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Is every symmetric monoidal combinatorial model category symmetric monoidally Quillen equivalent (by a zig-zag of symmetric monoidal Quillen equivalences) to a symmetric monoidal combinatorial simplicial model category?

Do we have a symmetric monoidal version of Dugger's theorem?

Is this true and if yes, where do I find a reference for that?

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Yes, this is proved in https://arxiv.org/abs/1506.01475 by Thomas Nikolaus and Steffen Sagave.

If you need a specific monoidal left Quillen equivalence, you can upgrade Dugger's result to the symmetric monoidal setting. One proceeds as in Dugger's proof. Start with the Reedy model structure on simplicial objects. Equip it with the degreewise monoidal product, which turns it into a monoidal model category (see, e.g., Ghazel-Kadhi). Now perform a left Bousfield localization with respect to hocolim-equivalences, as in Dugger's proof. The resulting model category is a monoidal because monoidal products preserve filtered colimits (see, e.g., Gorchinskiy-Guletskii). A symmetric monoidal Quillen equivalence connects it to the original category, like in Dugger's proof.

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    $\begingroup$ Not quite: That paper proves that any symmetric monoidal left adjoint functor between presentably symmetric monoidal $\infty$-categories is represented by a strong symmetric monoidal left Quillen functor between simplicial, combinatorial and left proper symmetric monoidal model categories (as stated in the abstract). As they recall in the introduction, the presentable $\infty$-category associated to a symmetric monoidal combinatorial model category is canonically endowed with a symmetric monoidal structure, to which we can associate a simplicial model category. But then? $\endgroup$
    – Andrea Gagna
    Commented May 9, 2018 at 14:51
  • $\begingroup$ So using the paper of Nikolaus and Sagave one can do the following: Starting with a symmetric monoidal combinatorial model category $\mathcal{M} $ one gets an underlying presentably symmetric monoidal ∞-category $\mathcal{M}_\infty $. By the main theorem of their paper $\mathcal{M}_\infty $ is modeled as a symmetric monoidal ∞-category by a symmetric monoidal combinatorial simplicial model category $\mathcal{N}$, i.e. there is an equivalence $\mathcal{M}_\infty \simeq \mathcal{N}_\infty$ of symmetric monoidal ∞-categories. $\endgroup$
    – Hadrian Heine
    Commented May 9, 2018 at 15:21
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    $\begingroup$ @AndreaGagna: I added an explicit construction. $\endgroup$
    – Dmitri Pavlov
    Commented May 9, 2018 at 19:44
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    $\begingroup$ @HadrianHeine: By the criterion of Gorchinskiy-Guletskii, in order to show that some left Bousfield localization L_S(C) is a monoidal model category it is necessary and sufficient that the derived monoidal product (in C) of any object in C and any morphism in S is a weak equivalence in L_S(C). In the case under consideration weak equivalences are hocolim-equivalences, so the above condition follows immediately from the fact that the left derived functor of X⊗− preserves filtered homotopy colimits. $\endgroup$
    – Dmitri Pavlov
    Commented May 10, 2018 at 1:37
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    $\begingroup$ @HadrianHeine: The explicit expression for generating cofibrations of the Reedy model structure guarantees this to be true. However, the only reason this assumption is made in Lemma 28 is to avoid cofibrantly replacing X. If you replace X by Q(X) in the statement, there is no need to assume this. $\endgroup$
    – Dmitri Pavlov
    Commented May 12, 2018 at 16:17

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