nLab
gerbe (general idea)

for related entries see


Idea

The concept gerbe is a categorification of the concept of principal bundle, together with a generalisation analogous to that from bundles to sheaves.

Recall that a GG-principal bundle (for GG a group) is a space PP equipped with a map PXP \to X to a base space XX, such that each fiber of PP looks like GG in a nice way.

A GG-gerbe is similarly a “space”, PXP \to X such that each fiber looks like BG\mathbf{B}G in a nice way. We can also replace GG with a sheaf of groups, or even with a 22-group.

Here “space” may mean ordinary topological space. In that case BG\mathbf{B} G is the classifying space of the group GG and the above describes the construction by Stasheff and Wirth of fibrations with fiber BGB G.

David Roberts: The list of axioms in Wirth-Stasheff about fibration theories is somewhat incomplete in my opinion (sorry Jim) - but only in a minor way. When it talks about the assignment of a category to each space, then goes on to talk about homotopies in that category, it seems to me we should be talking about (,1)(\infty,1)-categories. Even without such an extension, one needs to make sense of homotopies, and so should have the minimal structure required to talk about that - perhaps a category of fibrant objects?

More generally, “space” may refer to generalized spaces, called infinity-stacks: objects in any (infinity,1)-topos.

Recall from motivation for sheaves, cohomology and higher stacks that this is just heavy terminology for a very simple idea. The notion that a generalized space, also called an infinity-stack, is an object in an (,1)(\infty,1)-topos simplifies the situation conceptually by separating

  • conceptual structures (certain maps between spaces having certain fibers)

from their

  • implementation (details of what is regarded as a generalized space and how).

In particular, while gerbes are traditionally, originally by Giraud, introduced as 1-stacks with extra properties, one need not mention any details of stacks for describing the concept and behaviour of gerbes, all one needs is to remember that \infty-stacks are general notions of spaces, for which there is the familiar toolbox from homotopy theory of spaces, notably notions of homotopy pullback and fibration sequence. This makes transparent the relation between

  • gerbes
  • Stasheff–Wirth fibrations
  • nonabelian group extensions.

All of these are the same structure implemented in different contexts of generalized spaces.

For instance the last item here interprets extensions GHKG \to H \to K of a group KK by a group GG as a GG-gerbe over BK\mathbf{B}K, namely as a fibration BHBK\mathbf{B}H \to \mathbf{B}K with fiber BG\mathbf{B}G.

When the group GG in question is abelian, the theory of gerbes is very straightforwardly the generalization of that of principal bundles, because in this case the one-object groupoid BG\mathbf{B}G obtained by shifting GG in categorical degree (see the discussion at group) still has itself a group-structure: it is a 2-group. Much of what makes the discussion of nonabelian gerbes less than obvious is due to the fact that when GG is not abelian then the way in which BG\mathbf{B}G still relates to a group-like structure is slightly more involved and proceeds via the automorphism 2-group AUT(G)AUT(G) of GG.

Recall that an ordinary GG-principal bundle PXP \to X is a fibration of (generalized) spaces for which there is a morphism g:XBGg : X \to \mathbf{B} G such that PP is the homotopy pullback of the point along gg:

P * X BG. \array{ P &\to & {*} \\ \downarrow && \downarrow \\ X &\to& \mathbf{B}G } \,.

So PXBGP \to X \to \mathbf{B}G is a fibration sequence that extends for each choice of point *X{*} \to X of XX to the left to a fibration sequence GPXBGG \to P \to X \to \mathbf{B}G. This says that the fiber of PXP \to X over each point looks like the group GG. General nonsense implies then that the action of GG on itself induces an action of GG on all of PP and that this action is indeed principal.

When GG is an abelian group, so that BG\mathbf{B}G itself has a group structure, the object BBG\mathbf{B}\mathbf{B}G exists and the above statement has an immediate categorification:

A GG-gerbe for GG an abelian group is a fibration PXP \to X such that there is a morphism g:XBBGg : X \to \mathbf{B}\mathbf{B}G such that PP is the homotopy pullback of the point along this fibration.

P * X BBG. \array{ P &\to & {*} \\ \downarrow && \downarrow \\ X &\to& \mathbf{B}\mathbf{B}G } \,.

In this case for each point *X{*} \to X of XX this yields a fibration sequence

BGPXBBG \cdots \to \mathbf{B}G \to P \to X \to \mathbf{B}\mathbf{B} G

which says that the fiber of PXP \to X over each point of XX looks like BG\mathbf{B}G.

As in the previous case of ordinary bundles, general nonsense implies that PXP \to X comes with a principal BG\mathbf{B}G-action. PP is therefore also called a BG\mathbf{B}G-principal 2-bundle or a BG\mathbf{B}G-torsor. In its concrete incarnation as a stack, PP is called a GG-gerbe.

Moreover, since cohomology on XX with values in BBG\mathbf{B}\mathbf{B}G is nothing but the hom-set

H 2(X,G)=Ho(X,BBG) H^2(X,G) = Ho(X, \mathbf{B}\mathbf{B} G)

in the homotopy category of our generalized spaces, it is a tautology that these GG-gerbes are classified by H 2(X,G)H^2(X,G).

