G)→BOut(G). \mathbf{B} AUT(G) \to \mathbf{B} Out(G) \,.

By the above classification, this induces a morphism

Band:π 0GGerbe→H 𝒳 1(X,Out(G)) Band : \pi_0 G Gerbe \to H^1_{\mathcal{X}}(X, Out(G))

from GG-nn-gerbes to nonabelian cohomology in degree 1 with coefficients in Out(G)Out(G). For E∈GGerbeE \in G Gerbe the pair

(π n,Band(E)) (\pi_n, Band(E))

is called the band of EE. For [K]∈H 𝒳 1(X,Out(G))[K] \in H^1_{\mathcal{X}}(X, Out(G)) the (n+1)-groupoid GGerbe KG Gerbe_K of KK-banded GG-nn-gerbes is the homotopy pullback

GGerbe K → * ↓ ↓ K 𝒳(X,BAUT(G)) → 𝒳(X,BOut(G)). \array{ G Gerbe_K &\to& * \\ \downarrow && \downarrow^{\mathrlap{K}} \\ \mathcal{X}(X, \mathbf{B}AUT(G)) &\to& \mathcal{X}(X, \mathbf{B}Out(G)) } \,.

Write B 2Z(G)\mathbf{B}^2 Z(G) for the homotopy fiber of BAUT(G)→BOut(G)\mathbf{B}AUT(G) \to \mathbf{B}Out(G), producing a fiber sequence

B 2Z(G)→BAUT(G)→BOut(G). \mathbf{B}^2 Z(G) \to \mathbf{B} AUT(G) \to \mathbf{B}Out(G) \,.

We call Z(G)Z(G) the center of the infinity-group.

In terms of this the above GGerbe KG Gerbe_K is the cocycle (n+1)(n+1)-groupoid of the K-twisted Z(G)-cohomology in degree 2:

π 0GGerbe K≃H K 2(X,Z(G)). \pi_0 G Gerbe_K \simeq H^2_K(X,Z(G)) \,.

Notice that if Z(G)Z(G) itself is higher connected then H K 2(X,Z(G))H^2_K(X, Z(G)) is accordingly cohomology in higher degree.

Examples

Classification of abelian 2-gerbes

For XX a topological space, let 𝒳=Sh (∞,1)(X)\mathcal{X} = Sh_{(\infty,1)}(X) be its (∞,1)-category of (∞,1)-sheaves. Write

U(1)∈Grp(𝒳)⊂∞Grp(𝒳) U(1) \in Grp(\mathcal{X}) \subset \infty Grp(\mathcal{X})

for the sheaf of circle group-valued functions. And BU(1)\mathbf{B}U(1) for its delooping

Then

AUT(BU(1)):=Aut̲(B 2U(1))≃[U(1)→0U(1)→0ℤ 2], AUT(\mathbf{B}U(1)) := \underline{Aut}(\mathbf{B}^2 U(1)) \simeq [U(1) \stackrel{0}{\to} U(1) \stackrel{0}{\to} \mathbb{Z}_2] \,,

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 ℤ 2\mathbb{Z}_2 on either of the U(1)U(1)s is the canonical one given by mathbZ 2≃Aut(U(1))\mathb{Z}_2 \simeq Aut(U(1)) ). This is seen as follows: every invertible 2-functor F:B 2U(1)→B 2U(1)F : \mathbf{B}^2 U(1) \to \mathbf{B}^2 U(1) comes from a group auotmorphism of U(1)U(1), of which there are ℤ 2\mathbb{Z}_2. A pseudonatural transformation necessarily goes from any such FF to itself, and there are U(1)U(1) of them Similarly, any modification of these necessarily is an endo, and there are also U(1)U(1) of them.

Therefore BU(1)\mathbf{B}U(1)-2-gerbes are classified by the nonabelian cohomology

H 1(X,[U(1)→1→ℤ 2]). H^1(X,[U(1) \to 1 \to \mathbb{Z}_2]) \,.

This has a subgroup coming from the canonical inclusion

B 3U(1)=[U(1)→1→1]↪[U(1)→1→ℤ 2] \mathbf{B}^3 U(1) = [U(1)\to 1 \to 1] \hookrightarrow [U(1) \to 1 \to \mathbb{Z}_2]

on the trivial automorphism of U(1)U(1). The classification of U(1)U(1) EM-2-gerbes in HTT gives only this subgroup

H 3(X,U(1))↪H 1(X,[U(1)→1→ℤ 2]). H^3(X, U(1)) \hookrightarrow H^1(X, [U(1) \to 1 \to \mathbb{Z}_2]) \,.

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 / torsor / associated bundle

• principal 2-bundle / gerbe / bundle gerbe

• principal 3-bundle / 2-gerbe / bundle 2-gerbe

• principal ∞-bundle / associated ∞-bundle / ∞\infty-gerbe.

  • Super Gerbes

Notice that the notion of bundle gerbe and bundle 2-gerbe etc. is not unrelated, but is a priori a rather different notion.

References

The notion of “EM” ∞\infty-gerbes in an (∞,1)(\infty,1)-topos appears in section 7.2.2 of

• Jacob Lurie, Higher Topos Theory

The general notion (but without a concept of ambient (3,1)(3,1)-topos made explicit) for n=2n = 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 nn in an (∞,1)(\infty,1)-topos context is discussed in section 2.3 of

• Urs Schreiber, differential cohomology in a cohesive topos .

Discussion of the classification of GG-∞\infty-gerbes and more general fiber bundles in 1-localic (∞,1)(\infty,1)-toposes is in

• Matthias Wendt, Classifying spaces and fibrations of simplicial sheaves , Journal of Homotopy and Related Structures 6(1), 2011, pp. 1–38. (arXiv) (published version)