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.). 



Cartesian-closed category of spaces with the Whitehead property?
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?


