Stacks of fiber functors and Tannaka's reconstruction

Abstract: Given a quasi-compact category fibered in groupoids $\mathcal{X}$ and a monoidal subcategory $\mathcal{C}$ of its category of locally free sheaves $\text{Vect}(\mathcal{X})$, we are going to introduce the stack of fiber functors $\text{Fib}{\mathcal{X},\mathcal{C}}$ with source $\mathcal{C}$, which comes equipped with a map $\mathcal{P}{\mathcal{C}}\colon\mathcal{X}\to\text{Fib}{\mathcal{X},\mathcal{C}}$ and a functor $\mathcal{G}\colon\mathcal{C}\to\text{Vect}(\text{Fib}{\mathcal{X},\mathcal{C}})$. If $\mathcal{C}$ generates $\text{QCoh}(\mathcal{X})$ and $\mathcal{X}$ is an fpqc stack with quasi-affine diagonal, we show that $\mathcal{P}{\mathcal{C}}\colon\mathcal{X}\to\text{Fib}{\mathcal{X},\mathcal{C}}$ is an equivalence, as it happens by Tannaka's reconstruction when $\mathcal{X}$ is an affine gerbe over a field. In general, under mild assumption on $\mathcal{C}$, e.g. $\mathcal{C}=\text{Vect}(\mathcal{X})$, we show that $\text{Fib}{\mathcal{X},\mathcal{C}}$ is a quasi-compact fpqc stack with affine diagonal and that the image $\mathcal{G}(\mathcal{C})$ generates $\text{QCoh}(\text{Fib}{\mathcal{X},\mathcal{C}})$.