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.).