Path space (algebraic topology) explained
In algebraic topology, a branch of mathematics, the based path space
of a
pointed space
is the space that consists of all maps
from the interval
to
X such that
, called based
paths.
[1] In other words, it is the mapping space from
to
.
A space
of all maps from
to
X, with no distinguished point for the start of the paths, is called the
free path space of
X. The maps from
to
X are called free paths. The path space
is then the pullback of
XI\toX,\chi\mapsto\chi(0)
along
.
The natural map
is a fibration called the
path space fibration.
References
Further reading
- https://ncatlab.org/nlab/show/path+space
Notes and References
- Martin Frankland, Math 527 - Homotopy Theory - Fiber sequences
