In algebraic topology, the '''path space fibration''' over a pointed space <math>(X, *)</math><ref>Throughout the article, spaces are objects of the category of "reasonable" spaces; e.g., the category of compactly generated weak Hausdorff spaces.</ref> is a fibration of the form<ref>{{harvnb|Davis|Kirk|2001|loc=Theorem 6.15. 2.}}</ref> :<math>\Omega X \hookrightarrow PX \overset{\chi \mapsto \chi(1)}\to X</math> where *<math>PX</math> is the ''based'' path space of the pointed space <math>(X, *)</math>; that is, <math>PX = \{ f\colon I \to X \mid f \ \text{continuous}, f(0) = * \}</math> equipped with the compact-open topology. *<math>\Omega X</math> is the fiber of <math>\chi \mapsto \chi(1)</math> over the base point of <math>(X, *)</math>; thus it is the loop space of <math>(X, *)</math>.
The ''free'' path space of ''X'', that is, <math>\operatorname{Map}(I, X) = X^I</math>, consists of all maps from ''I'' to ''X'' that do not necessarily begin at a base point, and the fibration <math>X^I \to X</math> given by, say, <math>\chi \mapsto \chi(1)</math>, is called the '''free path space fibration'''.
The path space fibration can be understood to be dual to the mapping cone.{{clarify|more precise meaning|date=August 2022}} The fiber of the based fibration is called the mapping fiber or, equivalently, the homotopy fiber.
== Mapping path space == If <math>f\colon X\to Y</math> is any map, then the '''mapping path space''' <math>P_f</math> of <math>f</math> is the pullback of the fibration <math>Y^I \to Y, \, \chi \mapsto \chi(1)</math> along <math>f</math>. (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'''.<ref>{{harvnb|Davis|Kirk|2001|loc=§ 6.8.}}</ref>)
Since a fibration pulls back to a fibration, if ''Y'' is based, one has the fibration :<math>F_f \hookrightarrow P_f \overset{p}\to Y</math> where <math>p(x, \chi) = \chi(0)</math> and <math>F_f</math> is the homotopy fiber, the pullback of the fibration <math>PY \overset{\chi \mapsto \chi(1)}{\longrightarrow} Y</math> along <math>f</math>.
Note also <math>f</math> is the composition :<math>X \overset{\phi}\to P_f \overset{p}\to Y</math> where the first map <math>\phi</math> sends ''x'' to <math>(x, c_{f(x)})</math>; here <math>c_{f(x)}</math> denotes the constant path with value <math>f(x)</math>. Clearly, <math>\phi</math> is a homotopy equivalence; thus, the above decomposition says that any map is a fibration up to homotopy equivalence.
If <math>f</math> is a fibration to begin with, then the map <math>\phi\colon X \to P_f</math> is a fiber-homotopy equivalence and, consequently,<ref>using the change of fiber</ref> the fibers of <math>f</math> over the path-component of the base point are homotopy equivalent to the homotopy fiber <math>F_f</math> of <math>f</math>.
== 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 <math>\alpha, \beta</math> such that <math>\alpha(1) = \beta(0)</math> is the path <math>\beta \cdot \alpha\colon I \to X</math> given by: :<math>(\beta \cdot \alpha)(t)= \begin{cases} \alpha(2t) & \text{if } 0 \le t \le 1/2 \\ \beta(2t-1) & \text{if } 1/2 \le t \le 1 \\ \end{cases}</math>. This product, in general, fails to be associative on the nose: <math>(\gamma \cdot \beta) \cdot \alpha \ne \gamma \cdot (\beta \cdot \alpha)</math>, as seen directly. One solution to this failure is to pass to homotopy classes: one has <math>[(\gamma \cdot \beta) \cdot \alpha] = [\gamma \cdot (\beta \cdot \alpha)]</math>. 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.<ref>{{harvnb|Whitehead|1978|loc=Ch. III, § 2.}}</ref> (A more sophisticated solution is to ''rethink'' composition: work with an arbitrary family of compositions; see the introduction of Lurie's paper,<ref>{{cite web|first=Jacob|last=Lurie|authorlink=Jacob Lurie|url=http://www.math.harvard.edu/~lurie/papers/DAG-VI.pdf|title=Derived Algebraic Geometry VI: E[k]-Algebras|date=October 30, 2009}}</ref> leading to the notion of an operad.)
