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Let $C$ be a site, $\mathbf{S}$ some ($\infty$-? homotopy?) category of spaces. Question. What do you call a (covariant!) functor $F:C\to \mathbf{S}$ enjoying the following property: for every ...

Let $f\colon X\to Y$ be a proper continuous map of topological spaces which is a Serre's fibration. $X$ and $Y$ may be assumed to be locally compact, $Y$ is connected topological manifold of finite ...