nLab gerbe (general idea)

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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 $G$-principal bundle (for $G$ a group) is a space $P$ equipped with a map $P \to X$ to a base space $X$, such that each fiber of $P$ looks like $G$ in a nice way.

A $G$-gerbe is similarly a “space”, $P \to X$ such that each fiber looks like $\mathbf{B}G$ in a nice way. We can also replace $G$ with a sheaf of groups, or even with a $2$-group.

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

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 $(\infty,1)$-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 $(\infty,1)$-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 $G \to H \to K$ of a group $K$ by a group $G$ as a $G$-gerbe over $\mathbf{B}K$, namely as a fibration $\mathbf{B}H \to \mathbf{B}K$ with fiber $\mathbf{B}G$.

When the group $G$ 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 $\mathbf{B}G$ obtained by shifting $G$ 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 $G$ is not abelian then the way in which $\mathbf{B}G$ still relates to a group-like structure is slightly more involved and proceeds via the automorphism 2-group $AUT(G)$ of $G$.

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

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

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

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

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

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

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

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

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

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

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

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

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

Notice in particular that for $G = U(1)$ we have $H^2(X, U(1)) \simeq H^3(X, \mathbb{Z})$, for instance by a long exact sequence argument, so that $U(1)$-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 $G$ abelian, the entire concept of $G$-$(n-1)$-gerbe is straightforward: it is the $(n-1)$-stack incarnation of $\mathbf{B}^n G$-principal infinity-bundles, i.e. of fibrations $P \to X$ of (generalized) spaces that arise as homotopy pullbacks of the form

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

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

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

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

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

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

Now, a $G$-gerbe for $G$ nonabelian is supposed to be a fibration whose typical fiber is $\mathbf{B}G$. 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 $H$-principal 2-bundles as above and the notion of $G$-gerbe:

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

• $AUT(G)$ is the automorphism $2$-group of the groupoid $\mathbf{B}G$, i.e.

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

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

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

is a fibration sequence.

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

Notice in particular that when $G$ is abelian there is a canonical morphism

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

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

This is the general nonsense underlying the concept of gerbe.

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)

Last revised on June 29, 2009 at 16:25:11. See the history of this page for a list of all contributions to it.