Geodesic
A Pro-algebraic Fundamental Group for Topological Spaces
Informatics and Automation, Algebra and Arithmetic, Algebraic, and Complex Geometry, Tome 320 (2023), pp. 71-102.

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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. 
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Christopher Deninger. A Pro-algebraic Fundamental Group for Topological Spaces. Informatics and Automation, Algebra and Arithmetic, Algebraic, and Complex Geometry, Tome 320 (2023), pp. 71-102. http://geodesic.mathdoc.fr/item/TRSPY_2023_320_a4/

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