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.

isn'tfibered in groupoids. – Harry Gindi Jun 7 '10 at 22:30