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.

In this paper we will introduce a certain type of morphisms of log schemes (in the sense of Fontaine, Illusie and Kato) and study their moduli. Then by applying this we define the notion of toric algebraic stacks, which may be regarded as torus emebeddings in the framework of algebraic stacks and prove some fundamental properties. Also, we study the stack-theoretic analogue of toroidal embeddings.

In this paper, we show that there is an equivalence between the 2-category of smooth Deligne-Mumford stacks with torus embeddings and actions and the 1-category of stacky fans. To this end, we prove two main results. The first is related to a combinatorial aspect of the 2-category of toric algebraic stacks defined by I. Iwanari [Logarithmic geometry, minimal free resolutions and toric algebraic stacks, Preprint (2007)]; we establish an equivalence between the 2-category of toric algebraic stacks and the 1-category of stacky fans. The second result provides a geometric characterization of toric algebraic stacks. Logarithmic geometry in the sense of Fontaine-Illusie plays a central role in obtaining our results.

Abstract. 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. We consider the case of symmetric monoidal ∞-categories of perfect complexes on perfect derived stacks. The first main result especially 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 infinity-category of mixed Tate motives over a perfect field. We prove that its tannakization can be obtained from the G m -equivariant bar construction of a commutative differential graded algebra equipped with G m -action. Moreover, under 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], [21], [22].

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