Fibrations and homotopy lifting

A map $p:E\to B$ is a Hurewicz fibration when every homotopy $H:Y\times I\to B$ and every lift $\widetilde H_0:Y\to E$ of its initial map extend to a homotopy $\widetilde H:Y\times I\to E$ with $p\widetilde H=H$. The quantifier is over all parameter spaces Y in the usual convenient category of spaces. A Serre fibration requires the property for disks of every nonnegative dimension, equivalently for CW homotopy problems. The distinction matters for general spaces, although either notion gives the disk-lifting tools used for homotopy groups.

The property includes ordinary path lifting by taking Y to be a point. Unlike a covering, a general fibration need not give a unique lift with a chosen starting point. A path can move freely in a non-discrete fiber while projecting to the same base path. Nor does the lifting definition itself imply a local product chart. Fiber bundles over CW bases are standard examples of fibrations, while numerability or paracompactness hypotheses enter general bundle-to-Hurewicz statements. It is safer to name those settings than to assume every locally presented bundle automatically satisfies every global lifting claim.

For a product projection $B\times F\to B$, write the initial lift as $(H(y,0),a(y))$ and keep a(y) fixed while the base homotopy moves. This immediately gives a continuous lift. Allowing the fiber coordinate to vary continuously gives other lifts, exposing the difference from the discrete-fiber case. Pullbacks of fibrations are fibrations, since a lifted base homotopy and the prescribed point in the pullback remain compatible with its defining equation. These elementary examples are useful tests of any proposed general statement.

Fix $b_0\in B$ and let $P_{b_0}B$ be paths starting there, with endpoint evaluation to B. With the standard compact-open, compactly generated mapping-space convention, endpoint evaluation is a fibration; append a moving base-path segment to construct lifts. Its total space is contractible by shortening each path toward its initial endpoint, and the fiber over $b_0$ is the based loop space $\Omega B$. The projection reaches exactly the path component of $b_0$, so comparisons with all of B should either assume B path connected or restrict to that component.

For a Hurewicz fibration, a base path transports one fiber to another through a choice of lifted family. Reversing the path gives an inverse up to homotopy, so fibers over points in the same path component have the same homotopy type. For a Serre fibration, disk lifting supplies the corresponding identifications of homotopy groups; a homotopy equivalence of arbitrary fibers requires additional justification, such as suitable CW-type hypotheses. The transport is not generally a uniquely defined homeomorphism or pointwise canonical map. Going around a loop can produce nontrivial monodromy on fiber homotopy information. This recovers the discrete permutation picture of covering spaces as a special case, while allowing richer actions on continuous fibers.

Restricting a fibration to a chosen path component of the base is often necessary before writing a connected-base conclusion. A selected fiber can be empty if no point above that component has been supplied, and different base components need not have comparable fibers. For the long exact sequence we choose an actual point $e_0\in E$, set $b_0=p(e_0)$, and use the fiber over $b_0$. The based homotopy groups see the components containing those chosen points, while component sets near degree zero need their own pointed-set interpretation.