nLab
gerbe (as a stack)

This is a sub-entry for gerbe.

For related entries see

Idea

(The definitions of sheaf and stack are elsewhere in the nLab but we will have to review them briefly. There is also motivation for sheaves, cohomology and higher stacks, which provides other insights. For simplicity of exposition, we will initially look at sheaves, stacks and gerbes over a base space BB rather than in the more general setting of a topos.)

Naively a sheaf is a family of sets indexed ‘continuously’ by the points of a space BB. It corresponds either to a presheaf satisfying a ‘gluing condition’ (which is the version of descent in one of its simplest cases), or to an étale space over BB. The relationship between them is that the presheaf is the presheaf of local sections of the étale space, and the ‘gluing condition’ is that local sections that agree on the intersections of open sets can be glued uniquely to give a section over the union of the open sets.

An important class of sheaves are the torsors. Let GG be a sheaf of groups on BB. The category, Tors(B;G)Tors(B;G) of GG-torsors on BB is a groupoid. The notion of GG-torsor ‘localises’ well, so that if UU is an open set of BB, then we can restrict GG to a sheaf on UU, and look at the torsors over UU using the restricted GG. (We abuse notation and just write Tors(U;G)Tors(U;G) for the corresponding groupoid. If VV is another open set contained in UU, there is a restriction functor from Tors(U;G)Tors(U;G) to Tors(V;G)Tors(V;G), so it looks as if we have a presheaf of groupoids on BB, but things are not quite right here.

The assignment of Tors(G) UTors(G)_U to UU only gives a pseudofunctor not a functor from the category Open(B) opOpen(B)^{op} to the category of groupoids. It thus corresponds to a Grothendieck fibration or fibred category over Open(B)Open(B). It does have quite nice ‘gluing properties’ however, it is a stack of groupoids. (Roughly ‘morphisms glue, objects glue up to isomorphism’. For enlightenment note that there are almost presheaves of objects and of morphisms in this ‘pseudo’ presheaf.) This stack will be called 𝒯ors(G)\mathcal{T}ors(G).

There is a stackcompletion functor from fibred categories to stacks. If we take the sheaf of groups GG, think of it as a presheaf of groupoids BGBG, in the usual way, then it is a pseudo-functor from Open(B) opOpen(B)^{op} to the category of groupoids. If we stack complete it, we get … 𝒯ors(G)\mathcal{T}ors(G).

We thus can think of a stack of groupoids as a ‘lax’ generalisation of a sheaf of groupoids. What about gerbes?

(It should be mentioned that often in topological settings, the sheaf of groups is actually a constant sheaf and that in that case a GG-torsor is just a principal GG-bundle.)

Groups ‘are’ groupoids with a single object, but groupoids are not ‘groups with many objects’ (although that is a nice phrase to use when introducing them). A groupoid need not have any objects … if it is empty! Of course, we think of the vertex groups of a groupoid, but, if the groupoid is not connected, there may be many different non-isomorphic ones. So a group is a very special type of groupoid.

Similarly, the term ‘gerbe’ refers to a special sort of stack of groupoids.

A gerbe is to a general stack what, up to equivalence, a group is to a general groupoid. It is non-empty and connected.

David Roberts: This is reflected in the fact that the fibres of bundle gerbes (either abelian or GG-bundles gerbes) are transitive groupoids. I believe Tim and I had a bit of discussion of this sort of thing last year (or before - mists of time…). Or maybe this comment should go in gerbe (general idea)?

To make this precise we use some additional notions:

We will have a pseudofunctor F:Open(B) opGrpdF : Open(B)^{op}\to Grpd, and this will be a stack.

Definitions

  • A stack of groupoids, FF, on BB is locally non-empty if there is an open covering 𝒰\mathcal{U} of BB for which each groupoid F(U)F(U) is non-empty, for U𝒰U \in \mathcal{U}.

  • A stack of groupoids, FF, on BB is said to be locally connected if there is an open covering 𝒰\mathcal{U} of BB for which each groupoid F(U)F(U) is connected, for U𝒰U \in \mathcal{U}.

and finally:

  • A gerbe FF on BB is a locally non-empty, locally connected stack of groupoids on BB.

It is important to note that it does not state in the definition of a gerbe that the open cover that we have over which it is non-empty is or is not one over which it is connected.

