étalé space of sheaves on a differentiable stack

Question

If $F$ is a sheaf on a topological space $X$, the well-known étalé space contruction gives rise to a bundle $\Gamma F$ on $X$ such that $F$ is isomorphic to the sheaf of sections of $\Gamma F$.

On the other hand, if $\mathfrak X$ is a stack on $C$ the Grothendieck contruction gives an equivalence between the category of sheaves on $\mathfrak X$ and the category of faithful morphisms of stacks on $C$ with target $\mathfrak X$.

My question is:

1: Is the Grothendieck construction a generalisation of the étalé space construction?

In that case we could try to obtain the analogue result in the context of differentiables spaces or stacks:

2: If $\mathfrak X$ is now a differentiable space, is there an equivalence between the cat of sheaves on $\mathfrak X$ and the cat of faithful morphisms of differentiable spaces with target $\mathfrak X$?

3: Same question for $\mathfrak X$ a differentiable stack.

Answer 1

1). In what way is the Grothendieck construction a genralization of the étalé space construction?

Well, if $F$ is a sheaf on a topological space $X,$ this really means that $F$ is a sheaf on the poset $\Omega(X)$ if open subsets. The Grothendieck construction of $\int F$ is not the étalé space of course, but sheaves on $\int F$ (with the induced Grothendieck topology) is equivalent to sheaves on the total space of the étalé space of $F$.

What is really going on? Well, in general, if $F$ is a sheaf on a site $C,$ then we have an equivalence of topoi

$$Sh(C)/F \simeq Sh\left(\int_C F\right).$$

On the other hand, if $X$ is a topological space, we have that the canonical geometric morphism $$Sh(X)/F \to Sh(X)$$ is the image under the functor $$Sh:Top \to Topoi$$ of the étalé space $\Lambda(F) \to X$.

(You might want to read up on étale geometric morphisms for more info.)

You can unify these two ideas (namely that of taking the Grothendieck construction, and taking slice topoi) by replacing your site $C$ with $Sh(C)$ itself, endowing the latter with the canonical topology. Sheaves on $Sh(C)$ with respect to the canonical topology are equivalent to $Sh(C)$ itself (turns out if a preaheaf on $Sh(C)$ is a sheaf then it is representable). For a sheaf $F$ on $Sh(C),$ one has $$\int_{Sh\left(C\right)} F \simeq Sh(C)/F.$$

2): No. If $F$ is any sheaf, then the category of "sheaves over $F$" is the slice topos $Sh(C)/F,$ or equivalently, sheaves on $\int F$. This means they correspond exactly to a morphism $G \to F$ from another sheaf $G.$ If $F$ is a differentiable space, there is no reason $G$ needs to be one, e.g. take a sheaf $H$ which is not a differentiable space and take $H \times F \to F.$

3). The answer is still no (but below I will tell you how depending on what you mean, it CAN be yes) for a differentiable stack: Say $\mathfrak{X}$ is a differentiable stack, and $H$ is some sheaf which is not a differentiable stack, then $\mathfrak{X} \times H \to \mathfrak{X}$ is a sheaf on $\mathfrak{X},$ but the domain is not a differentiable stack.

However, if $\mathfrak{X}$ is an étale differentiable stack (so one coming from an étale Lie groupoid), then one can mean something different by a sheaf on $\mathfrak{X}$ than a sheaf on $\int \mathfrak{X}.$ For example, if $\mathfrak{X}=M$ is a manifold, a sheaf on $\int M$ is a sheaf on $\mathit{Mfd}/M$, sometimes called a large sheaf on $M,$ whereas usually one speaks of a sheaf on $M$ to be a sheaf on its poset of opens- small sheaves. It turns out that the category of small sheaves on an étale stack is equivalent to the category of representale étale maps $\mathfrak{Y} \to \mathfrak{X},$ with $\mathfrak{Y}$ another étale stack. (And if you relax sheaf to stack, you get all étale maps). See my paper: http://msp.warwick.ac.uk/agt/2013/13-02/p026.xhtml