Timeline for Is every symmetric monoidal combinatorial model category symmetric monoidally Quillen equivalent to a simplicial such?
Current License: CC BY-SA 4.0
15 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 12, 2018 at 16:17 | comment | added | Dmitri Pavlov | @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. | |
| May 12, 2018 at 13:55 | comment | added | Hadrian Heine | Thanks a lot. Before the authors state lemma 28, they assume that the domains and codomains of the generating cofibrations of the model category are cofibrant, in which the left Bousfield localization takes place.If the domains and codomains of the generating cofibrations of a model category are cofibrant, can we choose the generating cofibrations of the induced Reedy-model category such that their domains and codomains are cofibrant, too? | |
| May 11, 2018 at 2:00 | comment | added | Dmitri Pavlov | @HadrianHeine: This is Lemma 28 in arXiv:0907.0730v4. If you use derived monoidal products, there is no need to make any assumptions. For nonderived monoidal products all objects involved must be (made) cofibrant (by applying the cofibrant replacement functor, which amounts to deriving the monoidal product). | |
| May 10, 2018 at 23:12 | comment | added | Hadrian Heine | In which paper of Gorchinskiy-Guletskii do I find the criterion that it is enough to check that the derived monoidal product of any object in C and any morphism in S is a weak equivalence in L_S(C)? To apply this criterion do we need to assume that the domains and codomains of the generating cofibrations of C are cofibrant? | |
| May 10, 2018 at 1:37 | comment | added | Dmitri Pavlov | @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. | |
| May 9, 2018 at 21:48 | comment | added | Hadrian Heine | What is your argument that the resulting Bousfield-localization with respect to hocolim equivalences is monoidal using that monoidal products preserve filtered colimits? | |
| May 9, 2018 at 21:21 | comment | added | Hadrian Heine | Thanks a lot. Thats's great. And it should be easy to show that the canonical symmetric monoidal functor from simplicial sets to simplicial objects in the symmetric monoidal combinatorial model category we start with will preserve weak equivalences so that we get a simplicial symmetric monoidal model structure. | |
| May 9, 2018 at 19:45 | comment | added | Dmitri Pavlov | @HadrianHeine: I added an explicit construction. | |
| May 9, 2018 at 19:44 | history | edited | Dmitri Pavlov | CC BY-SA 4.0 |
added 575 characters in body
|
| May 9, 2018 at 19:44 | comment | added | Dmitri Pavlov | @AndreaGagna: I added an explicit construction. | |
| May 9, 2018 at 16:03 | comment | added | Andrea Gagna | I guess that is precisely what is missing and it turns out to require a theorem à la Dugger as you asked in the OP. | |
| May 9, 2018 at 15:23 | comment | added | Hadrian Heine | But how can one deduce then that $\mathcal{M}$ and $\mathcal{N}$ are linked by a zig-zag of symmetric monoidal Quillen equivalences? | |
| May 9, 2018 at 15:21 | comment | added | Hadrian Heine | 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. | |
| May 9, 2018 at 14:51 | comment | added | Andrea Gagna | 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? | |
| May 9, 2018 at 12:56 | history | answered | Dmitri Pavlov | CC BY-SA 4.0 |