Homotopy lifting property

Assume all maps are continuous functions between topological spaces. Given a map \pi\colon E \to B, and a space Y\,, one says that (Y, \pi) has the homotopy lifting property, or that \pi\, has the homotopy lifting property with respect to Y, if: • for any homotopy f_\bullet \colon Y \times I \to B, and • for any map \tilde{f}_0 \colon Y \to E lifting f_0 = f_\bullet|_{Y\times\{0\}} (i.e., so that f_\bullet\circ \iota_0 = f_0 = \pi\circ\tilde{f}_0), there exists a homotopy \tilde{f}_\bullet \colon Y \times I \to E lifting f_\bullet (i.e., so that f_\bullet = \pi\circ\tilde{f}_\bullet) which also satisfies \tilde{f}_0 = \left.\tilde{f}\right|_{Y\times\{0\}}. The following diagram depicts this situation: The outer square (without the dotted arrow) commutes if and only if the hypotheses of the lifting property are true. A lifting \tilde{f}_\bullet corresponds to a dotted arrow making the diagram commute. This diagram is dual to that of the homotopy extension property; this duality is loosely referred to as Eckmann–Hilton duality. If the map \pi satisfies the homotopy lifting property with respect to all spaces Y, then \pi is called a fibration, or one sometimes simply says that \pi has the homotopy lifting property. A weaker notion of fibration is Serre fibration, for which homotopy lifting is only required for all CW complexes Y. ==Generalization: homotopy lifting extension property==

There is a common generalization of the homotopy lifting property and the homotopy extension property. Given a pair of spaces X \supseteq Y, for simplicity we denote T \mathrel{:=} (X \times \{0\}) \cup (Y \times [0, 1]) \subseteq X\times [0, 1]. Given additionally a map \pi \colon E \to B, one says that (X, Y, \pi) has the homotopy lifting extension property if: • For any homotopy f \colon X \times [0, 1] \to B, and • For any lifting \tilde g \colon T \to E of g = f|_T, there exists a homotopy \tilde f \colon X \times [0, 1] \to E which covers f (i.e., such that \pi\tilde f = f) and extends \tilde g (i.e., such that \left.\tilde f\right|_T = \tilde g). The homotopy lifting property of (X, \pi) is obtained by taking Y = \emptyset, so that T above is simply X \times \{0\}. The homotopy extension property of (X, Y) is obtained by taking \pi to be a constant map, so that \pi is irrelevant in that every map to E is trivially the lift of a constant map to the image point of \pi. ==See also==