The homotopy fiber of over consists of pairs where is a path from to . It replaces an ordinary fiber by a homotopy-invariant construction.
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In algebraic topology, the concept of the homotopy fiber is a key tool used to study maps between topological spaces. It can be considered as a generalization of the notion of the fiber in the context of fibration, and it helps to understand the homotopical properties of the map in question.