A Pro-algebraic Fundamental Group for Topological Spaces

Consider a connected topological space $X$ with a point $x$ in $X$ and let $K$ be a field with the discrete topology. We study the Tannakian category of finite-dimensional (flat) vector bundles on $X$ and its Tannakian dual $\pi (X,x)$ with respect to the fiber functor in $x$. The maximal pro-étale quotient of $\pi (X,x)$ is the étale fundamental group of $X$ studied by Kucharczyk and Scholze. For well-behaved topological spaces, $\pi (X,x)$ is the pro-algebraic completion of the ordinary fundamental group. We obtain some structural results on $\pi (X,x)$ for very general topological spaces by studying (pseudo)torsors attached to its quotients. This approach uses ideas of Nori in algebraic geometry and a result of Deligne on Tannakian categories. We also calculate $\pi (X,x)$ for some generalized solenoids.