In mathematics, especially homotopy theory, the homotopy fiber is part of a construction that associates a fibration to an arbitrary continuous function of topological spaces .
In particular, given such a map, define to be the set of pairs where and is a path such that . We give a topology by giving it the subspace topology as a subset of (where is the space of paths in which as a function space has the compact-open topology). Then the map given by is a fibration. Furthermore, is homotopy equivalent to as follows: Embed as a subspace of by where is the constant path at . Then deformation retracts to this subspace by contracting the paths.
The fiber of this fibration (which is only well-defined up to homotopy equivalence) is the homotopy fiber , which can be defined as the set of all with and a path such that and , where is some fixed basepoint of .