Homotopy fiber

Remarks

The map $F_f \to X$, $(x, \gamma) \mapsto x$, is a fibration, and its fiber over $x$ is the space of paths from $f(x)$ to $y_0$. The ordinary fiber $f^{-1}(y_0)$ embeds in $F_f$ by $x \mapsto (x, c_{y_0})$, where $c_{y_0}$ is the constant path. If $f$ is a Serre fibration, this embedding is a weak homotopy equivalence.

Example: if $X = \set{x_0}$ and $f(x_0) = y_0$, then $F_f$ is the space of paths from $y_0$ to $y_0$, which is the loop space $\Omega Y$.

Non-example: the ordinary fiber need not be a homotopy fiber. Let $X = \set{x_0}$, $Y = \R$, $f(x_0) = 0$ and $y_0 = 1$. Then $f^{-1}(1)$ is empty, while $F_f$ is the space of paths in $\R$ from $0$ to $1$, which is nonempty and contractible.

The notations $\operatorname{hofib}(f)$ and $\operatorname{hofib}_{y_0}(f)$ are also used.