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A -gerbe is similarly a “space”, such that each fiber looks like in a nice way. We can also replace with a sheaf of groups, or even with a -group.
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 -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?
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 -topos simplifies the situation conceptually by separating
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 -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
All of these are the same structure implemented in different contexts of generalized spaces.
For instance the last item here interprets extensions of a group by a group as a -gerbe over , namely as a fibration with fiber .
When the group 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 obtained by shifting 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 is not abelian then the way in which still relates to a group-like structure is slightly more involved and proceeds via the automorphism 2-group of .
So is a fibration sequence that extends for each choice of point of to the left to a fibration sequence . This says that the fiber of over each point looks like the group . General nonsense implies then that the action of on itself induces an action of on all of and that this action is indeed principal.
When is an abelian group, so that itself has a group structure, the object exists and the above statement has an immediate categorification:
A -gerbe for an abelian group is a fibration such that there is a morphism such that is the homotopy pullback of the point along this fibration.
In this case for each point of this yields a fibration sequence
which says that the fiber of over each point of looks like .
As in the previous case of ordinary bundles, general nonsense implies that comes with a principal -action. is therefore also called a -principal 2-bundle or a -torsor. In its concrete incarnation as a stack, is called a -gerbe.
in the homotopy category of our generalized spaces, it is a tautology that these -gerbes are classified by .
Notice in particular that for we have , for instance by a long exact sequence argument, so that -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 abelian, the entire concept of --gerbe is straightforward: it is the -stack incarnation of -principal infinity-bundles, i.e. of fibrations of (generalized) spaces that arise as homotopy pullbacks of the form
Accordingly, such -gerbes for abelian are classified in cohomology by . (Another way to see why this is possible for an abelian group is that not only is a -group, but is an -group, which is what we need in general for an -gerbe.)
Moreover, for any pointed connected generalized space (any parameterized -groupoid with a single object), we may say that -principal -bundles are fibrations classified in this way by classifying morphisms
In particular, for any 2-group (not necessarily of the form for an abelian group as above) an -principal 2-bundle is a fibration in this sense classified by a morphism . The typical fiber of such a -bundle looks like .
Now, a -gerbe for nonabelian is supposed to be a fibration whose typical fiber is . 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 -principal 2-bundles as above and the notion of -gerbe:
For every group , there is the 2-group , defined equivalently as follows:
is the automorphism -group of the groupoid , i.e.
From the second description it is manifest that one can project out a copy of out of (the shifted copy): there is a morphism obtained simply by identifying all objects of . Indeed, is a groupoid extension of by the discrete groupoid on in that
is a fibration sequence.
This means that any -principal 2-bundle with typical fiber the groupoid has underlying it a fibration with typical fiber the one-object groupoid . This underlying object is the -gerbe.
Notice in particular that when is abelian there is a canonical morphism
which however is not an equivalence when has nontrivial automorphisms. Therefore -gerbes in the sense of nonabelian -gerbes classified by are even for abelian a bit more general than the things classified by just , which are however also often called -gerbes (in particular “bundle gerbes”).
This is the general nonsense underlying the concept of gerbe.
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