TITLE: Several simple questions on the geometry of higher stacks
QUESTION [13 upvotes]: I'm trying to understand definition/work out some examples. So, there are some simple questions about higher stacks.
For the simplicity assume that we are working with higher DM (Deligne-Mumford) stacks over $\mathbb{C}$.

Is there some analogue of Keel-Mori theorem about existance of coarse moduli space for higher stacks?
It is well known that ussual DM (1-)stacks with a point as a coarse moduli are quotients of a point by a finite group $G$ acting trivially, and coherent sheaves on it are just representations of $G$. So, what are the higher DM stacks, whose coarse moduli is just a point? What are the categories of coherent sheaves on such stacks?
What are the higher quotient stacks? What are the coherent sheaves on them? For example, what are the quotients of $\mathbb{A}^1$?
It is well known that ussual DM (1-)stacks etale-locally are quotient stacks. Is there some analogue for higher stacks?

REPLY [7 votes]: I don't have any knowledge about most of your questions, but I wanted to point out that asking for coherent sheaves is not a very good way to test higher stacks. By analogy, if I'm interested in rings of functions I won't get much of a sense for schemes or stacks - the ring of functions on a scheme or stack $X$ is the same as that of the affine scheme $Aff(X)=Spec \Gamma(O_X)$, the affinization of $X$. The same can be said for categories of (quasi)coherent sheaves -- they will factor through the "1-affinization" of $X$, which is the stack
recovered from $X$ by Tannakian reconstruction -- see this remarkable paper and DAG VIII for more on these notions. For example the group stack BG for G an abelian group doesn't have interesting actions on vector spaces (homomorphisms to the scheme $GL_n(R)$ , so there aren't any nontrivial coherent sheaves on $BBG$.