1
$\begingroup$

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.

$\endgroup$
2
  • $\begingroup$ 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? $\endgroup$ Jul 29, 2012 at 17:00
  • 1
    $\begingroup$ It is a shortcut for denoting the map $\mathcal{M}(Y,T)/\simeq \rightarrow \mathcal{M}(X,T)/\simeq$ if $f:X\rightarrow Y$. $\endgroup$ Jul 31, 2012 at 4:14

1 Answer 1

2
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.