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I am asking for references and discussions of statements of the form Every bounded hypercover can be refined by an ordinary cover By "bounded" I mean "finite height". E.g., are ...

Let $f:X\to Y$ be a morphism of affine varieties such that it is a fibre bundle with fibre $F$. Let $\pi_1(Y)=\Gamma$ be a free group (non abelian) of finite rank and $\pi_1(F)$ is a finite group $G$ ...