Gerbes and Z-graded symmetric monoidal categories

Question

Let me first recall a known fact. Suppose $X$ is a complex algebraic variety, $\mathscr{L}$ is a line bundle on $X$ and $L^\times$ is the total space of $\mathscr{L}$ with the zero section removed. Write $\pi:L^\times\longrightarrow X$ for the projection. Then $\mathscr{A}=\pi_*\mathscr{O}_{L^\times}$ is naturally a $\mathbb{Z}$-graded quasicoherent sheaf of commutative $\mathscr{O}_X$-algebras, and $\mathscr{L}$ can be recovered from $\mathscr{A}$ (because we can recover $L^\times$ together with the map $\pi$ and the action of the multiplicative group $\mathbb{G}_m$).

My question is whether this picture can be categorified along the following lines.

On the one hand, instead of line bundles let us look at $\mathbb{G}_m$-gerbes on $X$.

On the other hand, instead of $\mathscr{A}$ let us suppose (for instance) that we have a $\mathbb{Z}$-graded symmetric monoidal $\mathbb{C}$-linear abelian category $\mathscr{M}=\bigoplus_{d\in\mathbb{Z}}\mathscr{M}^d$, so that $\mathscr{M}^0$ is the category of coherent sheaves on $X$ with the usual tensor product.

Is there a setup of this sort where one gets a suitable equivalence between $\mathbb{G}_m$-gerbes on one side and a special class of $\mathbb{Z}$-graded symmetric monoidal categories on the other side? (It is fine with me if the description of the other side needs to be modified.)

Answer 1

This question can be approached abstractly through the general Tannakian formalism, as laid out e.g. here, or very concretely by hand. You construct maps in both directions. To a $G_m$ gerbe assign its tensor category $QC(Gerbe)$ of sheaves (I'll speak of quasicoherent sheaves out of force of habit - presumably you can work just with coherent). This is a commutative algebra over $QC(X)$, which locally is isomorphic to graded sheaves on $X$ (i.e. to $QC(X)\otimes Rep G_m$). Conversely to such a category assign its spectrum, the stack which to any ring attaches the groupoid of tensor functors from your category to modules. This carries a map to $X$ which is a $G_m$ gerbe.

Put another way, given a sheaf of tensor categories over $X$ (or commutative algebra over $QC(X)$) which is locally isomorphic to $Rep G$, you consider the stack (sheaf of groupoids) of isomorphisms of this sheaf of categories with $QC(X)\otimes Rep G$ [EDIT: Better and more Tannakian to say, the stack of fiber functors to $QC(X)$]. This is a $G$ gerbe (ie 's locally of the form $X\times BG$). This is of course just the usual Tannakian reconstruction as in Deligne, except that we have the base $X$ be a scheme (or geometric stack) instead of the spectrum of a field.

Of course you could also ask to give a more global characterization of such $Z$-graded commutative algebras over $QC(X)$. I think it's equivalent to characterize the module category given by sheaves of degree 1, aka twisted sheaves on the corresponding gerbe. This is your version of a categorified line bundle -- it's a module category locally isomorphic to sheaves. Presumably it can be characterized as an invertible module category -- one for which there exists (or maybe for which you specify - I'm conveniently pretending that everything has been taking place one level of categoricity down, which is fine if you only care about say a class in $H^2(X,G_m)$) an inverse with respect to tensor product of module categories over $QC(X)$. Then the above argument proves that such categorified line bundles (via Spec of the $Z$-graded $QC(X)$-algebra they generate) are equivalent to $G_m$-gerbes. [Edit:] You'll also need to make sure such invertible modules are locally trivial, ie again given by the same cohomology group. You might also want to think of things through a third perspective on this story after the gerby/Tannakian ones, namely that of Azumaya algebras. You want to know that your invertible module category has a generator as an $O_X$-linear category, and thus is equivalent to the category oif modules over a sheaf of algebras, namely the endomorphisms of this generator. And then appeal to a classification of these algebras up to Morita equivalence by the same cohomology group.

[By the way a very interesting recent paper about the derived version of this story is here.]

Answer 2

When $X={\rm spec}\; k$, the spectrum of a field, your question seems to be related to (non neutral) Tannaka duality : gerbes over a field are characterized by they categories of representations (see Deligne, P. Catégories tannakiennes. The Grothendieck Festschrift, Vol. II). This is not specific to $\mathbb G_m$, of course. Over an arbitrary basis, you are looking for a categorical interpretation of the cohomological Brauer group $H^2(X, \mathbb G_m)$, I don't know of a description along the lines you suggest. One could also understand your question in this way: it is possible to understand this $H^2$ as a $H^1$, arguing that "a $\mathbb G_m$-gerbe is a form of $B\mathbb G_m$", and this seems indeed feasible, see Breen, L Tannakian categories. Motives (Seattle, WA, 1991). To classify these forms, one has then to describe the automorphism group of $B\mathbb G_m$. This can be seen as categorical analogue of the set-theoretic fact that invertible sheaves are forms of $\mathcal O_X$, and as such are classified by $H^1(X,\mathbb G_m)$, where $\mathbb G _m$ is the automorphism group of $\mathcal O_X$.