Added:

The first mention of monoidal model categories (without the unit axiom) under this name is in

- William G. Dwyer, Philip S. Hirschhorn, Daniel M. Kan, Model Categories and More General Abstract Homotopy Theory, PDF. Mentioned in passing in Remark 55.10, with no definition given, but the preceding section discusses the pushout product axiom.

Added:

The first mention of monoidal model categories (without the unit axiom) under this name is in

Mark Hovey, Brooke Shipley, Jeff Smith,

*Symmetric spectra*, arXiv:math/9801077v1 (January 16, 1998).Stefan Schwede, Brooke Shipley,

*Algebras and modules in monoidal model categories*, arXiv:math/9801082v1 (January 19, 1998).{#Hovey98} Mark Hovey,

*Monoidal model categories*(arXiv:math/9803002v1, February 28, 1998).

Although the earliest mentions of terminology appear to be the sources indicated above, the notion itself is older. In particular, in his book Hovey credits the definition of a Quillen bifunctor to Dwyer–Hirschhorn–Kan–Smith, and this definition is itself based on the axiom SM7 in Quillen’s Homotopical Algebra.

The unit axiom together with the fact that the homotopy category is monoidal in this case is due to Hovey.

]]>Right, sure. Made a quick fix, but am just on my phone for the time being.

]]>The injective model structure (which is what you mentioned here and in the other place) is always monoidal, as long as M is monoidal. (Easy to see because cofibrations and acyclic cofibrations are defined objectwise.)

Proposition 7.9 in the cited work talks about the *projective* structure, for which one needs the domain category to have finite products.
(Also easy to see: the generating (acyclic) cofibrations have the form I⊗y(A), and the pushout product is (I◻J)⊗(y(A)⨯y(B)), so y(A)⨯y(B) must be projectively cofibrant
as a presheaf of sets, e.g., a retract of a representable presheaf.

added (here) a brief Examples-paragraph on “monoidal presheaves”, following discussion here

]]>Added:

Monoidal Reedy model structures are discussed in

- Moncef Ghazel, Fethi Kadhi,
*Reedy diagrams in V-model categories*, arXiv:1610.06535, doi:10.1007/s10485-019-09566-w.

I have split off from the existing remark (here) that cofibrancy of the tensor unit implies the unit axiom, a precursor remark (now here) making explicit that for cofibrant objects there is an internal-hom Quillen adjunction.

]]>added pointer to:

- Mark Hovey,
*Monoidal model categories*(arXiv:math/9803002)

added pointer to:

- Jacob Lurie, Def. A.3.1.2 in:
*Higher Topos Theory*, Annals of Mathematics Studies 170, Princeton University Press 2009 (pup:8957, pdf)

For completeness, I have added mentioning of the example of model categories of simplicial presheaves being Cartesian monoidal model categories if the site has finite products: here

]]>Yes to the first. The second should follow by inspecting how factorizations through derived functors are constructed in terms of $Q$.

]]>Mike, and I suppose braiding of the localization functor follows similarly?

And: do we have an argument that the monoidal structure that drops out of your abstract proof has the same derived associators $\alpha^L$ and derived unitors that one finds by the (more) explicit factorization here?

]]>Thanks!

]]>I have added your proof here.

]]>Thanks, Mike!

Originally I had been hoping for something more explicit that I could present to a crowd without heavy category theory background. But now that I see how slippery such a low-brow proof is going to be, I am glad you give me a solid argument at all!

Your argument should eventually go to *monoidal model category*.

(By the way, meanwhile I found Day’s article on *monoidal localisation*. He proves that localization in the presence of calculus of fractions is lax monoidal, but under the strong assumption that tensor product with every object preserves all the weak equivalences.)

