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Then the projection $$p:M_f \to [0,1]$$ is a Serre fibration. This follows because any map of a disc into $\mathbb{Q}$ factors through f. … However this projection is not a Dold fibration. It is easy to construct a diagram using $Y = \mathbb{Q}$ which will have no weak homotopy lift. …

Then $E_2 \to B$ is the identity map (a homeomorphism) and is a Serre fibration. The set-theoretic identity map $E_1 \to B$ is also continuous. … Since any map from a disk into $E_1$ or $B$ factors through a constant map, you can check that it is also a Serre fibration. …

However using cellular approximation I have been able to show that this map has a weaker fibration property; it is a sort of "Serre fibration version" of a Dold fibration (where we only have a weak homotopy … This is good enough for many applications, but it still leads me to ask my question: Is the counit map $\epsilon_X$ a Serre fibration? …

For example for any space $B$, the inclusion $$B \times \{0\} \to B \times [0,1]$$ has the left lifting property with respect to any Hurewicz fibration (by definition), hence this is the first example … Lemma: If $f: B' \to B$ is a trivial Hurewicz cofibration and $E \to B$ is a Hurewicz fibration, then we get an induced bijection: $$ f^*: \pi_0 \Gamma(B, E) \to \pi_0 \Gamma(B', f^*E) $$ Proof: First …