Given a based space <math>(X, *)</math>, we let :<math>P' X = \{ f\colon [0, r] \to X \mid r \ge 0, f(0) = * \}.</math> An element ''f'' of this set has a unique extension <math>\widetilde{f}</math> to the interval <math>[0, \infty)</math> such that <math>\widetilde{f}(t) = f(r),\, t \ge r</math>. Thus, the set can be identified as a subspace of <math>\operatorname{Map}([0, \infty), X)</math>. 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''': :<math>\Omega' X \hookrightarrow P'X \overset{p}\to X</math> where ''p'' sends each <math>f: [0, r] \to X</math> to <math>f(r)</math> and <math>\Omega' X = p^{-1}(*)</math> is the fiber. It turns out that <math>\Omega X</math> and <math>\Omega' X</math> are homotopy equivalent.
Now, we define the product map :<math>\mu: P' X \times \Omega' X \to P' X</math> by: for <math>f\colon [0, r] \to X</math> and <math>g\colon [0, s] \to X</math>, :<math>\mu(g, f)(t)= \begin{cases} f(t) & \text{if } 0 \le t \le r \\ g(t-r) & \text{if } r \le t \le s + r \\ \end{cases}</math>. 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, <math>p: P'X \to X</math> is an Ω<nowiki>'</nowiki>''X''-fibration.<ref>Let ''G'' = Ω{{'}}''X'' and ''P'' = ''P''{{'}}''X''. That ''G'' preserves the fibers is clear. To see, for each ''γ'' in ''P'', the map <math>G \to p^{-1}(p(\gamma)),\, g \mapsto \gamma g</math> is a weak equivalence, we can use the following lemma: {{math_theorem|name=Lemma|math_statement=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. *<math>f: p^{-1}(b) \to q^{-1}(b)</math> is a weak equivalence for some ''b'' in ''B''. *<math>f: p^{-1}(b) \to q^{-1}(b)</math> is a weak equivalence for every ''b'' in ''B''.}} We apply the lemma with <math>B = I, D = I \times G, E = I \times_X P, f(t, g) = (t, \alpha(t) g)</math> where ''α'' is a path in ''P'' and ''I'' → ''X'' is ''t'' → the end-point of ''α''(''t''). Since <math>p^{-1}(p(\gamma)) = G</math> 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.)</ref>
== Notes == {{reflist}}
== References == *{{cite book|first1=James F. |last1=Davis|first2= Paul|last2= Kirk|url=http://www.maths.ed.ac.uk/~aar/papers/davkir.pdf|title= Lecture Notes in Algebraic Topology|series=Graduate Studies in Mathematics|volume= 35|publisher=American Mathematical Society|location= Providence, RI|year= 2001|pages= xvi+367|isbn=0-8218-2160-1 |mr=1841974|doi=10.1090/gsm/035}} *{{cite book|last=May|first=J. Peter|authorlink=J. Peter May| url=http://www.math.uchicago.edu/~may/CONCISE/ConciseRevised.pdf |title=A Concise Course in Algebraic Topology|series=Chicago Lectures in Mathematics|publisher= University of Chicago Press|location= Chicago, IL|year= 1999|pages= x+243|isbn=0-226-51182-0|mr=1702278}} *{{cite book|first=George W.|last= Whitehead|authorlink=George W. Whitehead|title=Elements of homotopy theory|url=https://books.google.com/books?id=wlrvAAAAMAAJ|edition=3rd|series=Graduate Texts in Mathematics|volume=61|year=1978|publisher=Springer-Verlag|location=New York-Berlin|isbn=978-0-387-90336-1|pages=xxi+744|mr=0516508 }}
Category:Algebraic topology Category:Homotopy theory