9:48 AM
Hey, has anyone ever inspected SAG Theorem 1.4.8.1?
It seems kinda fishy to me.
There's a missing reference to the key step as well. Is this filled in elsewhere (like one of the DAG papers)?
The reason it seems fishy to me is that, at least if you're doing something really stupid and obvious, it's not true that the affine schemes étale over the spdm stack X are 0-truncated objects.
So what this seems to be saying is that if you have a spdm stack (X,O_X), and you take the spdm stack (X,π_0(O_X)), which is some kind of ordinary higher Deligne-Mumford stack, there always exists a 'coarse moduli DM-1-stack' for the discrete (but still higher-dimensional) Deligne-Mumford stack (X,π_0(O_X)).

5 hours later…
2:34 PM
@HarryGindi I think the missing reference is Theorem 2.3.13 in DAG5 (which implies the geometric morphism to the 1-localic reflection is étale).

3 hours later…
5:17 PM
@MarcHoyois Thanks!

5:33 PM
But yeah this says that every higher DM-stack is étale over a DM 1-stack, which is a bit surprising.
A similar result that I find even more surprising is that every bounded ∞-topos is pro-étale over its 2-localic reflection (this is the "structure theory" section in HTT), and as far as I know it is still an open question whether this holds if you replace 2 by 1.

6:24 PM
Does the map X→X^\heart for spdm stacks induce a bijection on minimal geometric points?
I have this intuition that it ought to (or at least a hope, in order to avoid some really nasty stuff I was doing with Gerbes)
I proved that Lurie's 'underlying space' is the sober reflection of the traditional topology on the geometric points of a DM stack, and if X and X^heart have the same geometric points, I can just use classical results to show that the spaces agree when the DM stack has quasicompact diagonal
Otherwise, I have to redo this whole stratification by gerbes business, which seems a bit tricky

6:46 PM
Actually, I think this theorem in DAG 5 actually implies it. It seems like it says that it's the slice over an n+2-connected object, which should be a 'gerbe' in this setting (in the generalized sense of 'gerbe')

3 hours later…
9:46 PM
Yep, it worked! Thanks again!