Bar Construction and Tannakization

Abstract

We continue our development of tannakizations of symmetric monoidal $\infty$-categories, begun in [19]. In this note we calculate the tannakizations of some examples of symmetric monoidal stable $\infty$-categories with fiber functors. We consider the case of symmetric monoidal $\infty$-categories of perfect complexes on perfect derived stacks. The first main result in particular 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 $\infty$-category of mixed Tate motives over a perfect field.We prove that its tannakization can be obtained from the ${\mathbb{G}}_m$-equivariant bar construction of a commutative differential graded algebra equipped with the ${\mathbb{G}}_m$-action. Moreover, under the Beilinson–Soulé vanishing conjecture, we prove that the underlying group scheme of the tannakization is the motivic Galois group for mixed Tate motives, constructed in [4], [25], [26]. The case of Artin motives is also included.