Bar construction and tannakization

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2 Citations (Scopus)


We continue our development of tannakizations of symmetric monoidal ∞-categories, begun in [19]. In this note we calculate the tannakizations of some examples of symmetric monoidal stable ∞-categories with fiber functors. We consider the case of symmetric monoidal ∞-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 ∞-category of mixed Tate motives over a perfect field. We prove that its tannakization can be obtained from the Gm-equivariant bar construction of a commutative differential graded algebra equipped with the Gm-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.

Original languageEnglish
Pages (from-to)515-568
Number of pages54
JournalPublications of the Research Institute for Mathematical Sciences
Issue number3
Publication statusPublished - 2014


  • Bar construction
  • Galois group
  • Motives
  • Tannaka duality
  • ∞-category

ASJC Scopus subject areas

  • Mathematics(all)


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