nLab monoidal homotopy category of a monoidal model category

Proposition

Let $(\mathcal{C}, \otimes, I)$ be a monoidal model category. Then the left derived functor of the tensor product exists and makes the homotopy category into a monoidal category $(Ho(\mathcal{C}), \otimes^L, \gamma(I))$.

If in in addition $(\mathcal{C}, \otimes)$ satisfies the monoid axiom, then the localization functor $\gamma\colon \mathcal{C}\to Ho(\mathcal{C})$ carries the structure of a lax monoidal functor

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

Proof

Consider the explicit model of $Ho(\mathcal{C})$ as the category of fibrant-cofibrant objects in $\mathcal{C}$ with left/right-homotopy classes of morphisms between them (discussed at homotopy category of a model category).

A derived functor exists if its restriction to this subcategory preserves weak equivalences. Now the pushout-product axiom implies that on the subcategory of cofibrant objects the functor $\otimes$ preserves acyclic cofibrations, and then the preservation of all weak equivalences follows by Ken Brown's lemma.

Hence $\otimes^L$ exists and its associativity follows simply by restriction. It remains to see its unitality.

To that end, consider the construction of the localization functor $\gamma$ via a fixed but arbitrary choice of (co-)fibrant replacements $Q$ and $R$, assumed to be the identity on (co-)fibrant objects. We fix notation as follows:

$$\emptyset \underoverset{\in Cof}{i_X}{\longrightarrow} Q X \underoverset{\in W \cap Fib}{p_x}{\longrightarrow} X \;\;\,,\;\; X \underoverset{\in W \cap Cof}{j_X}{\longrightarrow} R X \underoverset{\in Fib}{q_x}{\longrightarrow} \ast \,.$$

Now to see that $\gamma(I)$ is the tensor unit for $\otimes^L$, notice that in the zig-zag

$$(R Q I) \otimes (R Q X) \overset{j_{Q I} \otimes (R Q X)}{\longleftarrow} (Q I) \otimes (R Q X) \overset{(Q I)\otimes j_{Q X}}{\longleftarrow} (Q I) \otimes (Q X) \overset{p_I \otimes (Q X)}{\longrightarrow} I \otimes Q X \simeq Q X$$

all morphisms are weak equivalences: For the first two this is due to the pushout-product axiom, for the third this is due to the unit axiom on a monoidal model category. It follows that under $\gamma(-)$ this zig-zig gives an isomorphism

$$\gamma(I) \otimes^L \gamma(X)\simeq \gamma(X)$$

and similarly for tensoring with $\gamma(I)$ from the right.

To exhibit lax monoidal structure on $\gamma$, we need to construct a natural transformation

$$\gamma(X) \otimes^L \gamma(Y) \longrightarrow \gamma(X \otimes Y)$$

and show that it satisfies the the appropriate associativity and unitality condition.

By the definitions at homotopy category of a model category, the morphism in question is to be of the form

$$(R Q X) \otimes (R Q Y) \longrightarrow R Q (X\otimes Y)$$

To this end, consider the zig-zag

$$(R Q X) \otimes (R Q Y) \underoverset{\in Cof \cap W}{j_{Q X} \otimes R Q Y}{\longleftarrow} (Q X) \otimes (R Q Y) \underoverset{\in Cof \cap W}{(Q X) \otimes j_{Q Y} }{\longleftarrow} (Q X) \otimes (Q Y) \overset{p_X \otimes (Q Y)}{\longrightarrow} X \otimes (Q Y) \overset{Y \otimes p_Y}{\longrightarrow} X \otimes Y \,,$$

and observe that the two morphisms on the left are weak equivalences, as indicated, by the pushout-product axiom satisfied by $\otimes$.

Hence applying $\gamma$ to this zig-zag, which is given by the two horizontal part of the following digram

$$\array{ (R Q X) \otimes (R Q Y) &\longleftarrow& R( Q X \otimes Q Y ) &\longrightarrow& R Q (X \otimes Y) \\ \uparrow^{\mathrlap{id}} && \uparrow^{\mathrlap{j_{Q X \otimes Q Y}}} && \uparrow^{\mathrlap{j_{Q(X \otimes Y)}}} \\ \downarrow^{\mathrlap{id}} && \downarrow^{\mathrlap{id}} && \downarrow^{\mathrlap{p_{X\otimes Y}}} \\ (R Q X) \otimes (R Q Y) &\underoverset{\in Cof \cap W}{j_{Q X} \otimes j_{Q Y}}{\longleftarrow}& (Q X) \otimes (Q Y) &\overset{p_X \otimes p_Y}{\longrightarrow}& X \otimes Y } \,,$$

and inverting the first two morphisms, this yields a natural transformation as required.

To see that this satisfies associativity if the monoid axiom holds, tensor the entire diagram above on the right with $(R Q Z)$ and consider the following pasting composite:

$$\array{ (R Q X) \otimes (R Q Y) \otimes (R Q Z) &\longleftarrow& R( Q X \otimes Q Y ) \otimes (R Q Z) &\longrightarrow& (R Q (X \otimes Y)) \otimes (R Q Z) \\ \uparrow^{\mathrlap{id}} && \uparrow^{\mathrlap{j_{Q X \otimes Q Y} \otimes id }} && \uparrow^{\mathrlap{j_{Q(X \otimes Y)}\otimes id }} \\ && && Q(X \otimes Y) \otimes (R Q Z) &\overset{id \otimes j_{Q Z}}{\longleftarrow}& Q(X\otimes Y) \otimes (Q Z) \\ \downarrow^{\mathrlap{id}} && \downarrow^{\mathrlap{id}} && \downarrow^{\mathrlap{p_{(X\otimes Y)} \otimes id }} &(\star)& \downarrow^{\mathrlap{p_{(X \otimes Y)} \otimes id}} \\ (R Q X) \otimes (R Q Y) \otimes (R Q Z) &\underoverset{\in Cof \cap W}{j_{Q X} \otimes j_{Q Y} \otimes id}{\longleftarrow}& (Q X) \otimes (Q Y) \otimes (R Q Z) &\overset{p_X \otimes p_Y \otimes id}{\longrightarrow}& X \otimes Y \otimes (R Q Z) &\underset{id \otimes j_{Q Z}}{\longleftarrow}& X\otimes Y \otimes Q Z &\overset{id \otimes p_Z}{\longrightarrow}& X \otimes Y \otimes Z } \,,$$

Observe that under $\gamma$ the total top zig-zag in this diagram gives

$$(\gamma(X) \otimes^L \gamma(Y)) \otimes^L \gamma(Z) \to \gamma(X\otimes Y)\otimes^L \gamma(Z) \,.$$

Now by the monoid axiom (but not by the pushout-product axiom!), the horizontal maps in the square in the bottom right (labeled $\star$) are weak equivalences. This implies that the total horizontal part of the diagram is a zig-zag in the first place, and that under $\gamma$ the total top zig-zag is equal to the image of that total bottom zig-zag. But by functoriality of $\otimes$, that image of the bottom zig-zag is

$$\gamma(p_X \otimes p_Y \otimes p_Z) \circ \gamma(j_{Q X} \otimes j_{Q Y} \otimes j_{Q Z})^{-1} \,.$$

The same argument applies to left tensoring with $R Q Z$ instead of right tensoring, and so in both cases we reduce to the same morphism in the homotopy category, thus showing the associativity condition on the transformation that exhibits $\gamma$ as a lax monoidal functor.