Path space (algebraic topology)

In algebraic topology, a branch of mathematics, the path space $PX$ of a based space $(X,*)$ is the space that consists of all maps $f$ from the interval $I=[0,1]$ to X such that $f(0)=*$, called paths. In other words, it is the mapping space from $(I,0)$ to $(X,*)$. The space $X^I$ of all maps from $I$ to X (free paths or just paths) is called the free path space of X. The path space $PX$ can then be viewed as 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. Graduate Studies in Mathematics. 35. Providence, RI: American Mathematical Society. pp. xvi+367. doi:10.1090/gsm/035. ISBN 0-8218-2160-1. http://www.maths.ed.ac.uk/~aar/papers/davkir.pdf. 

Further reading

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