TITLE: Cartesian-closed category of spaces with the Whitehead property?
QUESTION [3 upvotes]: I'm not sure if this is standard, but we'll call the property that every weak homotopy equivalence is an honest homotopy equivalence the Whitehead property (from Whitehead's theorem for CW complexes).  
Then the question: Is there any (nontrivial) category of spaces that is cartesian-closed and has the Whitehead property?  If not, is there some counterexample we can construct to show that these properties are mutually exclusive?
I am familiar with the category of m-cofibrant spaces, but it is not cartesian closed (even though there is the weaker result allowing us to take loop spaces etc.).

REPLY [3 votes]: This is only a partial answer Harry, but, to my knowledge, all Cartesian-closed categories of topological spaces that I know of arise as the monocoreflective hull (in either Top or Hausdorff spaces etc.) of a productive class $C$ of generating exponentiable spaces. The requirement to be a productive class is that binary products of generators lie in the monocoreflective hull. Also, it is a well known theorem that a space is exponentiable if and only if it is core-compact. So, for example if $C$ is compact Hausdorff space, we get the category of compactly generated spaces as its monocoreflective hull. If you tried to use this machinery to produce a Cartesian-closed category of "whitehead spaces", I think it would fail: a natural choice would be compact Hausdorff m-cofibrant spaces for C- this should be a productive class (if I am not making a silly mistake), BUT, I don't think that being homotopy equivalent to a CW-complex is stable under colimits, as CW-complexes aren't. So, my guess of an answer would be "No." However, this is hardly a proof. By the way, what did you want to do with these spaces anyhow?
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