Let \(f\colon X\to Y\) be a continuous map, \(I=[0,1]\), and \(Y^I\) the space of paths \(I\to Y\) with the compact-open topology. The mapping path space of \(f\) is the subspace \[ E_f\coloneqq\ts{(x,\gamma)\in X\times Y^I \st \gamma(0)=f(x)} \] of \(X\times Y^I\) [@Hat02].
Let \(\iota\colon X\to E_f\) be \(x\mapsto(x,c_{f(x)})\), where \(c_y\) is the constant path at \(y\), and let \(p\colon E_f\to Y\) be \((x,\gamma)\mapsto\gamma(1)\). Then \(f=p\circ\iota\), the space \(E_f\) deformation retracts onto \(\iota(X)\), so \(\iota\) is a homotopy equivalence, and \(p\) is a fibration. Thus every continuous map factors as a homotopy equivalence followed by a fibration [@Hat02].
The mapping path space is the Eckmann–Hilton dual of the mapping cylinder, which factors a map as an embedding followed by a homotopy equivalence.