Path space fibration


In algebraic topology, the path space fibration over a based space is a fibration of the form
where
The space consists of all maps from I to X that may not preserve the base points; it is called the free path space of X and the fibration given by, say,, is called the free path space fibration.
The path space fibration can be understood to be dual to the mapping cone. The reduced fibration is called the mapping fiber or, equivalently, the homotopy fiber.

Mapping path space

If is any map, then the mapping path space of is the pullback of the fibration along. Since a fibration pullbacks to a fibration, if Y is based, one has the fibration
where and is the homotopy fiber, the pullback of the fibration along.
Note also is the composition
where the first map sends x to ; here denotes the constant path with value. Clearly, is a homotopy equivalence; thus, the above decomposition says that any map is a fibration up to homotopy equivalence.
If is a fibration to begin with, then the map is a fiber-homotopy equivalence and, consequently, the fibers of over the path-component of the base point are homotopy equivalent to the homotopy fiber of.

Moore's path space

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 such that is the path given by:
This product, in general, fails to be associative on the nose:, as seen directly. One solution to this failure is to pass to homotopy classes: one has. 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.
Given a based space, we let
An element f of this set has a unique extension to the interval such that. 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. Then, just as before, there is a fibration, Moore's path space fibration:
where p sends each f: → X to f and is the fiber. It turns out that and are homotopy equivalent.
Now, we define the product map:
by: for and,
This product is manifestly associative. In particular, with μ restricted to ΩX × ΩX, we have that ΩX is a topological monoid. Moreover, this monoid ΩX acts on PX through the original μ. In fact, is an Ω'X-fibration.