When is a category of groupoid schemes fibred over schemes?

2010-11-11T23:08:54Z

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.

When is a category of groupoid schemes fibred over schemes? - MathOverflow

When is a category of groupoid schemes fibred over schemes?