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started stub on Tannaka duality for geometric stacks, but need to interrupt now.
The theorem there can be read as justifying the point of view of derived noncommutative geometry to regard the 2-algebra $QC(X)$ as a valid replacement for the 1-algebra $\mathcal{O}(X)$.
I don’t think this line is right:
More generally, for $(S, \mathcal{O}_S)$ a ringed topos, we have
$Hom(S,X) \simeq Hom_\otimes(QC(X); A Mod) \,.$
Surely that should be $Hom_\otimes(QC(X); \mathcal{O}_S Mod)$?
clear case of copy-and-paste error. But check out the latest version, with lots more details.
I added the details on étale-locally ringed toposes that makes the statement of the main theorem the way I did it correct.
Have to quit now. Maybe more tomorrow.
Why is this result ’Tannakian’? Can it be seen as some way of reconstructing something from its actions?
Daniel Schäppi has Tannakian style results where essentially on is reconstructing a Hopf algebroid, ie a groupoid in affine schemes (ie2 a geometric stack) from its category of coherent sheaves. The original case reconstructing a group is the one-object version, BG, where coherent sheaves are more or less representations.
OK, so here the idea is that from $QC(X)$ (representation-like) and hom spaces to $QC(Spec A)$ one can reconstruct $X$, whereas classical Tannaka results reconstruct something from its category of representations and single underlying functor (maybe thought of as to the representations of the trivial object).
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