# When does a left Quillen functor preserve weak equivalences?

I am interested in an answer to the following question: suppose we have a left Quillen functor $$L: \mathcal{C} \rightarrow \mathcal{D}$$ between symmetric monoidal model categories $$\mathcal{C}$$ and $$\mathcal{D}$$ with symmetric monoidal product $$\otimes$$. I want to show that taking the "tensor product" with a cofibrant object $$X$$ and then applying the left Quillen functor preserves weak equivalences, i.e. if $$Y \rightarrow Z$$ is a weak equivalence in $$\mathcal{C}$$, I want $$L(Y \otimes X) \rightarrow L(Z \otimes X)$$ to be a weak equivalence in $$\mathcal{D}$$.

The reason I am interested in this is the orbit functor from the category of (naive) G-spectra to spectra: this is a left Quillen functor and I want to show that taking the smash product with a cofibrant spectrum $$\mathbf{X}$$ and then taking orbits preserves weak equivalences.

Of course if $$Y \otimes X$$ and $$Z \otimes X$$ are cofibrant, we are done, since left Quillen functors preserve weak equivalences between cofibrant objects. But I don't see why this should be true for arbitrary $$Y$$ and $$Z$$, since in general I would not expect $$Y \otimes X$$ or $$Z \otimes X$$ to be cofibrant if $$X$$ is. I am willing to assume that the functor $$- \otimes X$$ preserves weak equivalences. Is this assumption enough to prove the above, or are there additional assumptions needed?

A monoidal model category is flat if $$X⊗f$$ is a weak equivalence whenever $$X$$ is a cofibrant object and $$f:Y→Z$$ is a weak equivalence.
A symmetric monoidal model category is symmetric flat if $$X^{Σ_n}⊗_{Σ_n}f$$ is a weak equivalence whenever $$X$$ is a cofibrant object and $$f:Y→Z$$ is a weak equivalence, which is also $$Σ_n$$-equivariant with respect to some given $$Σ_n$$-actions on $$Y$$ and $$Z$$.
• Thanks, for the answer but this does not answer my question yet (maybe I have not formulated my question clear enough): Assuming that our category is flat, is it true that a left Quillen functor L maps the weak equivalence $Y \otimes X \rightarrow Z \otimes X$ to a weak equivalence? I haven't found a proof and I am not sure if this is possible in this generality or if there are there further assumptions needed. Commented Feb 23, 2022 at 15:55