Tannakization in derived algebraic geometry

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2 Citations (Scopus)

Abstract

In this paper we begin studying tannakian constructions in ∞-categories and combine them with the theory of motivic categories developed by Hanamura, Levine, and Voevodsky. This paper is the first in a series of papers. For the purposes above, we first construct a derived affine group scheme and its representation category from a symmetric monoidal ∞-category, which we shall call the tannakization of a symmetric monoidal ∞-category. It can be viewed as an ∞-categorical generalization of work of Joyal-Street and Nori. Next we apply it to the stable ∞-category of mixed motives equipped with the realization functor of a mixed Weil cohomology. We construct a derived motivic Galois group which represents the automorphism group of the realization functor, and whose representation category satisfies an appropriate universal property. As a consequence, we construct an underived motivic Galois group of mixed motives, which is a pro-algebraic group and has nice properties. Also, we present basic properties of derived affine group schemes in the Appendix.

Original languageEnglish
Pages (from-to)642-700
Number of pages59
JournalJournal of K-Theory
Volume14
Issue number3
DOIs
Publication statusPublished - 2014 Jul 8

Keywords

  • motives
  • symmetric monoidal ∞-category
  • tannakian construction

ASJC Scopus subject areas

  • Algebra and Number Theory
  • Geometry and Topology

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