gerbe in nLab

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Definition

General

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 τ ≤1𝒳↪𝒳\tau_{\leq 1 } \mathcal{X} \hookrightarrow \mathcal{X} inside 𝒳\mathcal{X}. This means that for CC any (∞,1)-site of definition for 𝒳\mathcal{X}, a gerbe is a (2,1)-sheaf on CC, 𝒢∈Sh (2,1)(C)\mathcal{G} \in Sh_{(2,1)}(C): a stack on CC.

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 𝒢→*\mathcal{G} \to * to the terminal object of 𝒳\mathcal{X} is an effective epimorphism);

• the 0th categorical homotopy group π 0𝒢\pi_0 \mathcal{G} is isomorphic to the terminal object ** as objects in the sheaf topos τ ≤0𝒳=Sh (1,1)(C)\tau_{\leq 0} \mathcal{X} = Sh_{(1,1)}(C). Here π 0𝒢\pi_0 \mathcal{G} is the sheafification of the presheaf of connected components of the groupoids that 𝒢:C op→Grpd↪∞Grpd\mathcal{G} : C^{op} \to Grpd \hookrightarrow \infty Grpd 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 τ ≤1𝒳\tau_{\leq 1} \mathcal{X} of a topological manifold or smooth manifold XX or a topological stack or differentiable stack XX. One then speaks of a gerbe over XX .

More precisely, we may associate to any X∈C:=X \in C := Top or X∈C:=X \in C := Diff the corresponding big site C/XC/X and form the (2,1)-topos τ leq𝒳:=Sh (2,1)(C/X)\tau_{leq} \mathcal{X} := Sh_{(2,1)}(C/X). In terms of this a gerbe is given by a collection of groupoids assigned to patches of XX, satisfying certain conditions.

Equivalent to this is the over-(2,1)-topos τ ≤1ℋ/j(X)\tau_{\leq 1} \mathcal{H}/j(X), where τ ≤1ℋ:=Sh (2,1)(C)\tau_{\leq 1}\mathcal{H} := Sh_{(2,1)}(C) is the big (2,1)-topos over CC (and jj denotes its (2,1)-Yoneda embedding).

Since this ℋ\mathcal{H} 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 .

GG-Gerbes

We discuss gerbes that have a “strucure group” GG akin to a principal bundle. Indeed, while not the same concept, these GG-gerbes are equivalent to AUT(G)AUT(G)-principal 2-bundles, for AUT(G)AUT(G) the automorphism 2-group of GG.

The following definition characterizes gerbes that are locally of the form of remark .

Properties

Equivalence of GG-gerbes to AUT(G)AUT(G)-2-bundles

Let 𝒳\mathcal{X} be any ambient (∞,1)-topos.

Let G∈Grp(𝒳)⊂∞Grpd(𝒳)G \in Grp(\mathcal{X}) \subset \infty Grpd(\mathcal{X}) be a group object (a 0-truncated ∞-group).

Write

GGerbe⊂𝒳 G Gerbe \subset \mathcal{X}

for the core of the full sub-(∞,1)-category on GG-gerbes in 𝒳\mathcal{X}.

Write

AUT(G):=Aut 𝒳 *(BG)∈2Grp(𝒳)⊂∞Grp(𝒳) AUT(G) := Aut_{\mathcal{X}_{*}}(\mathbf{B}G) \in 2 Grp(\mathcal{X}) \subset \infty Grp(\mathcal{X})

for the 2-group object called the automorphism 2-group of GG.

In the general perspective of (∞,1)-topos theory this appears as (Jardine & Luo, theorem 23).

Banded gerbes

For G∈Grp(𝒢)G \in Grp(\mathcal{G}), the automorphism 2-group AUT(G)AUT(G) has a canonical morphism to its 0-truncation, the ordinary outer automorphism group object of GG:

→AUT(G)→π 0(Aut(G))=:Out(G). \to AUT(G) \to \pi_0(Aut(G)) =: Out(G) \,.

Therefore every AUT(G)AUT(G)-cocycle has an underlying Out(G)Out(G)-cocycle (an Out(G)Out(G)-principal bundle):

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

By prop. this an assignment of Out(G)Out(G)-cohomology classes to GG-gerbes:

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

For P∈GGerbeP \in G Gerbe one says that Band(P)Band(P) is its band.

Sometimes in applications one considers not just the restriction from all gerbes to GG-gerbes for some GG, but further to KK-banded GG-gerbes for some K∈H 𝒳 1(X,Out(G))K \in H_{\mathcal{X}}^1(X,Out(G)).

The groupoid GGerbe K(X)G Gerbe_K(X) of KK-banded gerbes is the KK-twisted B 2Z(G)\mathbf{B}^2 Z(G)-cohomology of XX (where Z(G)Z(G) is the center of GG): it is the homotopy pullback

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

Sub-entries

More details on gerbes is at the following sub-entries:

• gerbe (as a stack)

• gerbe (in nonabelian cohomology)

• gerbe (in differential geometry)

• bundle gerbe

• gerbe (general idea)

Examples

• determinantal gerbe

• principal bundle / torsor / associated bundle

• principal 2-bundle / gerbe / bundle gerbe

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

• principal ∞-bundle / associated ∞-bundle / ∞-gerbe

References

The definition of gerbe goes back to (see also nonabelian cohomology)

• Jean Giraud, Cohomologie non abélienne , Springer (1971) (doi:10.1007/978-3-662-62103-5)

Review:

• Lawrence Breen, Notes on 1- and 2-gerbes in John Baez, Peter May (eds.) Towards Higher Categories, The IMA Volumes in Mathematics and its Applications, vol 152. Springer 2009 (arXiv:math/0611317, doi:10.1007/978-1-4419-1524-5_5).

• Ieke Moerdijk, Introduction to the language of stacks and gerbes (arXiv:math/0212266)

A discussion from via model structure on simplicial presheaves:

• John F. Jardine, Z. Luo: Higher order principal bundles, Mathematical Proceedings of the Cambridge Philosophical Society 140 2 (2006) 221-243 [doi:10.1017/S0305004105008911, web.archive:pdf]

The definition for nn-gerbes as nn-truncated and nn-connected objects (see ∞-gerbe) is in

• Jacob Lurie, Higher Topos Theory