Tannakization in derived algebraic geometry

TL;DR

This paper introduces a universal construction called tannakization for symmetric monoidal infinity-categories, generalizing classical concepts and applying it to derive a motivic Galois group in derived algebraic geometry.

Contribution

It develops a new infinity-categorical framework for tannakization and constructs a derived motivic Galois group with universal properties.

Findings

Constructed a derived affine group scheme from symmetric monoidal infinity-categories.

Applied tannakization to mixed motives to obtain a derived motivic Galois group.

Established properties of derived affine group schemes.

Abstract

We give a universal construction of a derived affine group scheme and its representation category from a symmetric monoidal infinity-category, which we shall call the tannnakization of a symmetric monoidal infinity-category. This can be viewed as infinity-categorical generalization of the work of Joyal-Street and Nori. We then apply it to the stable infinity-category of mixed motives equipped with the realization functor of a mixed Weil cohomology and obtain a derived motivic Galois group whose representation category has a universality, and which represents the automorphism group of the realization functor. Also, we present basic properties of derived affine group schemes in Appendix.