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Given a Quillen adjunction with left adjoint $L: \mathcal{C} \longrightarrow \mathcal{D}$ and right adjoint $R:\mathcal{D} \longrightarrow \mathcal{C}$, then we know that $L$ preserves cofibrations and acyclic cofibrations, and equivalently, $R$ preserves fibrations and acyclic fibrations.

My question is whether there is some condition on $R$ from which we can conclude that $L$ preserves weak equivalences or acyclic fibrations? In fact, the goal is to show that $L$ preserves cofibrant replacements.

I am in the situation where $R$ is fully faithful and preserves weak equivalences, fibrations and cofibrations. Do these properties also translate to $L$?

Thanks for any leads!

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  • $\begingroup$ We may be in the situation of the XY problem here. Why exactly do you want L to preserve cofibrant replacements? Could you provide the context for this problem? What is L and R in your case? $\endgroup$
    – Dmitri Pavlov
    Aug 4, 2022 at 21:22
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    $\begingroup$ @DmitriPavlov I wasn't aware of XY problem, but yes probably. I have a morphism of operads P -->Q and it induces a Quillen adjunction between the respective algebra cats Alg(P) |- Alg(Q). The right adjoint has a really easy description for which I can prove the above properties. The left adjoint has a more difficult description, especially cohomologically. However, I do really expect the left adjoint to preserve weak equivalences. P. Tamaroff provides some criterion for when this happens for morphisms of operads, unfortunately I do not expect this to be fulfilled. Thanks for considering tho! $\endgroup$
    – Lilolance
    Aug 5, 2022 at 12:31

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