Path space (algebraic topology)

In algebraic topology, a branch of mathematics, the path space [math]\displaystyle{ PX }[/math] of a based space [math]\displaystyle{ (X, *) }[/math] is the space that consists of all maps [math]\displaystyle{ f }[/math] from the interval [math]\displaystyle{ I = [0, 1] }[/math] to X such that [math]\displaystyle{ f(0) = * }[/math], called paths.^[1] In other words, it is the mapping space from [math]\displaystyle{ (I, 0) }[/math] to [math]\displaystyle{ (X, *) }[/math]. The space [math]\displaystyle{ X^I }[/math] of all maps from [math]\displaystyle{ I }[/math] to X (free paths or just paths) is called the free path space of X.^[2] The path space [math]\displaystyle{ PX }[/math] can then be viewed as the pullback of [math]\displaystyle{ X^I \to X, \, \chi \mapsto \chi(0) }[/math] along [math]\displaystyle{ * \hookrightarrow X }[/math].^[1]

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

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

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