Homotopy lifting property

Assume all maps are continuous functions between topological spaces. Given a map ${\displaystyle \pi \colon E\to B}$, and a space ${\displaystyle Y\,}$, one says that ${\displaystyle (Y,\pi )}$ has the homotopy lifting property,[1][2] or that ${\displaystyle \pi \,}$ has the homotopy lifting property with respect to ${\displaystyle Y}$, if:

• for any homotopy ${\displaystyle f_{\bullet }\colon Y\times I\to B}$, and
• for any map ${\displaystyle {\tilde {f}}_{0}\colon Y\to E}$ lifting ${\displaystyle f_{0}=f_{\bullet }|_{Y\times \{0\}}}$ (i.e., so that ${\displaystyle f_{\bullet }\circ \iota _{0}=f_{0}=\pi \circ {\tilde {f}}_{0}}$),

there exists a homotopy ${\displaystyle {\tilde {f}}_{\bullet }\colon Y\times I\to E}$ lifting ${\displaystyle f_{\bullet }}$ (i.e., so that ${\displaystyle f_{\bullet }=\pi \circ {\tilde {f}}_{\bullet }}$) which also satisfies ${\displaystyle {\tilde {f}}_{0}=\left.{\tilde {f}}\right|_{Y\times \{0\}}}$.

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 ${\displaystyle {\tilde {f}}_{\bullet }}$ 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 ${\displaystyle \pi }$ satisfies the homotopy lifting property with respect to all spaces ${\displaystyle Y}$, then ${\displaystyle \pi }$ is called a fibration, or one sometimes simply says that ${\displaystyle \pi }$ has the homotopy lifting property.

A weaker notion of fibration is Serre fibration, for which homotopy lifting is only required for all CW complexes ${\displaystyle Y}$.

There is a common generalization of the homotopy lifting property and the homotopy extension property. Given a pair of spaces ${\displaystyle X\supseteq Y}$, for simplicity we denote ${\displaystyle T\mathrel {:=} (X\times \{0\})\cup (Y\times [0,1])\subseteq X\times [0,1]}$. Given additionally a map ${\displaystyle \pi \colon E\to B}$, one says that ${\displaystyle (X,Y,\pi )}$ has the homotopy lifting extension property if:

• For any homotopy ${\displaystyle f\colon X\times [0,1]\to B}$, and
• For any lifting ${\displaystyle {\tilde {g}}\colon T\to E}$ of ${\displaystyle g=f|_{T}}$, there exists a homotopy ${\displaystyle {\tilde {f}}\colon X\times [0,1]\to E}$ which covers ${\displaystyle f}$ (i.e., such that ${\displaystyle \pi {\tilde {f}}=f}$) and extends ${\displaystyle {\tilde {g}}}$ (i.e., such that ${\displaystyle \left.{\tilde {f}}\right|_{T}={\tilde {g}}}$).

The homotopy lifting property of ${\displaystyle (X,\pi )}$ is obtained by taking ${\displaystyle Y=\emptyset }$, so that ${\displaystyle T}$ above is simply ${\displaystyle X\times \{0\}}$.

The homotopy extension property of ${\displaystyle (X,Y)}$ is obtained by taking ${\displaystyle \pi }$ to be a constant map, so that ${\displaystyle \pi }$ is irrelevant in that every map to E is trivially the lift of a constant map to the image point of ${\displaystyle \pi }$.

• Covering space
• Fibration

• Steenrod, Norman (1951). The Topology of Fibre Bundles. Princeton: Princeton University Press. ISBN 0-691-00548-6.
• Hu, Sze-Tsen (1959). Homotopy Theory (Third Printing, 1965 ed.). New York: Academic Press Inc. ISBN 0-12-358450-7.
• Husemoller, Dale (1994). Fibre Bundles (Third ed.). New York: Springer. ISBN 978-0-387-94087-8.
• Hatcher, Allen (2002), Algebraic Topology, Cambridge: Cambridge University Press, ISBN 0-521-79540-0.
• Jean-Pierre Marquis (2006) "A path to Epistemology of Mathematics: Homotopy theory", pages 239 to 260 in The Architecture of Modern Mathematics, J. Ferreiros & J.J. Gray, editors, Oxford University Press ISBN 978-0-19-856793-6

• A.V. Chernavskii (2001) [1994], "Covering homotopy", Encyclopedia of Mathematics, EMS Press
• homotopy lifting property at the nLab