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gerbe

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(,1)(\infty,1)-Topos Theory

(∞,1)-topos theory

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structures in a cohesive (∞,1)-topos

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Definition

General

In full generality, we have the following definition of gerbe .

Definition

Given an (∞,1)-topos 𝒳\mathcal{X}, a gerbe in 𝒳\mathcal{X} is an object 𝒢𝒳\mathcal{G} \in \mathcal{X} that is

  1. 1-truncated

  2. 1-connective (= connected).

The first condition says that a gerbe is an object in the (2,1)-topos τ 1𝒳𝒳\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, 𝒢Sh (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 opGrpdGrpd\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 XC:=X \in C := Top or XC:=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 1/j(X)\tau_{\leq 1} \mathcal{H}/j(X), where τ 1:=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 \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.

Remark

The definition 1 of gerbe is almost verbatim that of Eilenberg-MacLane object in degree 1. The only difference is that the latter is required to have not only the homotopy sheaf π 0=*\pi_0 = *, but even have a “global section” in the form of a morphism *P* \to P.

First consider this locally. A gerbe (as any 1-connected object) necessarily has local sections:

for

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

any topos point, the stalk functor x *x^*, being an inverse image is left exact and hence preserves homotopy sheaves and terminal objects. It follows that the 0th homotopy sheaf is trivial

π 1x *Px *π 1(P)x *** \pi_1 x^* P \simeq x^* \pi_1(P) \simeq x^* * \simeq *

as are all the degree-pp homotopy sheaves for p>1p \gt 1. Therefore x *Px^* P is a groupoid with a single object: the delooping groupoid of a group G xG_x:

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

More generally, by the discussion at looping and delooping we have in an equivalence of (∞,1)-categories

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

between the ∞-group objects in the ambient (∞,1)-topos 𝒳\mathcal{X} and the pointed connected objects.

It follows that for a gerbe PP that admits a global section *P* \to P the above relation holds not only stalk-wise, but globally: it is the delooping of its own first sheaf of homotopy groups

PBπ 1(P). P \simeq \mathbf{B} \pi_1(P) \,.

The following definition characterizes gerbes that are locally of the form of remark 1.

Definition

Let GGrp(𝒳)G \in Grp(\mathcal{X}) be a group object. A gerbe P𝒳P \in \mathcal{X} is a GG-gerbe if there exists an effective epimorphism U*U \to * and an equivalence

P UB(G U), P|_U \simeq \mathbf{B}(G|_U) \,,

where P U:=P×UP|_U := P \times U and G U:=G×UG|_U := G \times U.

Remark

In a typical application one considers gerbes over some topological space XX. In that case

  • 𝒳=Sh (,1)(Op(X))\mathcal{X} = Sh_{(\infty,1)}(Op(X)) is the (∞,1)-category of (∞,1)-sheaves on the category of open subsets of XX;

  • the terminal object of 𝒳\mathcal{X} is the space XX, regarded as an object in its own (,1)(\infty,1)-topos, hence we can write X:=*𝒳X := * \in \mathcal{X};

  • a group object G𝒳G \in \mathcal{X} is sheaf of groups on XX;

  • an effective epimorphism U*U \to *, hence U*U \to * is obtained from any open cover {U iX}\{U_i \to X\} by setting U:= iU iU := \coprod_i U_i;

  • with such a choice of effective epimorphism, G U= iG U iG|_U = \coprod_i G|_{U_i} is simply the restriction of the sheaf of groups GG to each open subset that is a member of the cover;

  • BG U𝒳/U\mathbf{B}G_{U} \in \mathcal{X}/U is the stack of G UG_{U}-principal bundles on UU.

Properties

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

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

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

Write

GGerbe𝒳 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.

Proposition

GG-gerbes in 𝒳\mathcal{X} are classified by first AUT(G)AUT(G)-nonabelian cohomology

π 0GGerbeπ 0𝒳(*,BAUT(G))=:H 𝒳 1(X,AUT(G)). \pi_0 G Gerbe \simeq \pi_0 \mathcal{X}(*, \mathbf{B} AUT(G)) =: H_{\mathcal{X}}^1(X,AUT(G)) \,.

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

Corollary

Since nonabelian cohomology with coefficients in AUT(G)AUT(G) also classified AUT(G)AUT(G)-principal 2-bundles it follows that also

π 0GGerbeAUT(G)2Bund(*). \pi_0 G Gerbe \simeq AUT(G) 2Bund(*) \,.

Notice that under this equivalence a GG-gerbe is not identified with the total space object of the corresponding AUT(G)AUT(G)-principal 2-bundle. The latter differs by an Aut(H)Aut(H)-factor. Where a GG-gerbe is locally equivalent to

B(G U)=G U* U \mathbf{B}(G|_U) = G|_U \stackrel{\to}{\to} *|_U

an AUT(G)AUT(G)-principal 2-bundle is locally equivalent to

AUT(G U)=Aut(G U)×Gp 1Ad(p 2)p 1Aut(G U). AUT(G|_U) = Aut(G|_U) \times G \stackrel{\overset{Ad(p_2) \cdot p_1}{\to}}{\underset{p_1}{\to}} Aut(G|_U) \,.

Instead, under the above equivalence a gerbe is identified with the associated ∞-bundle with fibers BG\mathbf{B}G that is associated via the canonical action of AUT(G)=Aut(BG)AUT(G) = Aut(\mathbf{B}G) on BG\mathbf{B}G.

Banded gerbes

For GGrp(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. 1 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 PGGerbeP \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 KH 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:

Examples

References

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

  • J. Giraud, Cohomologie non abélienne , Springer (1971)

Introductions include

A discussion from the point of view of (∞,1)-topos theory is in

  • Rick Jardine, Z. Luo, Higher order principal bundles , K-theory (2004) (web)

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

Revised on June 25, 2012 22:13:13 by Urs Schreiber (89.204.139.149)