Local connectedness can be well stated by saying that for the various UU, if xx and yy are local objects defined over UU, the set F(U)(x,y)F(U)(x,y) is not empty. (Translation: a ‘local object’, or ‘locally defined object’, of FF is a ‘local section’ of Ob(F)Ob(F), say, over UU, in other words, an element in Ob(F(U))Ob(F(U)).

Important example

Proposition

𝒯ors(G)\mathcal{T}ors(G) is a gerbe on BB.

Proof

To see locally non-empty: If UU is any open set in BB, then as 𝒯ors(G)(U)=Tors(U;G)\mathcal{T}ors(G)(U) = Tors(U;G), the category of G UG_U-torsors over UU, it has at least the trivial G-torsor (over UU) amongst its objects, so 𝒯ors(G)\mathcal{T}ors(G) is locally non-empty.

Next look at 𝒯ors(G)(U)\mathcal{T}ors(G)(U) again. Any two G UG_U-torsors are locally isomorphic to each other, since they are both locally isomorphic to the trivial G UG_U-torsor, so, if FF and F F^\prime are two G UG_U-torsors, there is an open cover such that over that cover FF and F F^\prime are isomorphic, hence 𝒯ors(G)\mathcal{T}ors(G) is locally connected. We thus have that 𝒯ors(G)\mathcal{T}ors(G) is a gerbe.

𝒜\mathcal{A}-Gerbes

The notion of a GG-gerbe arises in the article gerbe, but one needn’t use just a group GG. Fix a sheaf of abelian (possibly not necessary) groups 𝒜\mathcal{A} on BB. Then an 𝒜\mathcal{A}-gerbe is a gerbe FF on BB such that for any open UU on BB we have a functorial isomorphism 𝒜(U)Aut(s)\mathcal{A}(U)\stackrel{\sim}{\to} \text{Aut}(s) for all sF(U)s\in F(U).

Note that since FF is a stack, Aut(s)\text{Aut}(s) is a sheaf, so by isomorphism we mean an isomorphism as sheaves, and by functorial we mean given another object tF(U)t\in F(U), the isomorphism commutes

𝒜(U) Aut(s) Id 𝒜(U) Aut(t) \begin{matrix} \mathcal{A}(U) & \to & \text{Aut}(s) \\ Id \downarrow & & \downarrow \\ \mathcal{A}(U) & \to & \text{Aut}(t) \end{matrix}

In particular, we get that for any two objects C,DF(U)C, D\in F(U) we have that the sheaf Isom(C,D)Isom(C,D) is an 𝒜\mathcal{A}-torsor. This gives that if there is some object over UU, namely that F(U)F(U)\neq \emptyset, then the set of isomorphism classes of obects in F(U)F(U) is in natural bijection with H 1(U,𝒜 U)H^1(U, \mathcal{A}_U).

For example, consider the stack of rank 1 vector bundles on a scheme XX, Vect 1\text{Vect}_1. One can check that Vect 1\text{Vect}_1 is a 𝔾 m\mathbb{G}_m-gerbe by noting that the automorphism group of any vector bundle over UU will be precisely 𝒪 X(U) ×\mathcal{O}_X(U)^\times, and everything is functorial. By the interpretation in cohomology, we see that the global vector bundles (up to isomorphism) are in correspondence with H 1(X,𝒪 X ×)H^1(X, \mathcal{O}_X^\times) which is just Pic(X)\text{Pic}(X).

One can form the classifying stack, B𝒜B\mathcal{A} from the important example above by taking B𝒜(U)=𝒯ors(𝒜(U))B\mathcal{A}(U)=\mathcal{T}ors(\mathcal{A}(U)). A basic theorem about 𝒜\mathcal{A}-gerbes is that an 𝒜\mathcal{A}-gerbe, FF, is isomorphic to B𝒜B\mathcal{A} if and only if F(B)F(B)\neq \emptyset. This says that FF is isomorphic to the classifying stack if and only if it has a global object.

For information on how gerbes play a role in differential geometry see gerbe (in differential geometry).

References

There is a lengthier description of gerbes (at this level of generality) in the Menagerie notes that are available from Tim Porter's home page.

Other material available online includes the following:

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

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

Further references are given in the other entries on gerbes.

Revised on January 13, 2011 03:40:03 by hilbertthm90 (24.19.0.156)