# Homotopy extension property

In mathematics, in the area of algebraic topology, the homotopy extension property indicates which homotopies defined on a subspace can be extended to a homotopy defined on a larger space.

## Definition

Let ${\displaystyle X\,\!}$ be a topological space, and let ${\displaystyle A\subset X}$. We say that the pair ${\displaystyle (X,A)\,\!}$ has the homotopy extension property if, given a homotopy ${\displaystyle f_{t}\colon A\rightarrow Y}$ and a map ${\displaystyle F_{0}\colon X\rightarrow Y}$ such that ${\displaystyle F_{0}|_{A}=f_{0}}$, there exists an extension of ${\displaystyle F_{0}}$ to a homotopy ${\displaystyle F_{t}\colon X\rightarrow Y}$ such that ${\displaystyle F_{t}|_{A}=f_{t}}$. [1]

That is, the pair ${\displaystyle (X,A)\,\!}$ has the homotopy extension property if any map ${\displaystyle G\colon ((X\times \{0\})\cup (A\times I))\rightarrow Y}$ can be extended to a map ${\displaystyle G'\colon X\times I\rightarrow Y}$ (i.e. ${\displaystyle G\,\!}$ and ${\displaystyle G'\,\!}$ agree on their common domain).

If the pair has this property only for a certain codomain ${\displaystyle Y\,\!}$, we say that ${\displaystyle (X,A)\,\!}$ has the homotopy extension property with respect to ${\displaystyle Y\,\!}$.

## Visualisation

The homotopy extension property is depicted in the following diagram

If the above diagram (without the dashed map) commutes, which is equivalent to the conditions above, then there exists a map ${\displaystyle {\tilde {f}}}$ which makes the diagram commute. By currying, note that a map ${\displaystyle {\tilde {f}}\colon X\to Y^{I}}$ is the same as a map ${\displaystyle {\tilde {f}}\colon X\times I\to Y}$.

Also compare this to the visualization of the homotopy lifting property.

## Other

If ${\displaystyle \mathbf {\mathit {(X,A)}} }$ has the homotopy extension property, then the simple inclusion map ${\displaystyle i:A\to X}$ is a cofibration.

In fact, if you consider any cofibration ${\displaystyle i:Y\to Z}$, then we have that ${\displaystyle {\mathbf {\mathit {Y}} }}$ is homeomorphic to its image under ${\displaystyle {\mathbf {\mathit {i}} }}$. This implies that any cofibration can be treated as an inclusion map, and therefore it can be treated as having the homotopy extension property.