[[Topological retraction]]
# Deformation retraction

Let $Y \sube X$ be a subspace and $\iota : Y \to X$ the inclusion.
A **deformation retraction** $r: X \to Y$ is a
is a [[Topological retraction|retraction]], i.e. $r \iota = \id_{Y}$,
such that $\iota  r \simeq \id_{X}$ #m/def/homotopy 
$$
\begin{align*}
r\iota = \id_{Y} \qquad [\iota r] = [\id_{X}]
\end{align*}
$$
Hence in [[Category of topological spaces]], $r$ is a left inverse of $\iota$, but in [[Naïve homotopy category]] $[r]$ is a proper inverse of $[\iota]$.

## Properties

- Clearly if $Y$ is a deformation retract of $X$, $Y \simeq X$.
  Thus a deformation retraction is a stronger kind of [[Homotopy equivalence]].

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