path inside the mapping space as homotopy

Let \(X,Y\) be topological spaces, where \(X\) is locally compact and \(\varphi: A \rightarrow Y\) a continuous map for \(A \subseteq X\). Then a homotopy relative to \(\varphi\) is precisely given by a path

\begin{align*} \gamma: [0,1] \rightarrow \mathrm{Map}_A(X,Y) \end{align*}

by continuous map and restriction to the image, we may embed w.l.o.g. \(\mathrm{Map}_A(X,Y) \hookrightarrow \mathrm{Map}(X,Y)\) and thus also

\begin{align*} \mathrm{Map}([0,1], \mathrm{Map}_A(X,Y)) \hookrightarrow \mathrm{Map}([0,1], \mathrm{Map}(X,Y)) \end{align*}

(see: covariant compact open functor)

by currying of mapping spaces of locally compact spaces as homeomorphism we then get

\begin{align*} \mathrm{Map}([0,1], \mathrm{Map}_A(X,Y)) \cong \mathrm{Map}([0,1] \times X, Y) \end{align*}

Thus

\begin{align*} \mathrm{Map}([0,1], \mathrm{Map}_A(X,Y)) \arrow[hookrightarrow]{r}{\iota} & \mathrm{Map}([0,1], \mathrm{Map}(X,Y)) \arrow[]{r}{\cong}& \mathrm{Map}([0,1] \times X, Y) \end{align*}

We may again restrict the map to the image and get a Bijection

\begin{align*} \mathrm{Map}([0,1], \mathrm{Map}_A(X,Y)) \cong \mathrm{Map}_{[0,1] \times A}([0,1] \times X, Y) \end{align*}

Here the restiction is given by the construction of the homeomorphism