Path space (algebraic topology)

In algebraic topology, a branch of mathematics, the path space ${\displaystyle PX}$ of a based 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 paths.^[1] In other words, it is the mapping space from ${\displaystyle (I,0)}$ to ${\displaystyle (X,*)}$.

The space ${\displaystyle X^{I}}$ of all maps from ${\displaystyle I}$ to X (free paths or just paths) is called the free path space of X.^[2] The path space ${\displaystyle PX}$ can then be viewed as 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]

References

1. Martin Frankland, Math 527 - Homotopy Theory - Fiber sequences
2. Davis & Kirk 2001, Definition 6.14.
3. Davis & Kirk 2001, Theorem 6.15. 2.

• 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