Let $f:X\to Y$ be a map of topological spaces. Assume further that the homotopy fibre is contractible. We get a long exact sequence on the homotopy groups and if $X$ and $Y$ are connected $f$ is a weak equivalence.
If $X$ and $Y$ are CW complexes, then $f$ is automatically a homotopy equivalence.
Question: Can we show the existence of the homotopy inverse just using categorial methods (i.e. the homotopy pullback property) or do we really have to use Whitehead?
Moreover: If we don't assume that the spaces are CW-complexes we can't hope for $f$ to be a homotopy equivalence, can we?
Edit: Strangely enough I feel that the first part should work while the second statement also feels true. However if the first part works I don't see a reason why it should only work for CW-complexes.