When is a stack (NOT) geometric?

Question

Following the terminology of $n$-Lab, a geometric stack $\mathcal{X}$ on a site $\mathcal{(C,J)}$ is a stack for which there exists a representable epimorphism $X \to \mathcal{X}$ from an object $X$ of $\mathcal{C}$ (viewed as a representable presheaf). Equivalently, $\mathcal{X}$ is geometric if and only if there exists a (nice enough) groupoid object $\mathcal{G}$ in $\mathcal{C}$ such that $\mathcal{X}$ is (2-iso to) the stackification of the strict presheaf of groupoids $Hom(blank,\mathcal{G})$ (where nice enough essentially means that you can take enough iterated pullbacks in $\mathcal{C}$ to form a $\mathcal{C}$-enriched nerve).

My question is, is there a more intrinsic definition of geometric stack? By "more-intrinsic" I mean a definition that does not use the existential quantifier. For example, if our site is topological spaces, we know a presheaf is representable if and only if it sends colimits in $Top$ to limits in $Set$. Since geometric stacks are in some sense a natural generalization of representable presheaves, it would seem natural to expect a similar characterization of geometric stacks (at least in the case when our site is nice enough, like $Top$).

I ask this mostly because, although in some circumstances there is a natural atlas or a natural choice of representing groupoid object around to try to prove that something is a geometric stack, proving that a stack is NOT geometric becomes very difficult when the definition involves the EXISTENCE of a nice atlas.

If someone only knows the answer for certain sites, this is still interesting to me.

Answer 1

One problem -- or at least one characteristic aspect -- of the notion of geometric stack is of course that it makes explcit reference to a fixed chosen site. Different sites may give rise to equivalent toposes and still to different notions of geometric stacks.

One approach is to make that extra information an explicit extra piece of data in a controlled way. This is effectively what is achieved by the notion of geometry for a structured topos. In terms of this one can then characterize geometric stacks fairly intrinsically. For instance in Structured Spaces it is shown that with a standard choice for "geometry" a Deligne-Mumford stack is precisely a "2-scheme" in a suitable sense.

Going beyond that, one could ask which "geometries" in this sense are naturally associated to a given topos, without choosing them by hand, such that the corresponding 2-schemes are the natural notion of geometric stack.

I think a big step in that direction is achieved in Bertrand Toen's work Champs affine. As reviewed at rational homotopy theory in an (oo,1)-topos, Toen there shows that for stacks or higher stacks on the algebraic site, one can characterize "affine stacks" intrinsically, as the objects of the reflective sub-(oo,1)-category on objects that are local with respect to morphisms that induce isomorphisms in "rational cohomology", where "rational" is as seen by the ground field.

Using that intrinsic notion of "affine stack", Toen then gives in section 4 a definition of geometric oo-stacks.

This may or may not be exactly what you are asking for, but I think it does provide some noteworthy indications of the kind of approach that one should think about.