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infinity-gerbe

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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

Definition

Let 𝒳\mathcal{X} be an (∞,1)-topos.

For n¯n \in \overline{\mathbb{N}} an extended natural number,

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.

Definition

An \infty-gerbe in 𝒳\mathcal{X} is a connected object.

Write

Gerbe𝒳 \infty Gerbe \subset \mathcal{X}

for the core of the full-sub-(∞,1)-category on the \infty-gerbes.

Warning

The notion of morphisms between EM nn-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.

Remark

If one adds to the definition of “EM nn-gerbe” the condition that 𝒢\mathcal{G} has a “global section”, a global element *𝒢* \to \mathcal{G}, then it becomes the definition of an Eilenberg-MacLane object in degree nn. In other words, EM nn-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 𝒳\mathcal{X} are equivalently (the deloopings of) ∞-group objects: we have an equivalence of (∞,1)-categories

(ΩB):Grp(𝒳)BΩ𝒳 *,1. (\Omega \dashv \mathbf{B}) : \infty Grp(\mathcal{X}) \stackrel{\overset{\Omega}{\leftarrow}}{\underset{\mathbf{B}}{\to}} \mathcal{X}_{*, \geq 1} \,.

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

(x *x *):Grpdx *x *𝒳 (x^* \dashv x_*) : \infty Grpd \stackrel{\overset{x^*}{\leftarrow}}{\underset{x_*}{\to}} \mathcal{X}

any topos point and P𝒳P \in \mathcal{X} an \infty-gerbe, the stalk x *Px^* P \in ∞Grpd is connected, because the inverse image x *x^* preserves the finite limits that define categorical homotopy groups in an (∞,1)-topos and preserves the terminal object **:

x *PBG x. x^* P \simeq B G_x \,.

Therefore it is natural to consider the notion of an GG-\infty-gerbe for a fixed GGrp(𝒳)G \in \infty Grp(\mathcal{X}). This is done below.

Definition

A “restricted” nn-gerbe EE has, by definition, a single non-trivial homotopy sheaf?. For n2n \geq 2 this is an sheaf of abelian groups AA in the underlying topos

π nEA. \pi_n E \simeq A \,.

This AA is called the band of EE and that EE is banded by AA.

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.

Definition

For GGrp(𝒳)G \in \infty Grp(\mathcal{X}) an ∞-group, a GG-\infty-gerbe PP is

  • an \infty-gerbe, def. 2;

  • such that there exists an effective epimorphism U*U \to * in 𝒳\mathcal{X} and an equivalence

    P UBG U. P|_U \simeq \mathbf{B}G|_U \,.

Properties

Classification of EM nn-gerbes.

Let AGrp(𝒳)Grpd(𝒳)A \in Grp(\mathcal{X}) \subset \infty Grpd(\mathcal{X}) be an abelian group object and fix nn \in \mathbb{N}, n2n \geq 2.

Recall the notion of AA-banded nn-gerbes from def. 3.

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.

Proposition

There is a canonical bijection

π 0EMnGerbeH 𝒳 n+1(X,A). \pi_0 EM n Gerbe \simeq H^{n+1}_{\mathcal{X}}(X, A) \,.

This appears as HTT, cor. 7.2.2.27.

Warning

The morphisms of EM nn-gerbes in HTT, p. 576-577 are more restrictive than the morphisms of these objects regarded as general nn-gerbes.

By the discussion below, if EM nn-gerbes are regarded as special nn-gerbes, then their classification is by a nonabelian cohomology-extension of the above result.

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. 4 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)
Proposition

At least for 𝒳\mathcal{X} a 1-localic (∞,1)-topos we have a canonical bijection

π 0(GGerbe)H 1(𝒳,AUT(G)). \pi_0 (G Gerbe) \simeq H^1(\mathcal{X}, AUT(G)) \,.

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

Remark

The right hand side classifies als AUT(G)AUT(G)-principal ∞-bundles and this equivalence identifies GG-\infty-gerbes as the canonical AUT(G)AUT(G)-associated ∞-bundles.

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

Nonabelian banded \infty-gerbes

For GGrpd(𝒳)G \in \infty Grpd(\mathcal{X}) an nn-group (with n1n \geq 1) write

Out(G):=τ n1AUT(G)=τ n1Aut̲(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 0GGerbeH 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 EGGerbeE \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 KH 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)0 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 2\mathbb{Z}_2 on either of the U(1)U(1)s is the canonical one given by mathbZ 2Aut(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 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 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)11][U(1)1 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 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”.

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

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

The general notion for arbitrary nn in an (,1)(\infty,1)-topos context is discussed in section 2.3 of

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

Revised on October 10, 2013 03:38:52 by Urs Schreiber (89.204.135.176)