Assume we have an Artin stack $\mathcal X$ which we want to equip with some additional structure, let's say some sort of sheaf. Then the first choice we have to make is that of an appropriate site for our sheaf to live on. I want to consider
The "big étale site" $\mathcal X_{étale}$ with underlying category $\mathcal X$ and the topology induced by the étale topology
The "small étale site" $\mathcal X_{lisse\text{-}étale}$ which is the full subcategory of $\mathcal X_{étale}$ consisting of all $x\in\mathcal X$ such that $U\xrightarrow{x} \mathcal X$ is smooth, again with the topology induced by the étale topology
Neither of them is source of great happiness for me, (1) being way to big and (2) having rather nasty functorial behavior.
Fortunately, if the structure in question satisfies fppf-descent (e.g. when our sheaves have some coherence property) both sites should give the same definition because then we can work with atlases.
This is the statement I want to make precise: I want to use that a (cartesian) sheaf on $\mathcal X_\tau$ , where $\tau\in\{étale, lisse\text{-}étale\}$ is essentially the same as a morphism $\mathcal X_\tau \to (Sheaves)$ over $(Schemes)$, where $(Sheaves)$ is the category of sheaves on schemes, fibered in groupoids over $(Schemes)$. Then the precise statement I want to show is the following
Let $\mathcal S$ be a (not necessarily algebraic) fppf-stack fibered in groupoids over $(Schemes)$ Then the restriction $$ Hom(X_{étale} , \mathcal S) \to Hom(X_{lisse\text{-}étale},\mathcal S) $$ (morphisms of course taken over $(Schemes)$) is an equivalence of categories
My questions would be:
- Is this actually true?
- How to prove this nicely? I think I could do it using atlases but there are a lot of choices involved and it gets rather messy. I would love if it followed form some general abstract nonsense.