Gerbes and Stacks

Question

The definition of a gerbe on a smooth manifold that I know is that - after fixing an open cover $U_i$, a gerbe consists of the data of line bundles $L_{ij}$ on two-fold-intersections $U_{ij}$, isomorphisms $\alpha_{ijk}: L_{ij} \otimes L_{jk} \longrightarrow L_{ik}$ on three-fold intersections that satisfy a co-cycle condition on four-fold intersections.

A gerbe on a site is a stack $G$, such that for every object $U$, there exists a covering $U_i$ of $U$ such that $F_{U_i}$ is non-empty for every $i$ and for any two objects $x_1$, $x_2$ in $G_{U}$, there exists a covering $U_i$ of $U$ such that $x_1|_{U_i}$ and $x_2|_{U_i}$ are isomorphic (i.e. objects exist locally and they are locally isomorphic).

My question is that if these two notions are related or if it is just the same name for completely different things. In particular: Are gerbes on a manifold a special stack on the small site of that manifold? Is there a fully faithful functor of $2$-categories that sends gerbes over $M$ to stacks over (the small site of) $M$?

Answer 1

There is a canonical equivalence of $2$-categories

$$St\left(Man/M\right) \simeq St\left(Man\right)/M$$ between stacks on the large site of $M$ and stacks on the site of manifolds equipped with a map to $M$ (regarding $M$ as a representable sheaf). Given a map $\pi:\mathscr{Y} \to M$ for $\mathscr{Y}$ some stack on manifolds, it corresponds to the stack $\Gamma(\mathscr{Y})$ on $Man/M$ which assigns a map $f:N \to M$ the groupoid of sections $N \to \mathscr{Y}$ of $\pi$ over $f.$ Suppose that there is a cover of $U_i$ of $M$ such that each $U_i \times _M \mathscr{Y}\simeq U_i \times BU(1)$ (or if you prefer $U_i \times BGL(1)$). Then $\Gamma(\mathscr{Y})$ is easily seen to be a gerbe on the large site for $M$. By Dan Peterson's answer, we see that from the data of a bundle gerbe, one gets a stack $\pi:\mathscr{Y} \to M$ with this property. In fact, it is not hard to show that these are equivalent data, that is, given $\pi:\mathscr{Y} \to M$ such that there is a cover $U_i$ such that $U_i \times _M \mathscr{Y}\simeq U_i \times BU(1)$ is the same as giving a bundle gerbe on $M$. By taking each bundle gerbe $\pi:\mathscr{Y} \to M$ and sending it to $\Gamma(\mathscr{Y})$, one gets a fully faithful embedding of the $2$-category of bundles gerbes over $M$ into the $2$-category of gerbes over the large site of $M$ (which furthermore embeds fully faithfully into stacks on the large site of $M$). The essential image is precisely those gerbes on the site $Man/M$ which are banded by $U(1)$, as pointed out by Reimundo Heluani. It doesn't embed into the $2$-category of stacks on the small site of $M$ however.

Answer 2

Your question has been answered by Reimundo Heluani but let me spell out things in full detail. Suppose you have a gerbe on a smooth manifold $M$ in your sense, given by $(U_i,L_{ij},\alpha_{ijk})$. Here is a procedure for cooking up a stack out of this data:

i) Let $P_i = U_i \times \mathrm{BGL}(1)$ for all $i$. A map $X \to P_i$ is the same as a map to $U_i$ and a line bundle on $X$.

ii) For each pair of indices $i,j$, a map $X \to U_i \times_M P_j$ is a map $f \colon X \to U_i \cap U_j$ and a line bundle $L$ on $X$. Now define a $1$-morphism $\phi_{ij} \colon U_i \times_M P_j \to U_j \times_M P_i$ by the rule $(f,L) \mapsto (f,L \otimes f^\ast L_{ij})$.

iii) For each triple of indices $i,j,k$ there are two natural isomorphisms $ U_i \times_M U_j \times_M P_k \to U_k \times_M U_j \times_M P_i$ which we may abusively denote $\phi_{ik}$ and $\phi_{ij}\circ \phi_{jk}$, and the isomorphism $\alpha_{ijk}$ defines a $2$-morphism between these.

iv) For each quadruple overlap, the cocycle condition for the $\alpha$'s ensures that these $2$-morphisms commute strictly.

Thus we obtain gluing data that allows us to glue together all the $P_i \to U_i$ to a stack $P \to M$, which is locally a "$\mathrm{BGL}(1)$-bundle".

Answer 3

Here is one way to see it: gerbes on $M$ form a bigroupoid (= bicategory all of whose morphisms and 2-morphisms are invertible).

In particular, if $\mathcal{G}$ is a gerbe over $M$, and $\mathcal{I}_M$ denotes the trivial gerbe over $M$ (trivial cover $\{M\}$, trivial line bundles, trivial isomorphism) then we have a groupoid $Hom(\mathcal{G},\mathcal{I}_M)$ of "trivializations".

Define $$ U \mapsto Hom(\mathcal{G}|_U,\mathcal{I}_U). $$ This is your stack on the site of open sets of $M$.