I would like to know how to cite this theorem (which has a quite surprising consequence):

A model category $\mathcal{M}$ is right proper if and only if for any weak equivalence $f:A\to B$, the Quillen adjunction $\Sigma_f:\mathcal{M}/ A \leftrightarrows \mathcal{M} / B:f^*$ is a Quillen equivalence.

I had never heard of it (I never read that in any textbook I believe) before I browse the nLab site which refers to this blog comment from C. Rezk.

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  • $\begingroup$ There is a general (and quite useful) lemma saying that a Quillen adjunction $L:\mathcal{M}\leftrightarrows\mathcal{N}:R$ in which $L$ preserves and detects weak equivalences is a Quillen equivalence if and only if the counit map $v: L(R(X)) \to X$ is a weak equivalence for every fibrant object $X \in \mathcal{N}$. Now observe that $\Sigma_f$ preserves and detects weak equivalences and that $\mathcal{M}$ is right proper exactly when for every weak equivalence $f: A \to B$ in $\mathcal{M}$ and every fibrant $X \in \mathcal{M}/B$ the counit map $v: \Sigma_f(f^*(X)) \to X$ is a weak equivalence. $\endgroup$ – Yonatan Harpaz Sep 8 '15 at 11:15

This is Proposition 2.7 in Rezk's Every Homotopy Theory of Simplicial Algebras Admits a Proper Model

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    $\begingroup$ Proposition 2.5 in the published version. $\endgroup$ – Zhen Lin Sep 8 '15 at 12:46
  • $\begingroup$ Thanks; I didn't have access to the published version, which is why I linked the arxiv version above. $\endgroup$ – David White Sep 8 '15 at 18:14
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    $\begingroup$ It is freely available on the Web now (I mean legally freely available) : doi.org/10.1016/S0166-8641(01)00057-8 $\endgroup$ – Philippe Gaucher May 23 '17 at 7:43

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