Tannakization in derived algebraic geometry

Abstract

In this paper we begin studying tannakian constructions in 1-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 1-category, which we shall call the tannakization of a symmetric monoidal 1-category. It can be viewed as an 1-categorical generalization of work of Joyal-Street and Nori. Next we apply it to the stable 1-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.

Bar Construction and Tannakization

“…∞-categories. In this paper, we use theory of quasi-categories as in [17]. A quasicategory is a simplicial set which satisfies the weak Kan condition of Boardman-Vogt: A quasi-category S is a simplicial set such that for any 0 < i < n and any diagram n is the standard n-simplex.…”

“…a symmetric monoidal functor which preserves finite colimits). See [17,Section 3.2]. If R is a commutative ring spectrum, we refer to an object in CAlg( Cat…”

“…Roughly speaking, the underlying group scheme of MTG is obtained by truncating higher homotopy groups of valued points of MTG. In view of Theorem 2 and 3, we can say that the derived motivic Galois group constructed from DM ⊗ in [17] is a natural generalization of M T G to the whole mixed motives. This paper is organized as follows: In Section 2, we will review some of notions and notation which we need in this paper.…”

“…In [17] we have constructed tannakizations of stable symmetric monoidal ∞-categories. Let R be a commutative ring spectrum.…”

“…We continue our study of tannakizations of symmetric monoidal stable ∞-categories, begun in [17]. The issue treated in this paper is the calculation of tannakizations of examples of symmetric monoidal stable ∞-categories with fiber functors.…”