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What is the meaning of the statement "every map can be replaced by a weakly equivalent fibration"?

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It means that given any $f:A \to B$ we can find a space $E_f$ containing $A$ that is homotopy equivalent to $A$ and a fibration $p:E_f \to B$ such that $f = p \circ i$ where $i:A \to E_f$ is the inclusion. See Hatcher p. 407 ( for more details.

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