Tannakization of quasi-categories and monadic descent

TL;DR

This paper develops a framework connecting symmetric monoidal stable $$-categories, derived group schemes, and monadic descent, providing new insights into automorphism groups and descent theory in higher category contexts.

Contribution

It introduces a Tannakian formalism for quasi-categories, relating automorphism group schemes to monads via descent under finiteness conditions.

Findings

Constructed derived group schemes from fiber functors.

Established a comparison between representations of the group scheme and descent categories.

Provided conditions under which the comparison functor is an equivalence.

Abstract

Given a symmetric monoidal stable $\infty$-category $\mathcal{C}$ and a left adjoint symmetric monoidal fiber functor to $\operatorname{Mod}_A^{\otimes}$ for some $\mathbb{E}_{\infty}$-ring $A$, one can construct a derived group scheme $G$ of monoidal automorphisms of this functor. The left adjoint fiber functor also induces a monad on $\mathcal{C}$. Under some finiteness hypothesis on the fiber functor, we show there is a comparison functor from the category of representations of $G$ to the descent category of the induced monad on $\mathcal{C}$.