Also, here is an abstract reason for $\gamma$ to be lax monoidal. Let $V$ be a monoidal model category, and consider it as a “derivable category” in the sense of comparing composites (section 8) with $V_Q$ the subcategory of cofibrant objects and $V_R=V$. Then $\otimes :V\times V\to V$ is left derivable, i.e. it preserves the $Q$-subcategories and weak equivalences. Since deriving is pseudofunctorial (and product-preserving) on the 2-category of derivable categories and left derivable functors, it follows immediately that $Ho(V)$ is monoidal; this is Example 8.13 of *ibid*.

Now let $V_0$ denote the category $V$ with its trivial derivable structure: only isomorphisms are weak equivalences, and all objects are both $Q$ and $R$. Then of course $Ho(V_0) = V$, and $V_0$ is also a pseudomonoid in derivable categories. The identity functor $Id : V_0 \to V$ is not left derivable, since it does not preserve $Q$-objects; but it is *right* derivable, since we took all objects in $V$ to be $R$-objects (ignoring the fibrant objects in the model structure on $V$). Of course $Id$ is strong monoidal, and this monoidality constraint can be expressed as a square in the double category of *ibid* whose vertical arrows are left derivable functors and whose horizontal arrows are right derivable functors; moreover the axioms on a monoidal functor can be expressed using products and double-categorical pasting in this double category. Therefore, it is all preserved by the double pseudofunctor $Ho$; but $Ho(Id) = \gamma : V \to Ho(V)$. The only thing that is not visible to the double category is the invertibility of the monoidal constraint, and hence this is not preserved by the double pseudofunctor; thus $\gamma$ is only lax monoidal.

– Mike “let’s see how many times I can cite myself in the space of half an hour” Shulman

]]>$\gamma$ is strong monoidal as a functor defined on $C_c$, but only lax monoidal as a functor defined on $C$. I suppose there’s a sense in which it is “strong monoidal” on $C$ when you equip $C$ with a point-set-level derived tensor product, but the latter is not generally a point-set-level monoidal structure at all, so it’s not clear exactly what that means.

As for a reference, in section 17 of Homotopy limits, I wrote “Observe that by definition of the derived tensor product in $Ho(V_0)$, the localization functor $\gamma: V_0 \to Ho(V_0)$ is lax symmetric monoidal” but I didn’t give any sort of proof. I presume that I had something like #5 in mind.

]]>Also, γ’:C_c→Ho(C_c)=Ho(C) carries the structure of a strong monoidal functor γ:(C_c,⊗,1)⟶(Ho(C),⊗^L,γ(1)), i.e., the objects of the source are cofibrant.

Okay, that’s just what I am saying in #4 !

Okay, so then I suppose I am not making a mistake and the argument is sound. Still, I’d like a canonical citation. Which source says this clearly?

]]>Something must be wrong with the statement, looking at the formula (PQX)⊗(PQY)⟶PQ(X⊗Y) the expression X⊗Y does not in general compute the correct answer in the homotopy category, so the above map cannot be a weak equivalence.

What is always true is that the localization functor γ:C→Ho(C) carries the structure of a strong monoidal functor γ:(C,⊗^L,1)⟶(Ho(C),⊗^L,γ(1)), i.e., the tensor product on the left must also be derived.

Also, γ’:C_c→Ho(C_c)=Ho(C) carries the structure of a strong monoidal functor γ:(C_c,⊗,1)⟶(Ho(C),⊗^L,γ(1)), i.e., the objects of the source are cofibrant.

]]>Thanks. So with the formal argument above I would seem to always get that $\gamma$ is *strong* monoidal. Is that possible? It sounds too strong.

It seems to me that it should be fairly immediate like this, without needing the monoid axiom. If we use the definition of $Ho(C)$ that has the same objects as $C$, then $\gamma$ is the identity on objects, and since $\otimes^L$ can be defined using only cofibrant replacements, the morphism we need is of the form $Q X \otimes Q Y \to X\otimes Y$, so it can be just the tensor product of the two weak equivalences $Q X \to X$ and $Q Y \to Y$.

]]>Hm, but it’s simpler, isn’t it. Check if I am making a mistake in the following:

so let $\mathcal{C}$ be a monoidal model category with cofibrant tensor unit. Write $\mathcal{C}_c$ for the full subcategory on the cofibrant objects, and write $\gamma$ for localization at the weak equivalences.

