Is the homotopy category cartesian closed? The homotopy category $\mathsf{hTop}$ (topological spaces localized at the weak homotopy equivalences) doesn't have many limits / colimits, but it does have products, computed on the point-set level.
Now, on the point-set level, (compactly generated) spaces are cartesian closed. Cartesian closure in some sense only involves finite products, which are preserved in the passage to the homotopy category. So we might hope that the homotopy category has exponentials, computed on the point-set level.
 However, there seems to be a problem. Consider the unit interval $I$. Then $I \times (-): \mathsf{hTop} \to \mathsf{hTop}$ is isomorphic to the identity functor. But the exponential functor $(-)^I: \mathsf{hTop} \to \mathsf{hTop}$ sends $X$ to $X^I$, which is isomorphic in $\mathsf{hTop}$ to the discrete space on the path components of $X$, since the path space of a path-connected space is contractible. These functors are clearly not adjoint -- the adjoint of the identity functor is also the identity functor. So if there is an exponential on $\mathsf{hTop}$, it is not computed at the point-set level.
But perhaps this can be salvaged? Is the homotopy category cartesian closed?
EDIT As discussed in the comments, I'm simply confusing based and unbased path spaces, which are very, very different. So most of what I wrote was wrong: the unbased path space $X^I$ is homotopy equivalent to $X$ via the "constant path map" $X \to X^I$. So $I \times(-)$ and $(-)^I$, viewed as functors on $\mathsf{hTop}$, both become isomorphic to the identity functor and hence remain adjoint. So perhaps $\mathsf{hTop}$ does indeed have a cartesian closed structure computed as at the point-set level.
 A: Yes.
Let $i_0,i_1:*\to I$ be the inclusions of the endpoints of the interval, and denote composition with these two maps by $ev_0,ev_1:Top(A\times I,Z)\to Top(A,Z)$.
The diagram
$$\require{AMScd}
\begin{CD}
Top(A\times I,X\times Y) @>{ev_0}>> Top(A,X\times Y) \\
@V{-\circ id_A\times 1}VV \\
Top(A,X\times Y)
\end{CD}$$
is the same as the diagram
$$\begin{CD}
Top(A\times I,X)\times Top(A\times I,Y) @>{(ev_0,ev_0)}>> Top(A,X)\times Top(A,Y) \\
@V{(ev_1,ev_1)}VV \\
Top(A,X)\times Top(A,Y)
\end{CD}$$
The pushout of the first diagram is $Ho(Top)(A,X\times Y)$, and of the second diagram is $Ho(Top)(A,X)\times Ho(Top)(A,Y)$. This shows that $X\times Y$ still has the universal property of the prouct in the homotopy category.
Similarly, $Ho(Top)(A,X^Y)$ is a certain pushout over $Top(A\times I,X^Y)$, which using the cartesian closedness of $Top$ can be rewritten as the pushout yielding $Ho(Top)(A\times Y,X)$. Thus $X^Y$ has the exponential universal property in $Ho(Top)$ as well as in $Top$, and we conclude the homotopy category is cartesian closed.
