Homotopy fiber

The homotopy fiber of $f:X\to Y$ over $y_0$ consists of pairs $(x,\gamma )$ where $\gamma$ is a path from $f(x)$ to $y_0$. It replaces an ordinary fiber by a homotopy-invariant construction.

A principal fibration, in the homotopy-theoretic sense, is a homotopy pullback of a path-space fibration $PC\to C$ along a map $k:B\to C$. Its fibre is $\Omega C$ and acts on the fibre paths by concatenation, freely and transitively up to the relevant homotopies. Principal bundles for a topological group give this construction through its classifying space. When $C=K(A,n+1)$, the fibre is $K(A,n)$ and the principal fibration is classified by $H^{n+1}(B;A)$, yielding a principal Eilenberg–MacLane fibration.

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.