Path space (algebraic topology)

In algebraic topology, a branch of mathematics, the based path space ${\displaystyle PX}$ of a pointed space ${\displaystyle (X,*)}$ is the space that consists of all maps ${\displaystyle f}$ from the interval ${\displaystyle I=[0,1]}$ to X such that ${\displaystyle f(0)=*}$, called based paths.^[1] In other words, it is the mapping space from ${\displaystyle (I,0)}$ to ${\displaystyle (X,*)}$.

A space ${\displaystyle X^{I}}$ of all maps from ${\displaystyle I}$ to X, with no distinguished point for the start of the paths, is called the free path space of X.^[2] The maps from ${\displaystyle I}$ to X are called free paths. The path space ${\displaystyle PX}$ is then the pullback of ${\displaystyle X^{I}\to X,\,\chi \mapsto \chi (0)}$ along ${\displaystyle *\hookrightarrow X}$.^[1]

The natural map ${\displaystyle PX\to X,\,\chi \to \chi (1)}$ is a fibration called the path space fibration.^[3]

See also

• mapping space