notation about $hfib(f,y_0)$

Question

Let $f:X\rightarrow Y$ be a continuous map. Then we can define $hfib(f,y_0)$. What does $hfib(f,y_0)$ mean?

Answer 1

it is the homotopy fiber at the point $y_0$.

The usual fiber can be written as $\{(x,y) \in X \times Y \mid f(x)=y_0\}$, but the homotopy fiber instead consists of all points in $X$ together with a path in $Y$ connecting their image to $y_0$. That is

$hfib(f,y_0):=\{(x,\alpha) \in X \times Y^I \mid \alpha(0)=f(x), \alpha(1)=y_0\}$.

The most important part of this definition comes from the long exact sequence in homotopy groups and the fact that it is homotopy invariant in the sense that if another continuous map $g$ is homotopic to $f$, then the homotopy fibers are (weakly) homotopy equivalent.

Answer 2

Chances are high this denotes the homotopy fiber, i.e. the homotopy pullback $$\begin{array}{ccc} \operatorname{hfib}(f,y_0) & \rightarrow & X\\ \downarrow&&\downarrow_f\\ \ast&\overset{y_0}\rightarrow & Y \end{array}$$ For more see Wikipedia and nlab or your favorite book on algebraic topology.