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Schwede-Shipley introduced the notion of weak monoidal Quillen equivalences between monoidal model categories in "Equivalences of monoidal model categories". Are there any examples of such Quillen equivalences which are not the Dold-Kan correspondance or a strong monoidal Quillen equivalence?

Recall that a Quillen equivalence between two monoidal model categories $(\mathcal{A}, \otimes, \mathbb{I})$ and $(\mathcal{B}, \wedge, \mathbb{S})$ with left Quillen functor $L:\mathcal{A}\rightarrow \mathcal{B}$ (with right adjoint $R:\mathcal{B}\rightarrow \mathcal{A}$) is weak monoidal if $R$ is lax monoidal (hence $L$ is colax monoidal), such that:

  • for all $X$ and $Y$ cofibrant, the colax structure on $L$ gives weak equivalences $L(X\otimes Y)\stackrel{\sim}\rightarrow L(X) \wedge L(Y)$ in $\mathcal{B}$;
  • for any cofibrant replacement $\widetilde{\mathbb{I}}\stackrel{\sim}\longrightarrow \mathbb{I}$ of the unit in $\mathcal{A}$, the induced map $$ L(\widetilde{\mathbb{I}})\longrightarrow L(\mathbb{I}) \longrightarrow \mathbb{S}, $$ is a weak equivalence in $\mathcal{B}$ where the right arrow is induced by the colax structure on $L$.

I'm asking for examples where $L$ is not a strong monoidal functor.

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