Path space (algebraic topology)

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

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

The natural map $PX\to X,\,\chi \to \chi (1)$ is a fibration called the path space fibration.

References

• Davis, James F.; Kirk, Paul (2001). Lecture Notes in Algebraic Topology (PDF). Graduate Studies in Mathematics. Vol. 35. Providence, RI: American Mathematical Society. pp. xvi+367. doi:10.1090/gsm/035. ISBN 0-8218-2160-1. MR 1841974.

Further reading

• https://ncatlab.org/nlab/show/path+space