a fibration

A fibration is a map that behaves like a bundle of parallel fibers for the purposes of homotopy — a map p: E -> B you can think of as the base B with a copy of a 'fiber' hanging over each point, glued together so that paths and deformations downstairs can always be lifted upstairs. It is the homotopy-theoretic generalization of a covering space (which had discrete fibers) to fibers that can be whole spaces.

The defining property is the homotopy lifting property (HLP). A map p: E -> B is a fibration if, given any space Y, any map Y -> E, and any homotopy H: Y x [0,1] -> B of its projection, the homotopy lifts to a homotopy Y x [0,1] -> E starting from the given map and projecting to H. In words: any deformation in the base, together with a starting lift, can be carried along upstairs. Requiring HLP for all Y gives a Hurewicz fibration; requiring it only for cubes/CW complexes gives the weaker but more common Serre fibration. The fiber over a basepoint b_0 is F = p^{-1}(b_0); for a fibration the fibers over different points are all homotopy equivalent, so 'the fiber' is well-defined up to homotopy. Covering spaces are exactly the fibrations with discrete fibers; fiber bundles over reasonable bases are Serre (indeed Hurewicz) fibrations.

Fibrations are the central computational tool for homotopy groups, precisely because the homotopy lifting property forces a long exact sequence relating pi_n(F), pi_n(E), and pi_n(B). That sequence converts knowledge of two of the three spaces into knowledge of the third and is how virtually every nontrivial homotopy group gets computed. The Hopf fibration S^1 -> S^3 -> S^2, the path-loop fibration over a space, and the universal principal bundles are all worked through this way. An honest caveat: 'the fibers are homotopy equivalent' needs the base to be path-connected; over a disconnected base different components can carry genuinely different fibers, and the long exact sequence is stated per path-component.