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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.

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  • $\begingroup$ I haven't seen the notation M(f,T) before. What kind of object is it? I thought it should be the set of all commutative diagrams made of f: X -> Y and some X -> T and Y -> T.. but then it doesn't make much sense to me. Would you mind to explain this a little bit? $\endgroup$
    – Konrad Voelkel
    Commented Jul 29, 2012 at 17:00
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    $\begingroup$ It is a shortcut for denoting the map $\mathcal{M}(Y,T)/\simeq \rightarrow \mathcal{M}(X,T)/\simeq$ if $f:X\rightarrow Y$. $\endgroup$
    – Philippe Gaucher
    Commented Jul 31, 2012 at 4:14

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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.

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