Tannakization in derived algebraic geometry

Isamu Iwanari

Published 2011-12-08, updated 2012-08-17Version 3

We give a universal construction of a derived affine group scheme and its representation category from a symmetric monoidal infinity-category, which we shall call the tannnakization of a symmetric monoidal infinity-category. This can be viewed as infinity-categorical generalization of the work of Joyal-Street and Nori. We then apply it to the stable infinity-category of mixed motives equipped with the realization functor of a mixed Weil cohomology and obtain a derived motivic Galois group whose representation category has a universality, and which represents the automorphism group of the realization functor. Also, we present basic properties of derived affine group schemes in Appendix.