Suppose that (Sh(C), O) is a locally ringed topos. When is this the small etale topos of a DM stack? I think the condition is just that it is “locally isomorphic to the small etale topos of a scheme”. Here is why: (again I haven’t worked out all the details — so some of this may not work exactly as stated)

The condition means there exists a sheaf F in Sh(C) such that the localization (Sh(C)/F, O_F) is isomorphic to (Sh(U_{etale}), O_U) as a locally ringed topos. Consider the product sheaf F \times F and think of it as a sheaf over F via one of the projections. Via the isomorphism Sh(C)/F = Sh(U_{etale}) we can think of F \times F as an etale sheaf on U. Since every sheaf  on U_{etale} is representable by an algebraic space over U we conclude that (Sh(C)/F \times F, O_{F \times F}) is isomorphic to (Sh(R_{etale}), O_R) for some algebraic space R. By the fully faithfulness discussed in previous posts we obtain two morphisms s, t : R —> U. Moreover, we can do the same trick with F \times F \times F and obtain a composition morphism R \times_U R —> R (this will require a bit of work relating fibre products of etale morphisms of algebraic spaces to what happens on the side of small etale topoi, but I’m not worried). Hence (U, R, s, t, c) will be an etale groupoid algebraic space. The final step is to show that the DM-stack X = [U/R] has an associated locally ringed small etale topos (X_{etale}, O_X) which is equivalent to the locally ringed topos we started out with.

Note that [U/R] is a DM-stack since in the stacks project we work with algebraic stacks having no separation conditions whatsoever.

To characterize algebraic spaces among these will require a discussion of the inertia in terms of the language of locally ringed topoi as in the post dicussing the difference between Spec(R) and [Spec{C)/{+1, -1}].

Actually this sounds extremely familiar to me and I wouldn’t be surprised if I attended a talk or read an article/book which contained exactly this argument. There is after all an enormous literature on topoi, ringed topoi, etc (starting for example with Hakim, Topos anneles et schemas relatifs). It is also possible that somebody explained this to me in a conversation. Please think of it as being part of algebraic geometry already. But one of the things that is really fun about doing mathematics for me is the discovery process: As I work through material it feels as if I’m discovering it, even if I am reading a 100 year old text.

Certainly this argument will not become part of the stacks project until much later (if at all). My goal for the summer is to really start hacking away at basic properties of algebraic stacks.

PS: For experts on topoi: I keep writing Sh(C) whenever I mean topos since in the stacks project I define a topos to be Sh(C) where C is a site (a la Artin’s notes on Grothendieck topologies).