Homotopy fiber

Summary

The homotopy fiber of a map $f \colon X \to Y$ over $y_0$ is the space of pairs of a point $x \in X$ and a path from $f(x)$ to $y_0$. It replaces the ordinary fiber $f^{-1}(y_0)$ by a version that depends only on the homotopy class of $f$. For a Serre fibration the two fibers have the same homotopy groups. So homotopy fibers extend the long exact sequence of homotopy groups to arbitrary maps. The homotopy fiber of the inclusion of a point is the loop space.