Tannaka duality and stable infinity-categories

Abstract

We introduce a notion of fine Tannakian infinity-categories and prove Tannakian characterization results for symmetric monoidal stable infinity-categories over a field of characteristic zero. It connects derived quotient stacks with symmetric monoidal stable infinity-categories which satisfy a certain simple axiom. We also discuss several applications to examples.

Mixed motives and quotient stacks: Abelian varieties

“…• We apply results from derived Tannaka duality and techniques from derived algebraic geometry: DM ⊗ X is a fine ∞-category in the sense of [29], and there exist a derived quotient stack Z = [Spec A X /GL n ] and an equivalence DM ⊗ X ≃ QC ⊗ (Z) where QC ⊗ (Z) denotes the symmetric monoidal stable ∞-category of quasi-coherent complexes. Then the derived motivic Galois group is obtained from Z by the construction of the based loop space.…”

“…In Section 2 we recall some generalities concerning ∞-categories, ∞-operads, and spectra, etc. In Section 3, applying the Tannakian characterization in [29] to DM ⊗ X we discuss the consequences. We also include some basis definitions and facts about Chow and numerical motives, and mixed motives.…”

“…Let us briefly summarize our previous results obtained in [27], [28]. Let DM ⊗ be the symmetric monoidal stable presentable ∞-category of mixed motives (see Section 3 or [27], [28], [29]). By an ∞-category, we means an (∞, 1)-category (cf.…”

“…In this paper, we investigate the structure of the motivic Galois group of mixed motives tensor-generated by an abelian variety. In our papers [27], [28] and [29], we developed tannakian constructions for symmetric monoidal stable ∞-categories by means of derived algebraic geometry and proved Tannaka duality results for symmetric monoidal stable ∞-categories. We apply this theory to the symmetric monoidal stable ∞-category of mixed motives and find consequences for the structures of motivic Galois groups.…”

Motives for an elliptic curve

“…• Iwanari [11] uses derived Tannaka duality to describe the stable ∞-category of motives generated by a Kimura finite Chow motives as a symmetric monoidal stable ∞-category of quasi-coherent complexes on a derived quotient stack. In particular, motives for an elliptic curve are Kimura finite.…”