Path space fibration

In algebraic topology, the path space fibration over a based space formula_1 is a fibration of the form
where
The space formula_6 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 formula_7 given by, say, formula_5, 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 formula_9 is any map, then the mapping path space formula_10 of formula_11 is the pullback of the fibration formula_12 along formula_11. Since a fibration pullbacks to a fibration, if "Y" is based, one has the fibration
where formula_15 and formula_16 is the homotopy fiber, the pullback of the fibration formula_17 along formula_11.
Note also formula_11 is the composition
where the first map formula_21 sends "x" to formula_22; here formula_23 denotes the constant path with value formula_24. Clearly, formula_21 is a homotopy equivalence; thus, the above decomposition says that any map is a fibration up to homotopy equivalence.
If formula_11 is a fibration to begin with, then the map formula_27 is a fiber-homotopy equivalence and, consequently, the fibers of formula_11 over the path-component of the base point are homotopy equivalent to the homotopy fiber formula_16 of formula_11.
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 formula_31 such that formula_32 is the path formula_33 given by:
This product, in general, fails to be associative on the nose: formula_35, as seen directly. One solution to this failure is to pass to homotopy classes: one has formula_36. 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. (A more sophisticated solution is to "rethink" composition: work with an arbitrary family of compositions; see the introduction of Lurie's paper, leading to the notion of an operad.)
Given a based space formula_1, we let
An element "f" of this set has a unique extension formula_39 to the interval formula_40 such that formula_41. Thus, the set can be identified as a subspace of formula_42. 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": [0, "r"] → "X" to "f"("r") and formula_44 is the fiber. It turns out that formula_4 and formula_46 are homotopy equivalent.
Now, we define the product map:
by: for formula_48 and formula_49,
This product is manifestly associative. In particular, with "μ" restricted to Ω"X" × Ω"X", we have that Ω"X" is a topological monoid (in the category of all spaces). Moreover, this monoid Ω"X" acts on "P'X" through the original "μ". In fact, formula_51 is an Ω'"X"-fibration.