Consider a connected topological space $$X$$ with point $$x$$ in and let $$K$$ be field the discrete topology. We study Tannakian category of finite-dimensional (flat) vector bundles on its dual $$\pi(X,x)$$ respect to fiber functor . The maximal pro-étale quotient is étale fundamental group studied by Kucharczyk Scholze. For well-behaved spaces, pro-algebraic completion ordinary group. obtain ...