Path space fibration

In algebraic topology, the path space fibration over a pointed space $(X,*)$ ^[1] is a fibration of the form^[2]

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

where

$PX$ is the based path space of the pointed space $(X,*)$ ; that is, $PX=\{f\colon I\to X\mid f\ {\text{continuous}},f(0)=*\}$ equipped with the compact-open topology.
$\Omega X$ is the fiber of $\chi \mapsto \chi (1)$ over the base point of $(X,*)$ ; thus it is the loop space of $(X,*)$ .

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

Mapping path space

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

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

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

Note also $f$ is the composition

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

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

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

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

(\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: $(\gamma \cdot \beta )\cdot \alpha \neq \gamma \cdot (\beta \cdot \alpha )$ , as seen directly. One solution to this failure is to pass to homotopy classes: one has $[(\gamma \cdot \beta )\cdot \alpha ]=[\gamma \cdot (\beta \cdot \alpha )]$ . 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 $(X,*)$ , we let

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

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

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

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

Now, we define the product map

\mu :P'X\times \Omega 'X\to P'X

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

\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, $p:P'X\to X$ is an Ω'X-fibration.^[7]

Notes

^ Throughout the article, spaces are objects of the category of "reasonable" spaces; e.g., the category of compactly generated weak Hausdorff spaces.
^ Davis & Kirk 2001, Theorem 6.15. 2.
^ Davis & Kirk 2001, § 6.8.
^ using the change of fiber
^ Whitehead 1978, Ch. III, § 2.
^ Lurie, Jacob (October 30, 2009). "Derived Algebraic Geometry VI: E[k]-Algebras" (PDF).
^
Let G = Ω'X and P = P'X. That G preserves the fibers is clear. To see, for each γ in P, the map $G\to p^{-1}(p(\gamma )),\,g\mapsto \gamma g$ $G\to p^{-1}(p(\gamma )),\,g\mapsto \gamma g$ 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:
- f is a weak equivalence.
- $f:p^{-1}(b)\to q^{-1}(b)$ is a weak equivalence for some b in B.
- $f:p^{-1}(b)\to q^{-1}(b)$ is a weak equivalence for every b in B.
We apply the lemma with $B=I,D=I\times G,E=I\times _{X}P,f(t,g)=(t,\alpha (t)g)$ where α is a path in P and I → X is t → the end-point of α(t). Since $p^{-1}(p(\gamma ))=G$ 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.)

References

Davis, James F.; Kirk, Paul (2001). Lecture Notes in Algebraic Topology (PDF). Graduate Studies in Mathematics. Vol. 35. Providence, RI: American Mathematical Society. pp. xvi+367. doi:10.1090/gsm/035. ISBN 0-8218-2160-1. MR 1841974.
May, J. Peter (1999). A Concise Course in Algebraic Topology (PDF). Chicago Lectures in Mathematics. Chicago, IL: University of Chicago Press. pp. x+243. ISBN 0-226-51182-0. MR 1702278.
Whitehead, George W. (1978). Elements of homotopy theory. Graduate Texts in Mathematics. Vol. 61 (3rd ed.). New York-Berlin: Springer-Verlag. pp. xxi+744. ISBN 978-0-387-90336-1. MR 0516508.

[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 $G\to p^{-1}(p(\gamma )),\,g\mapsto \gamma g$ 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:

f is a weak equivalence.

$f:p^{-1}(b)\to q^{-1}(b)$ is a weak equivalence for some b in B.

$f:p^{-1}(b)\to q^{-1}(b)$ is a weak equivalence for every b in B.

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

[8] s a weak equivalence.

[9] $f:p^{-1}(b)\to q^{-1}(b)$ is a weak equivalence for some b in B.

[10] $f:p^{-1}(b)\to q^{-1}(b)$ is a weak equivalence for every b in B.

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