Constructing a stack (gerbe) from a connected groupoid

Question

Let $\mathcal{G}=(A\rightrightarrows X)$ be a groupoid. Here $X={\rm Ob}(\mathcal{G})$, $A={\rm Ar}(\mathcal{G})$, and we have 5 maps: $s,t\colon A\to X$ (the source and the target, surjective), $m\colon A\times_X A\to A$ (multiplication of composable arrows), ${\rm id}\colon X\to A$ ($x\mapsto{\rm id}_x$, injective), and $i\colon A\to A$ ($a\mapsto a^{-1}$), satisfying the usual axioms. I say that my groupoid is connected if for any two objects $x,y\in X$ there exists an arrow $a\colon x\to y$.

Assume that a finite group $\Gamma$ acts on $\mathcal{G}$, i.e., it acts on $X$ and $A$ such all the 5 maps are $\Gamma$-equivariant. We say that $\mathcal{G}$ is a $\Gamma$-groupoid.

Now I want to construct a fibered category (gerbe) $\mathbb{G}$ over the category (site) of finite $\Gamma$-sets, starting from a connected $\Gamma$-groupoid $\mathcal{G}$. In other words, for any finite $\Gamma$-set $S$, I want to construct a groupoid $\mathbb{G}(S)$, and for a morphism $S\to T$ of finite $\Gamma$-sets, I want to define a restriction functor $\mathbb{G}(T)\to \mathbb{G}(S)$. How can I do that? I could not find this in Giraud's book.

Answer 1

The thing that Simon gives in his first comment is just a prestack, it needs to be stackified. You can do this by letting $\mathbb{G}(S)$ be the groupoid of principal $\mathcal{G}$-bundles in $\Gamma Set$, in other words, $\Gamma$-equivariant $\mathcal{G}$-bundles in Set.

In more detail, I'm assuming you are considering the topology on $\Gamma Set$ to be the one where a covering family is a surjective map. Then we define a principal $\mathcal{G}$-bundle over a $\Gamma$-set $S$ to be a surjective map of $\Gamma$-sets (all maps from now on will be $\Gamma$-equivariant), $\pi\colon P\to S$, a map $b\colon P\to X$ and an action map $a\colon P\times_{X,s} A \to P$ such that $\pi(a(p,f)) = \pi(p)$ (the action is fibrewise); in addition, we demand that the map $P\times_X A \to P\times_S P$ is an isomorphism.

Additionally, we demand there is a cover $j\colon T\to S$ and a map $\sigma\colon T \to P$ such that the pulled back bundle $T\times_S P \to T$ is isomorphic to the pullback of the canonical $\mathcal{G}$-bundle $ t\colon A \to X$ (the action is by composition in the groupoid) along $b\circ \sigma\colon T \to P \to X$

The groupoid $\mathbb{G}(S)$ is that of principal $\Gamma$-bundles with bundle maps between them (they preserve fibres and actions, including the data of the map to $X$). The functor $\mathbb{G}(S) \to \mathbb{G}(T)$ is given by pullback.

Answer 2

David Roberts already gave a nice answer, which you accepted. However, let me give you another perspective. Your groupoid $\mathcal{G}$ has an action of $\Gamma,$ i.e. is a groupoid object in $Set^{B\Gamma}\cong \Gamma-Set$ (where $B\Gamma$ is $\Gamma$ viewed as a one object category). So $$\mathcal{G} \in Gpd\left(Set^{B\Gamma}\right) \cong Gpd^{B\Gamma}.$$ I.e., your groupoid $\mathcal{G}$ is the same data as a funtor $\mathbb{G}:B\Gamma \to Gpd$. Explicitly, it sends the one object to the underlying groupoid in sets of $\Gamma,$ and $g \in \Gamma$ gets sent to the functor $\mathcal{G} \to \mathcal{G}$ induced by the action.

Now, you wanted to arrive at a gerbe over the large site $\Gamma-Set$ with surjections as covers. Well, this is the canonical topology on the topos $\Gamma-Set$, and sheaves over $\Gamma-Set$ with respect to this topology are equivalent to $\Gamma-Set$ itself, so, stacks over this site, are equivalent to the bicategory of (weak) functors $B\Gamma \to Gpd$, of which $\mathbb{G}$ is an example. So it corresponds canonically to a stack on $\Gamma-Set$. Explicitly, given a $\Gamma$-set $X,$ $X$ may viewed as a functor $X:B\Gamma \to Set$ and hence also as a functor $X^{id}:\Gamma \to Gpd,$ where we view a set as a groupoid with all identity arrows. Then the stack on $\Gamma-Set$ that $\mathbb{G}$ corresponds to, sends $X$ to the groupoid of maps $Hom\left(X^{id},\mathbb{G}\right).$ Provided that $\pi_0\mathcal{G}=*$ this stack is a gerbe.

If you prefer to get this stack as a fibered category, there is another approach. Consider the Grothendieck construction of $\mathbb{G}$ $$\pi:\int_{B\Gamma} \mathbb{G} \to B\Gamma.$$ It can be canonically identified with the action groupoid $\Gamma \ltimes \mathcal{G} \to B\Gamma.$ (See e.g. the "generalized action groupoid" construction in http://arxiv.org/abs/1011.6070, or see the homework assignment I gave my topos theory class: http://people.mpim-bonn.mpg.de/carchedi/HW1.pdf). Anyway, this is a fibered category over $\Gamma$ describing the gerbe associated to $\mathcal{G}.$ The fibered category over $\Gamma-Set$ it corresponds to is "sheaves over $\int_{B\Gamma} \mathbb{G}$ with the induced Grothendieck topology" which is easily seen to be the same as $Set^{\Gamma \ltimes \mathcal{G}},$ (since the topology becomes the canonical topology again) which becomes a fibered category over $Set^{\Gamma}$ via the functor $$\pi_!:Set^{\Gamma \ltimes \mathcal{G}} \to Set^{\Gamma},$$ where $\pi_!$ is left adjoint to the restriction functor.