# étalé space of sheaves on a differentiable stack

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.

• Do you mean for $C$ to be a site? May 16 '14 at 14:28
• Yes, $C$ is a site, $\mathfrak X$ has the induced topology (that is covering families are $x_i\rightarrow x$ which are sent to covering families of $C$). The notion of faithful morphism of stack is clear as one sees stack as categories fibered in groupoids. May 16 '14 at 15:06

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 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