Suppose that we have a simplicial model category $M$. The simplicial enrichment will be denoted by $map_{M}$. Let $f:A\rightarrow B$ be a morphism in the category $M$ such that $A$ is cofibrant. Suppose that for any fibrant object $R$, the induced map $map_{M}(B,R)\rightarrow map_{M}(A,R)$ is a weak homotopy equivalence of simplicial sets. Do we have that $f$ is a weak equivalence ?
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Yes. This is Proposition 9.7.1 in Hirschhorn's book. You don't even need $A$ to be cofibrant.
answered Aug 31, 2018 at 17:51
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$\begingroup$ Thank you! Just before accepting the answer, I would like to clarify some points. To conclude that $f:A\rightarrow B$ is a weak equivalence do we need to test for all fibrant objects R or it is enough to take one fibrant object in each equivalence classe? I do mean the following, $f:A\rightarrow B$ is a weak equivalence if the induced map $map_{M}(B,R)\rightarrow map_{M}(A,R)$ is a weak homotopy equivalence for any representative fibrant object $R$ ? $\endgroup$– LetCommented Aug 31, 2018 at 18:51
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$\begingroup$ Now that you know this subject is treated so nicely in Hirschhorn's Chapter 9, why don't you read it? The answer to your followup question is very close to the proposition in my answer. $\endgroup$ Commented Sep 1, 2018 at 3:17