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Path space fibration

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In algebraic topology, the path space fibration over a pointed space [1] is a fibration of the form[2]

where

  • is the based path space of the pointed space ; that is, equipped with the compact-open topology.
  • is the fiber of over the base point of ; thus it is the loop space of .

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 given by, say, , is called the free path space fibration.

The path space fibration can be understood to be dual to the mapping cone.[clarification needed] The fiber of the based fibration is called the mapping fiber or, equivalently, the homotopy fiber.

Mapping path space

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If is any map, then the mapping path space of is the pullback of the fibration along . (A mapping path space satisfies the universal property that is dual to that of a mapping cylinder, which is a push-out. Because of this, a mapping path space is also called a mapping cocylinder.[3])

Since a fibration pulls back 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,[4] the fibers of over the path-component of the base point are homotopy equivalent to the homotopy fiber of .

Moore's path space

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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.[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.)

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 to 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 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, is an Ω'X-fibration.[7]

Notes

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  1. ^ Throughout the article, spaces are objects of the category of "reasonable" spaces; e.g., the category of compactly generated weak Hausdorff spaces.
  2. ^ Davis & Kirk 2001, Theorem 6.15. 2.
  3. ^ Davis & Kirk 2001, § 6.8.
  4. ^ using the change of fiber
  5. ^ Whitehead 1978, Ch. III, § 2.
  6. ^ Lurie, Jacob (October 30, 2009). "Derived Algebraic Geometry VI: E[k]-Algebras" (PDF).
  7. ^ Let G = Ω'X and P = P'X. That G preserves the fibers is clear. To see, for each γ in P, the map is a weak equivalence, we can use the following lemma:

    Lemma — Let p: DB, q: EB be fibrations over an unbased space B, f: DE a map over B. If B is path-connected, then the following are equivalent:

    • f is a weak equivalence.
    • is a weak equivalence for some b in B.
    • is a weak equivalence for every b in B.

    We apply the lemma with where α is a path in P and IX is t → the end-point of α(t). Since 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.)

References

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