In algebraic topology, the path space fibration over a pointed space
[1] is a fibration of the form[2] where The free path space of X, that is,
, consists of all maps from I to X that do not necessarily begin at a base point, and the fibration
, is called the free path space fibration.
The path space fibration can be understood to be dual to the mapping cone.
(A mapping path space satisfies the universal property that is dual to that of a mapping cylinder, which is a push-out.
is the homotopy fiber, the pullback of the fibration
denotes the constant path with value
is a homotopy equivalence; thus, the above decomposition says that any map is a fibration up to homotopy equivalence.
is a fibration to begin with, then the map
is a fiber-homotopy equivalence and, consequently,[4] the fibers of
over the path-component of the base point are homotopy equivalent to the homotopy fiber
By definition, a path in a space X is a map from the unit interval I to X.
Again by definition, the product of two paths
given by: This product, in general, fails to be associative on the nose:
( γ ⋅ β ) ⋅ α ≠ γ ⋅ ( β ⋅ α )
One solution to this failure is to pass to homotopy classes: one has
[ ( γ ⋅ β ) ⋅ α ] = [ γ ⋅ ( β ⋅ α ) ]
Another solution is to work with paths of arbitrary lengths, leading to the notions of Moore's path space and Moore's path space fibration, described below.
[5] (A more sophisticated solution is to rethink composition: work with an arbitrary family of compositions; see the introduction of Lurie's paper,[6] leading to the notion of an operad.)
, we let An element f of this set has a unique extension
Thus, the set can be identified as a subspace of
The resulting space is called the Moore path space of X, after John Coleman Moore, who introduced the concept.
Now, we define the product map by: for
, This product is manifestly associative.
Moreover, this monoid Ω'X acts on P'X through the original μ.
[7] Lemma — Let p: D → B, q: E → B be fibrations over an unbased space B, f: D → E a map over B.
If B is path-connected, then the following are equivalent: We apply the lemma with
if γ is the constant path, the claim follows from the lemma.
(In a nutshell, the lemma follows from the long exact homotopy sequence and the five lemma.)