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Path space fibration - Wikipedia

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In algebraic topology, the path space fibration over a pointed space {\displaystyle (X,*)}[1] is a fibration of the form[2]

{\displaystyle \Omega X\hookrightarrow PX{\overset {\chi \mapsto \chi (1)}{\to }}X}

where

The free path space of X, that is, {\displaystyle \operatorname {Map} (I,X)=X^{I}}, consists of all maps from I to X that do not necessarily begin at a base point, and the fibration {\displaystyle X^{I}\to X} given by, say, {\displaystyle \chi \mapsto \chi (1)}, 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.

If {\displaystyle f\colon X\to Y} is any map, then the mapping path space {\displaystyle P_{f}} of {\displaystyle f} is the pullback of the fibration {\displaystyle Y^{I}\to Y,\,\chi \mapsto \chi (1)} along {\displaystyle f}. (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

{\displaystyle F_{f}\hookrightarrow P_{f}{\overset {p}{\to }}Y}

where {\displaystyle p(x,\chi )=\chi (0)} and {\displaystyle F_{f}} is the homotopy fiber, the pullback of the fibration {\displaystyle PY{\overset {\chi \mapsto \chi (1)}{\longrightarrow }}Y} along {\displaystyle f}.

Note also {\displaystyle f} is the composition

{\displaystyle X{\overset {\phi }{\to }}P_{f}{\overset {p}{\to }}Y}

where the first map {\displaystyle \phi } sends x to {\displaystyle (x,c_{f(x)})}; here {\displaystyle c_{f(x)}} denotes the constant path with value {\displaystyle f(x)}. Clearly, {\displaystyle \phi } is a homotopy equivalence; thus, the above decomposition says that any map is a fibration up to homotopy equivalence.

If {\displaystyle f} is a fibration to begin with, then the map {\displaystyle \phi \colon X\to P_{f}} is a fiber-homotopy equivalence and, consequently,[4] the fibers of {\displaystyle f} over the path-component of the base point are homotopy equivalent to the homotopy fiber {\displaystyle F_{f}} of {\displaystyle f}.

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 {\displaystyle \alpha ,\beta } such that {\displaystyle \alpha (1)=\beta (0)} is the path {\displaystyle \beta \cdot \alpha \colon I\to X} given by:

{\displaystyle (\beta \cdot \alpha )(t)={\begin{cases}\alpha (2t)&{\text{if }}0\leq t\leq 1/2\\\beta (2t-1)&{\text{if }}1/2\leq t\leq 1\\\end{cases}}}.

This product, in general, fails to be associative on the nose: {\displaystyle (\gamma \cdot \beta )\cdot \alpha \neq \gamma \cdot (\beta \cdot \alpha )}, as seen directly. One solution to this failure is to pass to homotopy classes: one has {\displaystyle [(\gamma \cdot \beta )\cdot \alpha ]=[\gamma \cdot (\beta \cdot \alpha )]}. 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 {\displaystyle (X,*)}, we let

{\displaystyle P'X=\{f\colon [0,r]\to X\mid r\geq 0,f(0)=*\}.}

An element f of this set has a unique extension {\displaystyle {\widetilde {f}}} to the interval {\displaystyle [0,\infty )} such that {\displaystyle {\widetilde {f}}(t)=f(r),\,t\geq r}. Thus, the set can be identified as a subspace of {\displaystyle \operatorname {Map} ([0,\infty ),X)}. 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:

{\displaystyle \Omega 'X\hookrightarrow P'X{\overset {p}{\to }}X}

where p sends each {\displaystyle f:[0,r]\to X} to {\displaystyle f(r)} and {\displaystyle \Omega 'X=p^{-1}(*)} is the fiber. It turns out that {\displaystyle \Omega X} and {\displaystyle \Omega 'X} are homotopy equivalent.

Now, we define the product map

{\displaystyle \mu :P'X\times \Omega 'X\to P'X}

by: for {\displaystyle f\colon [0,r]\to X} and {\displaystyle g\colon [0,s]\to X},

{\displaystyle \mu (g,f)(t)={\begin{cases}f(t)&{\text{if }}0\leq t\leq r\\g(t-r)&{\text{if }}r\leq t\leq s+r\\\end{cases}}}.

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, {\displaystyle p:P'X\to X} is an Ω'X-fibration.[7]

  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 {\displaystyle G\to p^{-1}(p(\gamma )),\,g\mapsto \gamma g} is a weak equivalence, we can use the following lemma:

    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 {\displaystyle B=I,D=I\times G,E=I\times _{X}P,f(t,g)=(t,\alpha (t)g)} where α is a path in P and I → X is t → the end-point of α(t). Since {\displaystyle p^{-1}(p(\gamma ))=G} 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.)