The answer is no, even for the sheaf-case.

First of all, you would have to make some assumptions such as assume that X_{λ}→ Spec(B_{λ}) is locally of finite presentation. For simplicity also assume that the algebraic space X is of finite presentation over Spec(B). Then if U→X is an étale presentation, it descends to a *morphism* U_{λ}→X_{λ}. Now the main problem is that even if this morphism is étale after base change to X, we cannot guarantee that it eventually becomes étale. Indeed, this is related to the openness of versality in Artin's algebraization theorem.

For example, let B_{0} be a ring and let B=colim B_{λ} be a direct limit of essentially étale algebras such that B is henselian.
If openness of versality holds for X_{λ}, then U_{λ}→X_{λ} is smooth in a neighborhood of the image of U→U_{λ} and (at least if B_{0} is noetherian) it follows that U_{λ}→X_{λ} is smooth for sufficiently large λ.

But ... openness of versality does not hold for general X_{λ}. Artin has given a nice bunch of examples where everything but this holds (see "The implicit function theorem in algebraic geometry", S5). For example, let B_{0}=k[x] be the affine line and let B be the henselization (or localization) at the origin. Let X_{0} be a "bad" sheaf, e.g., the following (Ex. 5.10):

Let X_{0}=colim_{k} X_{0,k} where X_{0,i}=Spec(k[x,y]/y(x-1)(x-2)...(x-k)) — the union of the x-axis and k vertical lines. This is an example of an ind-space, so by definition:

X_{0}(T)=colim_{k} X_{0,k}(T)

for any scheme T. Then clearly, X=X_{0}×_{B0} Spec(B) is isomorphic to Spec(B) but X_{λ} is not an algebraic space for any λ.

If you assume that openness of versality holds for X_{λ} and that the Spec(B)→Spec(B_{λ}) are smooth, then the answer to your question is likely "yes". However, openness of versality is the most subtle of Artin's criteria so it is probably difficult to assert.

`$\{X_{\lambda}\}$`

of others, and studying the circumstances under which one may translate useful properties between $X$ and a cofinal subsystem of the $X_{\lambda}$. But I think you're asking something slightly different: if we have a fixed system of stacks approximating an algebraic one, can we conclude that a cofinal system ofthoseis algebraic? $\endgroup$ – Clark Barwick Feb 8 '10 at 12:27