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  1. The induced map most likely isn’t a homeomorphism when X,YX, 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,YX, Y are compact Hausdorff though.

    Adam

    diff, v13, current

    • CommentRowNumber2.
    • CommentAuthorTodd_Trimble
    • CommentTimeDec 22nd 2020

    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 XX is locally compact, but I’ll have a closer look.

    • CommentRowNumber3.
    • CommentAuthorTodd_Trimble
    • CommentTimeDec 23rd 2020
    • (edited Dec 23rd 2020)

    Following on #2, let me try to say a few things. According to Cagliari, if XX is a pointed space, then

    X:Top *Top *X \wedge -: Top_\ast \to Top_\ast

    preserves colimits provided that X×:TopTopX \times -: Top \to Top preserves colimits (now forgetting the basepoint of XX). It is well-known that X×:TopTopX \times -: Top \to Top preserves colimits if XX is locally compact (the precise necessary and sufficient condition is that XX is core compact). Now Top *Top_\ast is topological over Set *Set_\ast. Since Set *Set_\ast is a total category (being for example monadic over SetSet), it would follow that Top *Top_\ast is as well. From that it would follow from that X:Top *Top *X \wedge -: Top_\ast \to Top_\ast, being cocontinuous, has a right adjoint GG. We can calculate GG is at the underlying set-level: if we put Y={0,1}Y = \{0, 1\} with basepoint 11, then

    Top *(X,Z)Top *(XY,Z)Top *(Y,GZ)|GZ|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 GZG Z may be identified with basepoint-preserving maps XZX \to Z. The topology on GZG Z is of course uniquely determined from the natural isomorphism

    Top *(X,Z)Top *(,GZ)Top_\ast(X \wedge -, Z) \cong Top_\ast(-, G Z)

    Let us denote the pointed space GZG Z by Map *(X,Z)Map_\ast(X, Z). There is a question of whether we can lift the natural isomorphism of hom-sets

    Top *(XY,Z)Top *(Y,Map *(X,Z)Top_\ast(X \wedge Y, Z) \cong Top_\ast(Y, Map_\ast(X, Z)

    to the spatial level, where we have

    Map *(XY,Z)Map *(Y,Map *(X,Z))(*)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,XYX, 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,YX, Y are locally compact: the smash product XYX \wedge Y is likely only compactly generated. But if X,YX, Y are compact, it should be fine, since XYX \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×YX \times Y needed to form the quotient X×YXYX \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.

    • CommentRowNumber4.
    • CommentAuthorTodd_Trimble
    • CommentTimeDec 23rd 2020

    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).

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTime7 days ago

    added these pointers:


    Discussion of exponential objects in slice categories of compactly generated topological spaces (towards local cartesian closure):

    diff, v15, current

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTime6 days ago

    While I was at it, I have added missing data, links and formatting to the existing list of references, and organized by date of appearance.

    diff, v17, current

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