The fibred category $\mathcal A$ of algebraic spaces over a scheme $S$ is a stack (over the category of affine schemes with the etale topology). This is proved in Laumon and Moret-Bailly's book (see (1.6.4) and (3.4.6)).


Let $\mathcal G$ be the fibred category of **group** algebraic spaces over a scheme $S$. Is $\mathcal G$ a stack? 


My guess is that the forgetful functor $\mathcal G\to \mathcal A$ is "representable" in some sense (as the stabilizers of $\mathcal G$ are smaller than those of $\mathcal A$). But I can't see how to make this rigorous.


**Sidenote.** Note that $\mathcal A$ is not an algebraic stack (see Claim 3.1 in <http://arxiv.org/pdf/math/0602646v1.pdf>). 



The stack of group algebraic spaces
Here's a sketch proof. First, recall that inside the category of presheaves, sheaves are closed under taking limits. Second, inside the set of maps between the underlying algebraic spaces, the group homomorphisms are constructed by taking an equaliser:
$$
Hom\_{Grp}(G,H) \to Hom\_{Sp}(G,H) \rightrightarrows Hom\_{Sp}(G\times G,H) \times Hom\_{Sp}(\ast,H).
$$
Thus the condition that the hom-presheaf is a sheaf holds for the fibred category of group algebraic spaces, and so at the very least it is a prestack.


I claim that one can use the existence part of the universal property of descent (consider the map of descent data that is the multiplication map on the underlying groups) to show that given descent data for group algebraic spaces, the descended underlying algebraic space inherits a multiplication, and again by universality (uniqueness, this time), this makes the multiplication that of a group object. I can fill in details if needed, just not right now.


