The theory of dendroidal sets has been developed to serve as a combinatorial model for homotopy coherent operads, see [MW07, CM13b]. An ∞-operad is a dendroidal set D satisfying certain lifting conditions.

In this paper we give a definition of K-groups Kn (D) for a dendroidal set D. These groups generalize the K-theory of symmetric monoidal (resp. permutative) categories and algebraic K-theory of rings. We establish some useful properties like invariance under the appropriate equivalences and long exact sequences which allow us to compute these groups in some examples. Using results from [Heu11b] and [BN12] we show that the K-theory groups of D can be realized as homotopy groups of a K-theory spectrum .

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.

We introduce a notion of n-quasi-categories as fibrant objects of a model category structure on presheaves on Joyal's n-cell category Θn. Our definition comes from an idea of Cisinski and Joyal. However, we show that this idea has to be slightly modified to get a reasonable notion. We construct two Quillen equivalences between the model category of n-quasi-categories and the model category of Rezk Θn-spaces, showing that n-quasi-categories are a model for (∞, n)-categories. For n = 1, we recover the two Quillen equivalences defined by Joyal and Tierney between quasi-categories and complete Segal spaces.