Mapping path space

In algebraic topology, the path space fibration over a based space (X, *) is a fibration of the form

where

The space ${\displaystyle X^{I}}$ consists of all maps from I to X that may not preserve the base points; it is called the free path space of X and the fibration ${\displaystyle X^{I}\to X}$ given by, say, ${\displaystyle \chi \mapsto \chi (1)}$, is called the free path space fibration.

The path space fibration can be understood to be dual to the mapping cone. The reduced fibration is called the mapping fiber or, equivalently, the homotopy fiber.

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