Path space fibration

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.) 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, 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 \le t \le 1/2 \\ \beta(2t-1) & \text{if } 1/2 \le t \le 1 \\ \end{cases}. This product, in general, fails to be associative on the nose: (\gamma \cdot \beta) \cdot \alpha \ne \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. (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 (X, *), we let :P' X = \{ f\colon [0, r] \to X \mid r \ge 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 \ge 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 \le t \le r \\ g(t-r) & \text{if } r \le t \le s + r \\ \end{cases}. This product is manifestly associative. In particular, with μ restricted to ΩX × ΩX, we have that ΩX is a topological monoid (in the category of all spaces). Moreover, this monoid ΩX acts on PX through the original μ. In fact, p: P'X \to X is an Ω'X-fibration. == Notes ==