Tannaka duality revisited

TL;DR

This paper strengthens Lurie's Tannaka duality theorem for spectral algebraic stacks, providing more general conditions under which maps between stacks correspond to specific functors between categories of complexes.

Contribution

It introduces several enhanced versions of Tannaka duality, broadening its applicability to spectral algebraic stacks with new functorial characterizations.

Findings

Established strengthened Tannaka duality theorems

Identified maps with exact symmetric monoidal functors preserving certain complexes

Extended duality to broader classes of spectral stacks

Abstract

We establish several strengthened versions of Lurie's Tannaka duality theorem for certain classes of spectral algebraic stacks. Our most general version of Tannaka duality identifies maps between stacks with exact symmetric monoidal functors between $\infty$-categories of quasi-coherent complexes which preserve connective and pseudo-coherent complexes.