I need a bibliographical reference for this fact: let $\mathcal{M}$ be a model category such that all objects are cofibrant; then the class of weak equivalences is the class of maps f such that $\mathcal{M}(f,T)/\simeq$ is a bijection for any fibrant object $T$ where $\simeq$ is the homotopy relation. I would prefer a reference in Hirschhorn's book (I have it but I cannot find where it is proved). Thanks in advance.
Take the 2minute tour
×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

This is Theorem 7.8.6 on page 133 of Hirschhorn. The first direction (that any weak equivalence $f$ gives a bijection $\mathcal{M}(f,T)/\sim$, for $T$ fibrant) is Corollary 7.7.4(1). So the proof of the theorem is really just the proof of the other implication. 