Notice in particular that for G=U(1)G = U(1) we have H 2(X,U(1))H 3(X,)H^2(X, U(1)) \simeq H^3(X, \mathbb{Z}), for instance by a long exact sequence argument, so that U(1)U(1)-gerbes in the above sense are classified by third integral cohomology. This classification statement was the main motivation for the study of the realization of the notion of gerbe that goes by the name bundle gerbe.

In this fashion, for GG abelian, the entire concept of GG-(n1)(n-1)-gerbe is straightforward: it is the (n1)(n-1)-stack incarnation of B nG\mathbf{B}^n G-principal infinity-bundles, i.e. of fibrations PXP \to X of (generalized) spaces that arise as homotopy pullbacks of the form

P * X BB nG. \array{ P &\to & {*} \\ \downarrow && \downarrow \\ X &\to& \mathbf{B}\mathbf{B}^n G } \,.

Accordingly, such (n1)(n-1)-gerbes for GG abelian are classified in cohomology by H n+2(X,G)H^{n+2}(X,G). (Another way to see why this is possible for GG an abelian group is that not only is BG\mathbf{B}G a 22-group, but B nG\mathbf{B}^n G is an (n1)(n-1)-group, which is what we need in general for an nn-gerbe.)

Moreover, for AA any pointed connected generalized space (any parameterized \infty-groupoid with a single object), we may say that AA-principal \infty-bundles are fibrations PXP \to X classified in this way by classifying morphisms XAX \to A

P * X A. \array{ P &\to & {*} \\ \downarrow && \downarrow \\ X &\to& A } \,.

The fiber of such an \infty-bundle is the loop space object ΩA\Omega A. The classifying morphism XAX \to A is then called a cocycle in nonabelian cohomology.

In particular, for HH any 2-group (not necessarily of the form H=BGH = \mathbf{B}G for GG an abelian group as above) an HH-principal 2-bundle is a fibration in this sense classified by a morphism XBHX \to \mathbf{B} H. The typical fiber of such a 22-bundle looks like HH.

Now, a GG-gerbe for GG nonabelian is supposed to be a fibration whose typical fiber is BG\mathbf{B}G. Since this is not a 2-group, one has to say what one wants to mean by this.

This now is the crucial fact that translates between the straightforward definition of HH-principal 2-bundles as above and the notion of GG-gerbe:

For every group GG, there is the 2-group AUT(G)AUT(G), defined equivalently as follows:

  • AUT(G)AUT(G) is the automorphism 22-group of the groupoid BG\mathbf{B}G, i.e.

    AUT(G)=Aut Grpd(BG); AUT(G) = Aut_{Grpd}(\mathbf{B}G) \,;
  • AUT(G)AUT(G) is the 2-group corresponding to the crossed module given by the sequence (GAdAut(G))(G \stackrel{Ad}{\to} Aut(G)) of groups, with the canonical action of Aut(G)Aut(G) on GG.

From the second description it is manifest that one can project out a copy of BG\mathbf{B}G out of AUT(G)AUT(G) (the shifted copy): there is a morphism AUT(G)BGAUT(G) \to \mathbf{B}G obtained simply by identifying all objects of AUT(G)AUT(G). Indeed, AUT(G)AUT(G) is a groupoid extension of BG\mathbf{B}G by the discrete groupoid on Aut(G)Aut(G) in that

Aut(G)AUT(G)BG Aut(G) \to AUT(G) \to \mathbf{B}G

is a fibration sequence.

This means that any AUT(G)AUT(G)-principal 2-bundle with typical fiber the groupoid AUT(G)AUT(G) has underlying it a fibration with typical fiber the one-object groupoid BG\mathbf{B}G. This underlying object is the GG-gerbe.

Notice in particular that when GG is abelian there is a canonical morphism

BBGBAUT(G) \mathbf{B}\mathbf{B} G \to \mathbf{B} AUT(G)

which however is not an equivalence when GG has nontrivial automorphisms. Therefore GG-gerbes in the sense of nonabelian GG-gerbes classified by H(X,BAUT(G))H(X,\mathbf{B}AUT(G)) are even for GG abelian a bit more general than the things classified by just H(X,B 2G)H(X, \mathbf{B}^2 G), which are however also often called GG-gerbes (in particular “bundle gerbes”).

This is the general nonsense underlying the concept of gerbe.

See also

References

There is a lengthier description of gerbes, concentrating on the low-dimensional aspects, in the Menagerie notes that are available from Tim Porter's home page.

Other material available online includes the following:

  • J. Duskin, An outline of non-abelian cohomology in a topos : (I) The theory of bouquets and gerbes, Cahiers de Top. et Géom. Diff. Categ. 23 no. 2 (1982), p. 165-191 (numdam)

  • I. Moerdijk, Introduction to the language of stacks and gerbes (arXiv)

  • Larry Breen, Notes on 1- and 2-gerbes (arXiv)

See the references given there for more, in particular also the reference to the work by Jardine which relates to the discussion of gerbes in the context of infinity-stacks using the model structure on simplicial presheaves.

The work by Stasheff and Wirth mentioned at the beginning is

  • James Wirth & Jim Stashef, Homotopy Transition Cocycles (arXiv blog)
Revised on June 29, 2009 16:25:11 by Tim Porter (62.136.207.112)