nLab gerbe (as a stack)

This is a sub-entry for gerbe.

For related entries see

• bundle gerbe
• gerbe (general idea)

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

The assignment of $Tors(G)_U$ to $U$ only gives a pseudofunctor not a functor from the category $Open(B)^{op}$ to the category of groupoids. It thus corresponds to a Grothendieck fibration or fibred category over $Open(B)$. 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 $\mathcal{T}ors(G)$.

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

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 $G$-torsor is just a principal $G$-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 $G$-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 $F : Open(B)^{op}\to Grpd$, and this will be a stack.

Definitions

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

• A stack of groupoids, $F$, on $B$ is said to be locally connected if for each open $U$ in $B$ there is an open covering $\mathcal{U}$ of $B$ such that all elements in $F(U)$ become connected in all $F(U_i)$.

and finally:

• A gerbe $F$ on $B$ is a locally non-empty, locally connected stack of groupoids on $B$.

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 $U$, if $x$ and $y$ are local objects defined over $U$, the set $F(U_i)(x,y)$ is not empty. (Translation: a ‘local object’, or ‘locally defined object’, of $F$ is a ‘local section’ of $Ob(F)$, say, over $U$, in other words, an element in $Ob(F(U))$.

Important example

Proposition

$\mathcal{T}ors(G)$ is a gerbe on $B$.

Proof

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

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

$\mathcal{A}$-Gerbes

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

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

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

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

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

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