# detecting weak equivalences in a simplicial model category

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 ?

Yes. This is Proposition 9.7.1 in Hirschhorn's book. You don't even need $A$ to be cofibrant.
• 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 $R$ ? – TTip Aug 31 '18 at 18:51