Fibration

In topology, a branch of mathematics, a fibration is a generalization of the notion of a fiber bundle. A fiber bundle makes precise the idea of one topological space (called a fiber) being "parameterized" by another topological space (called a base). A fibration is like a fiber bundle, except that the fibers need not be the same space, nor even homeomorphic; rather, they are just homotopy equivalent. Weak fibrations discard even this equivalence for a more technical property.

Fibrations do not necessarily have the local Cartesian product structure that defines the more restricted fiber bundle case, but something weaker that still allows "sideways" movement from fiber to fiber. Fiber bundles have a particularly simple homotopy theory that allows topological information about the bundle to be inferred from information about one or both of these constituent spaces. A fibration satisfies an additional condition (the homotopy lifting property) guaranteeing that it will behave like a fiber bundle from the point of view of homotopy theory.

Fibrations are dual to cofibrations, with a correspondingly dual notion of the homotopy extension property; this is loosely known as Eckmann–Hilton duality.

A fibration (or Hurewicz fibration or Hurewicz fiber space) is a continuous mapping p : E → B satisfying the homotopy lifting property with respect to any space. Fiber bundles (over paracompact bases) constitute important examples. In homotopy theory, any mapping is 'as good as' a fibration—i.e. any map can be decomposed as a homotopy equivalence into a "mapping path space" followed by a fibration into homotopy fibers.

...
Wikipedia