The category of topological categories $Cat(Top)$ is fibred over $Top$ - the functor sending a groupoid $X_1 \rightrightarrows X_0$ to its object space $X_0$ is a Grothendieck fibration. Now one can restrict to interesting subcategories, such as that of topological groupoids $Gpd(Top)$, or of etale groupoids (source and target are local homeomorphisms), or proper etale groupoids (additionally with $(s,t):X_1 \to X_0^2$ proper), and these are all still fibred over $Top$. This arises because the source and target maps have various pullback stability properties, but it is not automatic from usual pullback stability.
The cartesian lift of $M \to X_0$ for a groupoid (say) $X_1 \rightrightarrows X_0$ has arrow space $X_1 \times_{X_0^2} M^2$, so we can talk about $Obj:Gpd(S) \to S$ being a fibration for any finitely complete category (and even some non-finitely complete categories like $Diff$). If you have to 'interesting' full subcategories $C_1,C_2$ of $Gpd(S)$ that are fibred over $S$ via the object functor, then their intersection is again fibred (this is a trivial observation).
In particular, if we consider $S = Sch$ (or even relative schemes), then we get a fibration $Gpd(Sch) \to Sch$. But often one is interested in stacks of various kinds, or descent with respect to certain Grothenieck (pre)topologies, and so want to consider subcategories of $Gpd(Sch)$. My question is this:
What are some subcategories of $Gpd(Sch)$ that are fibred over $Sch$ (resp. with $Sch$ replaced with $Sch/S$)?
I'm interesting in subcategories that are defined by placing conditions on the source and target maps, like smoothness, flatness, tamely ramified or anything. Results that pertain to stacks are especially welcome.
