A fibration is a map satisfying the homotopy lifting property. This is easily seen to be equivalent to the following:

A map $f:X \to Y$ is a fibration if and only if there is a continuous function which given a path, $\phi$ , in $Y$ and a point, $x$ , lying above $\phi(0)$ , returns a lift of $\phi$ , starting at $x$ .

Let $D^2$ denote the set of complex numbers with modulus less than or equal to 1. An example of a fibration is the map $g: D^2 \to [-1,1]$ sending a complex number $z$ to $re(z)$ .

Note that if we restrict $g$ to the boundary of $D^2$ , we do not get a fibration. Although we may still lift any path to begin at a prescribed point, we cannot make this assignment continuously.

Another class of fibrations are found in fibre bundles.