Path-space fibration ($\Omega X\to PX\to X$)

Let $PX$ consist of paths starting at the chosen basepoint of $X$. Endpoint evaluation is a fibration $\Omega X\to PX\to X$, and $PX$ is contractible by shortening each path. Pulling it back along a map $Y\to X$ gives its homotopy fibre, after reversing the chosen path direction if needed. This makes an acyclic relative Sullivan algebra modelling $PX$ useful for explicit rational fibre calculations.