Path space fibration

In algebraic topology, the path space fibration over a pointed space ${\displaystyle (X,*)}$^[1] is a fibration of the form^[2]

${\displaystyle \Omega X\hookrightarrow PX{\overset {\chi \mapsto \chi (1)}{\to }}X}$

where

• ${\displaystyle PX}$ is the based path space of the pointed space ${\displaystyle (X,*)}$; that is, ${\displaystyle PX=\{f\colon I\to X\mid f\ {\text{continuous}},f(0)=*\}}$ equipped with the compact-open topology.
• ${\displaystyle \Omega X}$ is the fiber of ${\displaystyle \chi \mapsto \chi (1)}$ over the base point of ${\displaystyle (X,*)}$; thus it is the loop space of ${\displaystyle (X,*)}$.

The free path space of X, that is, ${\displaystyle \operatorname {Map} (I,X)=X^{I}}$, consists of all maps from I to X that do not necessarily begin at a base point, 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.^{[clarification needed]} The fiber of the based fibration is called the mapping fiber or, equivalently, the homotopy fiber.