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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 $\mathbb{C}$.

  1. Is there some analogue of Keel-Mori theorem about existance of coarse moduli space for higher stacks?
  2. 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?
  3. What are the higher quotient stacks? What are the coherent sheaves on them? For example, what are the quotients of $\mathbb{A}^1$?
  4. It is well known that ussual DM (1-)stacks etale-locally are quotient stacks. Is there some analogue for higher stacks?
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I'd love to add some questions to yours! –  Fernando Muro Mar 13 at 22:11

1 Answer 1

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 $BBG$.

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Do you mean $G$ an abelian variety? –  Will Sawin Mar 31 at 18:52
    
No, eg G could be finite - the algebraic stack BZ/2 has no interesting homomorphisms to a group scheme, eg GLn. For G an abelian variety the same is true one level down even, so G has no interesting reps and BG has no interesting coherent sheaves –  David Ben-Zvi Mar 31 at 21:17
    
Ah, I missed the second $B$ in $BBG$, –  Will Sawin Mar 31 at 22:19
    
Hi David, is there some standard higher-categorical thing that replaces quasicoherent sheaves then? –  JBorger Apr 1 at 8:59
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Yes one can consider quasicoherent sheaves of categories (QC(X)-modules), QC(X)-mod-mod, QC(X)-mod-mod-mod etc. I think one (very rough and naive) expectation might be that rings O_X detect affine schemes, QC=O_X-mod detects stacks with affine diagonal (see Gaitsgory link), sheaves of categories (QC-mod) detect stacks whose diagonal has affine diagonal etc. Certainly that picture lines up well with stacks of the form B...BG (G affine and abelian) –  David Ben-Zvi Apr 1 at 17:18

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