In algebraic topology, the path space fibration over a pointed space [1] is a fibration of the form[2]
where
- is the based path space of the pointed space ; that is, equipped with the compact-open topology.
- is the fiber of over the base point of ; thus it is the loop space of .
The free path space of X, that is, , consists of all maps from I to X that do not necessarily begin at a base point, 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.[clarification needed] The fiber of the based fibration is called the mapping fiber or, equivalently, the homotopy fiber.
- ^ 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.