If $(C, \tau)$ is a site with pullbacks and $\tau$ subcanonical, it is well known that these things are essentially equivalent:
- Groupoids $s,t: U_1 \to U_0$ where $s,t$ are covering for the $\tau$-topology
- Geometric stacks, that is stacks (in groupoids) over $C$ with a representable $\tau$-covering from an object of $C$.
- Etendues, that is a topos over $\mathrm{Sh}_{\tau}(C)$ which is locally isomorphic to the small topos $\mathrm{Sh}(U_\tau)$ of coverings of $U \in C$.
Essentially to $U: U_1 \to U_0 \times U_0$ we associate the stackification of the groupoid functor associated to $U$. If $X$ is a geometric stack, and $U_0 \to X$ a presentation, we associate the groupoid $U_1=U_0 \times_X U_0 \to U_0 \times U_0$. If $X$ is an étendue, and $F \to 1$ is a well supported object such that $X/F=\mathrm{Sh}(U_\tau)$ with $U \in C$, then via $F \times F \to F$, we get (I think) that $X/(F \times F)$ also represents $\mathrm{Sh}(U'_\tau)$. And $U' \to U$ is the corresponding groupoid.
Main question: is this completely spelled out in the litterature? The closest I have been able to find is Dorette A. Pronk, Etendues and stacks as bicategory of fractions, but she only treat some particular cases like topological stacks, differentiable stacks and algebraic stacks. What I am looking for is some kind of axiomatic treatment of geometric stacks that outlines the three approaches (and what are the exact conditions for them to be equivalent).
Second question: Pronk assumes in her statement on $DM$-stacks as étendues that $X/(F \times F)$ comes from a $U'$. Does this not follow from the hypotheses (see also this Stack project blog post) ?
The situation is a bit subtle if we consider theses objects as part of a $2$-category. For example for groupoids we really want to invert weak isomorphisms. (For that I like the theory of anafunctors as in David Roberts, Internal categories, anafunctors and localisations.) For étendues it is important to consider the topos over the big topos; algebraically this is the same as looking at the topos with its structure sheaf (ie see it as a locally ringed topos in the case of algebraic stacks).
The papers by David Roberts and David Carchedi seem interesting, but I have not been able to find the full link between étendues and geometric stacks completely explained. The link between stacks and groupoids is a lot more standard. For instance the post Link between internal groupoids and stacks on a topos ? by Carchedi explains well the link between (standard) stacks over a base site and internal groupoids (or categories) in the topos.
I have a (third and final) related question about geometric stacks: is the correct definition the one I gave, or should we require the stronger condition that the diagonal of $X$ has to be representable as well? For algebraic stacks this is equivalent, since a surjective representable cover implies that the diagonal is representable by effectivity of fppf descent for separated locally quasi-finite morphisms. What happens for geometric stacks when this descent condition is not satisfied? Does the iterated construction even converge?
Edit: for the third question, in HAG2, Töen and Vezzosi show that the link between geometric stacks and groupoids is still valid in derived algebraic geometry. This is their Proposition 1.3.4.2. In their Assumption 1.3.2.2, they do assume that a $\tau$-covering satisfy the descent condition.
I should introduce some motivations for this question: I'd like to see a treatment of algebraic (DM) stacks from the internal point of view of their étale topoi. Standard references for schemes are Hakim's thesis, and more recently Ingo Blechschmidt thesis, Using the internal language of toposes in algebraic geometry.
It seems the étendue point of view would be the best suited from an internal logic perspective. Of course the ultimate goal would probably be an internal description of (derived) algebraic geometry expressed in the logic of their $\infty$-topos via HoTT, but meanwhile I would also be happy for just an internal description of standard algebraic stacks.
