# A pro-algebraic fundamental group for topological spaces and an arithmetic application (live stream)

Using the Tannakian formalism we defined and studied a pro-algebraic fundamental group for connected topological spaces. Using ideas of Nori and a result of Deligne on fibre functors of Tannakian categories we also defined a pseudo-torsor under this fundamental group which can serve as a replacement for the universal covering space in this generality. We introduce amalgamated products of pro-algebraic groups in order to prove a Seifert van Kampen theorem. The group of connected components of the pro-algebraic fundamental group is isomorphic to the pro-étale fundamental group used by Kucharczyk and Scholze to exhibit the absolute Galois group of a field $K$ of characteristic zero containing all roots of unity as the fundamental group of an ordinary topological space $Y_K$. We calculate the pro-algebraic fundamental group of $Y_K$ and show that $Y_K$ also carries a little bit of information about the motivic Galois group of $K$. If time permits, we also mention categorical criteria derived from the work of Coulembier for the Tannakian dual of a neutral Tannaka category over a perfect field to be reduced or even perfect. Part of this work is joint with Michael Wibmer.

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