Path space fibration

In algebraic topology, the path space fibration of a pointed space maps its based path space PX to X by sending each path to its endpoint. Its fiber over the base point is the loop space ΩX. A related free path space fibration uses all maps from the interval I to X, without requiring paths to begin at a base point. The fiber of the based fibration is also called the mapping fiber or homotopy fiber.

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Endpoint map

Maps each based path to its value at 1

Based path space

Continuous paths from I to X that start at the base point

Fiber

The loop space ΩX