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The induced map most likely isn’t a homeomorphism when $X, Y$ are locally compact Hausdorff.
The original statement was in monograph by Postnikov without proof.
Not only that, in the current form it couldn’t possibly be true, since the map could lack to be bijective.
For more details see here: https://math.stackexchange.com/questions/3934265/adjunction-of-pointed-maps-is-a-homeomorphism .
I’ve added a reference in the case when $X, Y$ are compact Hausdorff though.
Adam
Thanks for the correction at the end, Adam (it was a simple oversight from whoever wrote that). I would have expected that the statement you are talking (this is in the context of pointed spaces and maps) would go through assuming $X$ is locally compact, but I’ll have a closer look.
Following on #2, let me try to say a few things. According to Cagliari, if $X$ is a pointed space, then
$X \wedge -: Top_\ast \to Top_\ast$preserves colimits provided that $X \times -: Top \to Top$ preserves colimits (now forgetting the basepoint of $X$). It is well-known that $X \times -: Top \to Top$ preserves colimits if $X$ is locally compact (the precise necessary and sufficient condition is that $X$ is core compact). Now $Top_\ast$ is topological over $Set_\ast$. Since $Set_\ast$ is a total category (being for example monadic over $Set$), it would follow that $Top_\ast$ is as well. From that it would follow from that $X \wedge -: Top_\ast \to Top_\ast$, being cocontinuous, has a right adjoint $G$. We can calculate $G$ is at the underlying set-level: if we put $Y = \{0, 1\}$ with basepoint $1$, then
$Top_{\ast}(X, Z) \cong Top_\ast(X \wedge Y, Z) \cong Top_\ast(Y, G Z) \cong {|G Z|}$so that the points of $G Z$ may be identified with basepoint-preserving maps $X \to Z$. The topology on $G Z$ is of course uniquely determined from the natural isomorphism
$Top_\ast(X \wedge -, Z) \cong Top_\ast(-, G Z)$Let us denote the pointed space $G Z$ by $Map_\ast(X, Z)$. There is a question of whether we can lift the natural isomorphism of hom-sets
$Top_\ast(X \wedge Y, Z) \cong Top_\ast(Y, Map_\ast(X, Z)$to the spatial level, where we have
$Map_\ast(X \wedge Y, Z) \cong Map_\ast(Y, Map_\ast(X, Z)) \qquad (\ast)$The answer would be ’yes’ if all three of $X, Y, X \wedge Y$ are locally compact – this would be a simple Yoneda argument, together with associativity of smash product in the presence of the local compactness assumptions. I don’t think the answer would necessarily be ’yes’ if only $X, Y$ are locally compact: the smash product $X \wedge Y$ is likely only compactly generated. But if $X, Y$ are compact, it should be fine, since $X \wedge Y$ will again be compact. I don’t believe Hausdorffness needs to enter the fray, although it’s true that the smash product of two compact Hausdorff spaces is again compact Hausdorff, because the equivalence relation on $X \times Y$ needed to form the quotient $X \times Y \to X \wedge Y$ is a closed equivalence relation.
Of course, a lot of this discussion is nudging us in the direction of working with a convenient category of topological spaces in the first place, such as some variant of compactly generated spaces, where the topologies can be probed by maps from compact spaces. Over at smash product, some work of Elmendorf and Mandell is quoted which would assure the isomorphism $(\ast)$ in such a convenient setting.
All this being said, however, I think some more study is needed before making further edits to the page. In particular, I want to see how this material for the pointed space section jibes with the material of the previous section (which I’m a little rusty on myself).
added these pointers:
Discussion of exponential objects in slice categories of compactly generated topological spaces (towards local cartesian closure):
Peter I. Booth, Ronnie Brown, Spaces of partial maps, fibred mapping spaces and the compact-open topology, General Topology and its Applications 8 2 (1978) 181-195 $[$doi:10.1016/0016-660X(78)90049-1$]$
Peter I. Booth, Ronnie Brown, On the application of fibred mapping spaces to exponential laws for bundles, ex-spaces and other categories of maps, General Topology and its Applications 8 2 (1978) 165-179 $[$doi:10.1016/0016-660X(78)90048-X$]$
Peter May, Johann Sigurdsson, §1.3.7-§1.3.9 in: Parametrized Homotopy Theory, Mathematical Surveys and Monographs, vol. 132, AMS 2006 (ISBN:978-0-8218-3922-5, arXiv:math/0411656, pdf)
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