Stacks of fiber functors and Tannaka's reconstruction

TL;DR

This paper develops a stack of fiber functors associated with a fibered category and demonstrates that, under certain conditions, it provides a Tannaka-type reconstruction of the original category, extending classical results.

Contribution

It introduces the stack of fiber functors for fibered categories and proves its equivalence to the original category under specific conditions, generalizing Tannaka's reconstruction.

Findings

The stack of fiber functors is equivalent to the original category when generated by a subcategory.

The stack $ ext{Fib}_{ ext{X}, ext{C}}$ is quasi-compact with affine diagonal.

The functor $ ext{G}$ generates the quasi-coherent sheaves on the stack.

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 $\text{QCoh}(\text{Fib}_{\mathcal{X},\mathcal{C}})$.