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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeApr 4th 2016
    • (edited Apr 4th 2016)

    I have added to monoidal model category statement and proof (here) of the basic statement:


    Let (𝒞,)(\mathcal{C}, \otimes) be a monoidal model category. Then 1) the left derived functor of the tensor product exsists and makes the homotopy category into a monoidal category (Ho(𝒞), L,γ(I))(Ho(\mathcal{C}), \otimes^L, \gamma(I)). If in in addition (𝒞,)(\mathcal{C}, \otimes) satisfies the monoid axiom, then 2) the localization functor γ:𝒞Ho(𝒞)\gamma\colon \mathcal{C}\to Ho(\mathcal{C}) carries the structure of a lax monoidal functor

    γ:(𝒞,,I)(Ho(𝒞), L,γ(I)). \gamma \;\colon\; (\mathcal{C}, \otimes, I) \longrightarrow (Ho(\mathcal{C}), \otimes^L , \gamma(I)) \,.

    The first part is immediate and is what all authors mention. But this is useful in practice typically only with the second part.

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeJun 21st 2016

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

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeJun 21st 2016

    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.

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeJun 21st 2016
    • (edited Jun 21st 2016)

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

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

    Then the inverse of the natural iso filling

    𝒞 c×𝒞 c = 𝒞 c×𝒞 c ()() 𝒞 γ 𝒞×𝒞 γ 𝒞×γ 𝒞 γ 𝒞 Ho(𝒞×𝒞) Ho(𝒞)×Ho(𝒞) () L() Ho(𝒞) \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

    μ X,Y:γ(X) Lγ(Y)γ(XY). \mu_{X,Y} \;\colon\; \gamma(X) \otimes^L \gamma(Y) \longrightarrow \gamma(X \otimes Y) \,.

    Moreover the associativity condition for this to exhibit γ\gamma as a monoidal functor is the equation

    ()𝒞 c×𝒞 c×𝒞 c (()())() 𝒞 = α = 𝒞 c×𝒞 c×𝒞 c ()(()()) 𝒞 γ 𝒞×𝒞×𝒞 η γ 𝒞 Ho(𝒞)×Ho(𝒞)×Ho(𝒞) () L(() L()) Ho(𝒞)𝒞 c×𝒞 c×𝒞 c (()())() 𝒞 γ 𝒞×𝒞×𝒞 η γ 𝒞 Ho(𝒞)×Ho(𝒞)×Ho(𝒞) (() L()) L() Ho(𝒞) = α L = Ho(𝒞)×Ho(𝒞)×Ho(𝒞) () L(() L()) Ho(𝒞), (\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 α L\alpha^L is induced from the composite on the left via the universal property of the localization.

    So γ\gamma is monoidal.

    Am I making a mistake here?

    • CommentRowNumber5.
    • CommentAuthorMike Shulman
    • CommentTimeJun 22nd 2016

    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)Ho(C) that has the same objects as CC, then γ\gamma is the identity on objects, and since L\otimes^L can be defined using only cofibrant replacements, the morphism we need is of the form QXQYXYQ X \otimes Q Y \to X\otimes Y, so it can be just the tensor product of the two weak equivalences QXXQ X \to X and QYYQ Y \to Y.

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeJun 22nd 2016

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

    • CommentRowNumber7.
    • CommentAuthorDmitri Pavlov
    • CommentTimeJun 22nd 2016
    • (edited Jun 22nd 2016)

    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.

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeJun 22nd 2016

    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?

    • CommentRowNumber9.
    • CommentAuthorMike Shulman
    • CommentTimeJun 22nd 2016

    γ\gamma is strong monoidal as a functor defined on C cC_c, but only lax monoidal as a functor defined on CC. I suppose there’s a sense in which it is “strong monoidal” on CC when you equip CC 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)Ho(V_0), the localization functor γ:V 0Ho(V 0)\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.

    • CommentRowNumber10.
    • CommentAuthorMike Shulman
    • CommentTimeJun 22nd 2016

    Also, here is an abstract reason for γ\gamma to be lax monoidal. Let VV be a monoidal model category, and consider it as a “derivable category” in the sense of comparing composites (section 8) with V QV_Q the subcategory of cofibrant objects and V R=VV_R=V. Then :V×VV\otimes :V\times V\to V is left derivable, i.e. it preserves the QQ-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)Ho(V) is monoidal; this is Example 8.13 of ibid.

    Now let V 0V_0 denote the category VV with its trivial derivable structure: only isomorphisms are weak equivalences, and all objects are both QQ and RR. Then of course Ho(V 0)=VHo(V_0) = V, and V 0V_0 is also a pseudomonoid in derivable categories. The identity functor Id:V 0VId : V_0 \to V is not left derivable, since it does not preserve QQ-objects; but it is right derivable, since we took all objects in VV to be RR-objects (ignoring the fibrant objects in the model structure on VV). Of course IdId 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 HoHo; but Ho(Id)=γ:VHo(V)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 γ\gamma is only lax monoidal.

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

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeJun 22nd 2016

    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.)

    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeJun 22nd 2016

    I have added your proof here.

    • CommentRowNumber13.
    • CommentAuthorMike Shulman
    • CommentTimeJun 22nd 2016

    Thanks!

    • CommentRowNumber14.
    • CommentAuthorUrs
    • CommentTimeJun 23rd 2016

    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 α L\alpha^L and derived unitors that one finds by the (more) explicit factorization here?

    • CommentRowNumber15.
    • CommentAuthorMike Shulman
    • CommentTimeJun 23rd 2016

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

    • CommentRowNumber16.
    • CommentAuthorUrs
    • CommentTimeOct 4th 2017

    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