The proalgebraic fundamental group of a connected topological space X, recently introduced by the first author, is an affine group scheme whose representations classify local systems of finite-dimensional vector spaces on X. In this article, we further develop the theory of the proalgebraic fundamental group, in particular, we establish homotopy invariance and a Seifert–van Kampen theorem. To facilitate the latter, we study amalgamated free product of affine group schemes. We also compute the proalgebraic fundamental group of the arithmetically relevant Kucharcyzk–Scholze spaces and compare it to the motivic Galois group.