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In Lurie's paper Tannaka Duality for Geometric Stacks, it is essentially shown that specifying a morphism of geometric objects

$$ f \colon X \to Y$$

is equivalent to giving a corresponding pullback morphism, which is a symmetric monoidal functor

$$ f^* \colon \mathrm{QC}(Y) \to \mathrm{QC}(X),$$

where $\mathrm{QC}$ indicates the category of quasi-coherent sheaves.

In actuality, Lurie proves the following result.

Theorem 5.11. Suppose that $(S,\mathcal{O}_S)$ is a ringed topos which is local for the étale topology, and that $X$ is a geometric stack. Then the functor

$$ T \colon f \mapsto f^* $$

induces an equivalence of categories

$$ T \colon \mathrm{Hom}( (S, \mathcal{O}_s), \mathrm{Sh}(X_\mathrm{\acute{e}t})) \to \mathrm{Hom}(\mathrm{QC}(X), \mathcal{O}_S\mathrm{-Mod}).$$

Short explanation:
Here the Hom functor on the left is taken in the (2-)category of locally ringed toposes, and on the right it corresponds to the tensor functors, which are symmetric monoidal functors which also preserve finite colimits, with an additional tameness condition.
(A stack is said to be geometric if it is quasi-compact, and the diagonal morphism is representable and affine.)

Now I don't see how to recover from this the usual Tannakian reconstruction results, for instance that given a neutral Tannakian category I can recover an algebraic group (by looking at automorphisms of the fibre functor) whose category of representations is this category I started with.

In his helpful answer, David Ben-Zvi mentions that from Lurie's result, we can find that having a faithful fibre functor from our Tannakian category $\mathcal{C}$ to $K$-vector spaces means that $\mathcal{C}$ is the category of quasi-coherent sheaves on some classifying stack $\mathbf{B}G$ (or, more accurately, on some $G$-gerbe if we are not just working over an algebraically closed field). However, I fail to see why this is so; in particular, Lurie does not seem to have a result about essential surjectivity of the above functor $T$, and I don't see how one would deduce such a thing from the previously mentioned theorem.

Sorry if this is all too trivial, David Ben-Zvi's answer left me wanting for a fuller understanding of the picture.

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1 Answer 1

up vote 10 down vote accepted

Tannaka duality makes two complementary statements: first, that an affine algebraic group can be recovered from its category of representations (the "reconstruction problem"), and second, that certain categories are the categories of representations of an affine algebraic group (the "recognition problem").

Lurie's paper generalizes the first statement, but not the second. The generalization of the second statement to Lurie's setting (in fact, to a slightly more general setting) is contained in Daniel Schäppi's paper, A characterization of categories of coherent sheaves of certain algebraic stacks. In this paper, Schäppi defines the notion of a weakly Tannakian category, and shows that weakly Tannakian categories are precisely the categories of coherent sheaves for "coherent algebraic stacks with the resolution property." This result specializes to the classical recognition result on affine algebraic groups.

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