nLab infinity-gerbe

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 $\mathcal{X}$ be an (∞,1)-topos.

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

• a $n$-gerbe is an object $\mathcal{G} \in \mathbf{G}$ that is

  1. 1-connective;

  2. n-truncated.

• an EM $n$-gerbe is an an $n$-gerbe that is

  1. n-connective

  2. n-truncated.

The definition of “EM $n$-gerbes” appears as HTT, def. 7.2.2.20. For $n = 2$ the definition of “general $n$-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 $\mathcal{X}$ is a connected object.

Write

$$\infty Gerbe \subset \mathcal{X}$$

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

Warning

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

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

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

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

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

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

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

Definition

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

$$\pi_n E \simeq A \,.$$

This $A$ is called the band of $E$ and that $E$ is banded by $A$.

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

$G$-$\infty$-Gerbes

Let $\mathcal{X}$ be an (∞,1)-topos.

Definition

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

• an $\infty$-gerbe, def. ;

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

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

Properties

Classification of EM $n$-gerbes.

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

Recall the notion of $A$-banded $n$-gerbes from def. .

Write

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

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

Proposition

There is a canonical bijection

$$\pi_0 EM n Gerbe_A \simeq H^{n+1}_{\mathcal{X}}(X, A) \,,$$

where $EM n Gerbe_A$ is the category of all $EM$ $n$-gerbes banded over $A$.

This appears as HTT, cor. 7.2.2.27.

Warning

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

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

Classification of $G$-$\infty$-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 $G$-$\infty$-gerbes defines “locally trivial $\mathbf{B}G$-fibrations”. By the main theorem there, these are classified by cohomology in $\mathcal{X}$ with coefficients in the internal automorphism ∞-group

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

Proposition

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

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

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

Remark

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

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

Nonabelian banded $\infty$-gerbes

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

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

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

$$\mathbf{B} AUT(G) \to \mathbf{B} Out(G) \,.$$

By the above classification, this induces a morphism

$$Band : \pi_0 G Gerbe \to H^1_{\mathcal{X}}(X, Out(G))$$

from $G$-$n$-gerbes to nonabelian cohomology in degree 1 with coefficients in $Out(G)$. For $E \in G Gerbe$ the pair

$$(\pi_n, Band(E))$$

is called the band of $E$. For $[K] \in H^1_{\mathcal{X}}(X, Out(G))$ the (n+1)-groupoid $G Gerbe_K$ of $K$-banded $G$-$n$-gerbes is the homotopy pullback

$$\array{ G Gerbe_K &\to& * \\ \downarrow && \downarrow^{\mathrlap{K}} \\ \mathcal{X}(X, \mathbf{B}AUT(G)) &\to& \mathcal{X}(X, \mathbf{B}Out(G)) } \,.$$

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

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

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

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

$$\pi_0 G Gerbe_K \simeq H^2_K(X,Z(G)) \,.$$

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

Examples

Classification of abelian 2-gerbes

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

$$U(1) \in Grp(\mathcal{X}) \subset \infty Grp(\mathcal{X})$$

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

Then

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

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

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

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

This has a subgroup coming from the canonical inclusion

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

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

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

In the terminology of orientifold/Jandl gerbes the more general objects on the right are the “Jandl 2-gerbes” or “orientifold 2-gerbes”.

Related entries

• 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 $(\infty,1)$-topos appears in section 7.2.2 of

• Jacob Lurie, Higher Topos Theory

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

• Urs Schreiber, differential cohomology in a cohesive topos .

Discussion of the classification of $G$-$\infty$-gerbes and more general fiber bundles in 1-localic $(\infty,1)$-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)