Path space fibration

In algebraic topology, the path space fibration over a based space is a fibration of the form where is the path space of X; i.e., equipped with the compact-open topology. is the fiber of over the base point of X; thus it is the loop space of X. The space consists of all maps from I to X that may not preserve the base points; it is called the free path space of X 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. The reduced fibration is called the mapping fiber or, equivalently, the homotopy fiber. 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.) 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, the fibers of over the path-component of the base point are homotopy equivalent to the homotopy fiber of . 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. (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.