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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?

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1 Answer 1

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This condition is known as flatness.

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 $Z$.

Various monoidal model categories of spectra are known to be flat and symmetric flat. For symmetric spectra valued in symmetric monoidal model categories is shown as Proposition 3.5.1 of arXiv:1410.5699v2.

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  • $\begingroup$ 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. $\endgroup$
    – user389759
    Commented Feb 23, 2022 at 15:55
  • $\begingroup$ @user389759: This is false for arbitrary L, but in your case it follows from what I wrote: projectively cofibrant G-objects have a free action of G, so G-coinvariants of such objects are also homotopy G-coinvariants. Since the map between them is a weak equivalence (as explained in my answer), this implies that the map between G-coinvariants is also a weak equivalence. $\endgroup$
    – Dmitri P.
    Commented Feb 23, 2022 at 17:48
  • $\begingroup$ I see. Thanks for the clarification! $\endgroup$
    – user389759
    Commented Feb 24, 2022 at 11:58

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