Bar construction and tannakization

TL;DR

This paper advances the understanding of tannakizations in symmetric monoidal infinity-categories, linking bar constructions with motivic Galois groups, and providing explicit calculations for categories of motives.

Contribution

It demonstrates that tannakizations include bar constructions for ring spectra and applies this to compute motivic Galois groups for mixed Tate motives.

Findings

Tannakization includes bar constructions for augmented ring spectra.

Tannakization of mixed Tate motives relates to torus-equivariant bar constructions.

Under conjectures, the motivic Galois group matches classical groups.

Abstract

In this note we continue our development of tannakizations of symmetric monoidal infinity-categories, begun in our previous paper. The issue treated in this paper is the calculation of tannakizations of examples of symmetric monoidal stable infinity-categories with fiber functors. We consider the case of symmetric monoidal infinity-categories of perfect complexes on perfect derived stacks. The first main result especially says that our tannakization includes the bar construction for an augmented commutative ring spectrum and its equivariant version as a special case. We apply it to the study of the tannakization of the stable infinity-category of mixed Tate motives over a perfect field. We prove that its tannakization can be obtained from the torus-equivariant bar construction of a commutative differential graded algebra equipped with torus-action. Moreover, under Beilinson-Soule vanishing conjecture, we prove that the underlying group scheme of the tannakization is the conventional motivic Galois group for mixed Tate motives. The case of Artin motives is also included.