Moduli 'space' of stacks?

Question

In algebraic geometry, we are frequently interested in parametrizing geometric objects. Formally, parametrization of geometric objects having some property can be viewed as a functor $F:Sch\rightarrow Set$ from the category of schemes to the category of sets (which assigns to a scheme the set of families of geometric objects over this scheme). The problem of finding the moduli space of geometric objects can then be considered (up to distinction between fine and coarse moduli spaces) as the problem of finding an object corepresenting $F$.

It sometimes happens that $F$ can not be corepresented within the category of schemes (as in the case of the moduli problem for smooth projective curves). We are then forced to extend our category somehow; for smooth projective curves, it is enough to extend the category of schemes to the category of Deligne--Mumford stacks.

My question is: what are examples of geometrically interesting moduli problems for stacks which are not representable within the category of stacks? If there are such examples, what are the resulting extensions of the category of stacks?

P.S.: I am being deliberately ambiguous about what kind of stacks we consider (DM, Artin, etc.) because I do not know a priori where interesting examples will come from.

Answer 1

Such a moduli problem for stacks is expected to be a $2$-stack.

For example, consider the stack of line bundles on $X$, whose objects are parameterized by $H^1(X, \mathbb{G}_m)$. This is a (trivial) example of a $\mathbb{G}_m$-gerbe on $X$; these in turn are parameterized by $H^2(X, \mathbb{G}_m)$. The collection of those forms a $2$-stack.

See the introduction to Lurie's Higher Topos Theory for a nice discussion.

Answer 2

Fix group $G$ (could be a finite group, could be an algebraic group). The collection of all stacks isomorphic to $BG$ is naturally a $2$-stack.

Its $\pi_1$ is $Out(G)$, and its $\pi_2$ is $Z(G)$.

The $k$-invariant in $H^3(Out(G),Z(G))$ which classifies the $2$-stack up to isomorphism is the obstruction to the existence of a short exact sequence of groups $$ 0 \to Z(G) \to E \to Out(G) \to 0 $$

It's not very easy to come up with an example of a group $G$ where the obstruction is non-zero (by which I mean it's not very easy to compute the obstruction). I seem to remember that the obstruction is non-zero for the dihedral group with $16$ elements.