 Homotopy fiber

The homotopy fiber of $f:X\to Y$ over $y_0$ consists of pairs $(x,\gamma )$ where $\gamma$ is a path from $f(x)$ to $y_0$. It replaces an ordinary fiber by a homotopy-invariant construction.

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.