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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 \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$, \mathscr{M}=\bigoplus_{d\in\mathbb{Z}}\mathscr{M}^d$, 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.)

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# Gerbes and Z-graded symmetric monoidal categories

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.)