Then the inverse of the natural iso filling

$\array{ \mathcal{C}_c \times \mathcal{C}_c &=& \mathcal{C}_c \times \mathcal{C}_c &\overset{(-)\otimes (-)}{\longrightarrow}& \mathcal{C} \\ {}^{\mathllap{\gamma_{\mathcal{C} \times \mathcal{C}}}}\downarrow && {}^{\mathllap{\gamma_{\mathcal{C}} \times \gamma_{\mathcal{C}}}}\downarrow &\swArrow_\simeq& \downarrow^{\mathrlap{\gamma_{\mathcal{C}}}} \\ Ho(\mathcal{C} \times \mathcal{C}) &\simeq& Ho(\mathcal{C})\times Ho(\mathcal{C}) &\underset{(-)\otimes^L (-)}{\longrightarrow}& Ho(\mathcal{C}) }$gives the structure morphism

$\mu_{X,Y} \;\colon\; \gamma(X) \otimes^L \gamma(Y) \longrightarrow \gamma(X \otimes Y) \,.$Moreover the associativity condition for this to exhibit $\gamma$ as a monoidal functor is the equation

$(\star) \;\;\;\;\;\; \array{ \mathcal{C}_c \times \mathcal{C}_c \times \mathcal{C}_c &\overset{((-)\otimes(-))\otimes (-)}{\longrightarrow}& \mathcal{C} \\ {}^{\mathllap{=}}\downarrow &\swArrow_{\alpha}& \downarrow^{\mathrlap{=}} \\ \mathcal{C}_c \times \mathcal{C}_c \times \mathcal{C}_c &\overset{ (-) \otimes ( (-) \otimes (-) ) }{\longrightarrow}& \mathcal{C} \\ {}^{\mathllap{\gamma_{\mathcal{C} \times \mathcal{C} \times \mathcal{C}}}}\downarrow &\swArrow_\eta& \downarrow^{\mathrlap{\gamma_{\mathcal{C}}}} \\ Ho(\mathcal{C}) \times Ho(\mathcal{C}) \times Ho(\mathcal{C}) &\overset{ (-) \otimes^L ( (-) \otimes^L (-) ) }{\longrightarrow}& Ho(\mathcal{C}) } \;\;\;\;\;\; \simeq \;\;\;\;\;\; \array{ \mathcal{C}_c \times \mathcal{C}_c \times \mathcal{C}_c &\overset{((-)\otimes(-))\otimes (-)}{\longrightarrow}& \mathcal{C} \\ {}^{\mathllap{\gamma_{\mathcal{C} \times \mathcal{C} \times \mathcal{C}}}}\downarrow &\swArrow_{\eta'}& \downarrow^{\mathrlap{\gamma_{\mathcal{C}}}} \\ Ho(\mathcal{C}) \times Ho(\mathcal{C}) \times Ho(\mathcal{C}) &\overset{((-)\otimes^L(-))\otimes^L (-)}{\longrightarrow}& Ho(\mathcal{C}) \\ {}^{\mathllap{=}}\downarrow &\swArrow_{\alpha^L}& \downarrow^{\mathrlap{=}} \\ Ho(\mathcal{C})\times Ho(\mathcal{C})\times Ho(\mathcal{C}) &\underset{(-)\otimes^L((-)\otimes^L (-))}{\longrightarrow}& Ho(\mathcal{C}) } \,,$where the derived associator $\alpha^L$ is induced from the composite on the left via the universal property of the localization.

So $\gamma$ is monoidal.

Am I making a mistake here?

]]>It seems I lost my literature citation for that statement, that the localization functor on a monoidal model category satisfying the monoid axioms is lax monoidal. I wanted to add a citation to the entry. Anyone remembers it? It’s not stated in Hovey’s book, as far as I see.

]]>I have expanded a good bit the writeup of the proof of that statement here, making the associators and unitors explicit.

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