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 ${\mathcal{P}}_{\mathcal{C}}:\mathcal{X}\to {\text{Fib}}_{\mathcal{X},\mathcal{C}}$ and a functor $\mathcal{G}:\mathcal{C}\to \text{Vect}({\text{Fib}}_{\mathcal{X},\mathcal{C}})$. If $\mathcal{C}$ generates $\text{QCoh}(\mathcal{X})$ and $\mathcal{X}$ is an fpqc stack with quasi-affine diagonal, we show that ${\mathcal{P}}_{\mathcal{C}}:\mathcal{X}\to {\text{Fib}}_{\mathcal{X},\mathcal{C}}$ 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 $\text{QCoh}({\text{Fib}}_{\mathcal{X},\mathcal{C}})$.

28 pages. arXiv admin note: text overlap with arXiv:1409.4073

References in corpus (3)

• Tannaka Duality for Geometric Stacks
• Tensor categorical foundations of algebraic geometry
• A characterization of categories of coherent sheaves of certain algebraic stacks