Lecture 5

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5.1 Monoidal model categories

Definition 5.1.1. A symmetric monoidal model category is a model category $\mathcal{C}$ equipped with a closed symmetric monoidal structure $(\mathcal{C}, \otimes , \mathbb {I})$ such that the two following compatibility conditions are satisfied:

  1. (Pushout-product axiom) For every pair of cofibrations $f \colon X \to Y$ and $f' \colon X' \to Y'$, their pushout-product

    \[ f \square f' \colon (X \otimes Y') \coprod _{X \otimes X'} (Y \otimes X')\to Y \otimes Y' \]

    is also a cofibration. Moreover, it is a acyclic cofibration if either $f$ or $f'$ is.

  2. (Unit axiom) For every cofibrant object $X$ and every cofibrant resolution $\emptyset \hookrightarrow Q\mathbb {I} \to \mathbb {I}$ of the tensor unit, the induced morphism $Q \mathbb {I} \otimes X \to \mathbb {I} \otimes X$ is a weak equivalence.

Remark 5.1.2.The pushout-product axiom appearing above is equivalent to asking that the tensor product $\otimes \colon \mathcal{C} \times \mathcal{C} \to \mathcal{C}$ is a Quillen bifunctor. We also highlight that if the model category is cofibrantly generated then it is enough by [Proposition 4.2.5, Hov99] to just check the pushout-product axiom on the generating sets of cofibrations and acyclic cofibrations.

Proposition 5.1.3 ([Theorem 4.3.2, Hov99]). Let $\mathcal{C}$ be a symmetric monoidal model category, then $\operatorname {Ho}(\mathcal{C})$ can be given the structure of a closed symmetric monoidal category with tensor structure $(\otimes ^{\mathbb {L}}, R \mathbb {I})$, with $R \mathbb {I}$ the fibrant replacement of the tensor unit in $\mathcal{C}$.

5.2 Chain complexes

Lemma 5.2.1 ([Proposition 4.2.13, Hov99]). Let $R$ be a commutative ring, then the projective model structure on $\textbf{Ch}(R)$ is a monoidal model category.

Example 5.2.2.For $R$ a commutative ring, the injective model structure need not be a monoidal model category in general. Take $R=\mathbb {Z}$ and equip $\textbf{Ch}(\mathbb {Z})$ with the injective model structure. There there are injective cofibrations $\mathbb {Z} \to \mathbb {Q}$ and $0 \to \mathbb {Z}/2 \mathbb {Z}$. The pushout-product of these morphism is the morphism $\mathbb {Z}/2 \mathbb {Z} \to 0$ which is not an injective cofibration. As such, $\textbf{Ch}(\mathbb {Z})_\mathrm {inj}$ is not a monoidal model category.

Lemma 5.2.3. Let $R$ be a ring, then the identity functor is a Quillen equivalence:

\[ \operatorname {id} : \textbf{Ch}(R)_{\mathrm{proj}} \rightleftarrows \textbf{Ch}(R)_{\mathrm{inj}} :\operatorname {id}. \]

Note that in particular the above result demonstrates that Quillen equivalent models on the same underlying category need not share the same properties, the projective model can be monoidal while the injective is not.

5.3 Arrow categories

We now give an example of two separate monoidal structures on a given category, each of which is part of a monoidal model structure.

Definition 5.3.1.Let $\mathcal{C}$ be a small category. Then its arrow category $\textbf{Arr}(\mathcal{C})$ is the functor category $\mathcal{C}^ I$. That is, the objects in the arrow category are the morphisms in $\mathcal{C}$ and a morphism $\alpha \colon f \to g$ is a commuting square

\[ \xymatrix{ X_0 \ar[r]^{\alpha _0} \ar[d]_{f} & Y_0 \ar[d]^{g} \\ X_1 \ar[r]_{\alpha _1} & Y_1 } \]

in $\mathcal{C}$. We will write $\operatorname {Ev}_ i\alpha = \alpha _ i$ for $i \in \{ 0,1\} $.

Definition 5.3.2.Suppose that $\mathcal{C}$ carries a symmetric monoidal structure $(\otimes , \mathbb {I})$.

As we have a functor category, we can equip it with both he projective and injective model structures.

Proposition 5.3.3 ([Proposition 2.2, Hov14]). Let $\mathcal{C}$ be a symmetric monoidal model category. Then $\textbf{Arr}(\mathcal{C})_{\mathrm{inj}}$ is a symmetric monoidal model category with respect to the tensor product monoidal structure and moreover satisfies the monoid axiom if $\mathcal{C}$ does.

Proposition 5.3.4 ([Proposition 2.2, WY19]). Let $\mathcal{C}$ be a symmetric monoidal model category. Then $\textbf{Arr}(\mathcal{C})_{\mathrm{proj}}$ is a symmetric monoidal model category with respect to the pushout-product monoidal structure and moreover satisfies the monoid axiom if $\mathcal{C}$ does.

5.4 Preservation under left Bousfield localization

Left Bousfield localization does not always preserve the property of being a monoidal model category as the next example will show.

Proposition 5.4.1 ([Proposition 4.2.15, Hov99]). Let $F$ be a Frobenius ring which is also a finite-dimensional Hopf algebra over some field $k$, then $R \textbf{-Mod}_{\mathrm{st}}$ is a monoidal model category with monoidal structure $M \otimes _ k N$ and $R$ acts via the diagonal $R \to R \otimes _ k R$. The monoidal structure is symmetric if and only if $R$ is cocommutative.

Example 5.4.2 ([Example 4.1, Whi18]). Let $R = \mathbb {F}_2[\Sigma _3]$, then an $R$-module is an $\mathbb {F}_2$-vector space along with an action of the symmetric group $\Sigma _3$. The ring $R$ is a Hopf algebra over the field $\mathbb {F}_2$, and therefore $R \textbf{-Mod}_{\mathrm{st}}$ is a monoidal model category.

We shall localize $R \textbf{-Mod}_{\mathrm{st}}$ with respect to the morphism $f \colon 0 \to \mathbb {F}_2$ where the codomain has a trivial $\Sigma _3$-action. We will now demonstrate that $L_ f R \textbf{-Mod}_{\mathrm{st}}$ cannot be monoidal.

An object will be $f$-trivial if and only if it has no $\Sigma _3$-fixed points if and only if it does not admit $\Sigma _3$-equivariant morphisms from $\mathbb {F}_2$.

To show that the pushout-product axiom fails in $L_ f R \textbf{-Mod}_{\mathrm{st}}$ it is enough to find an $f$-locally trivial object $N$ such that $N \otimes _{\mathbb {F}_2} N$ is not $f$-locally trivial. We let $N \cong \mathbb {F}_2 \oplus \mathbb {F}_2$ where the element $(12) \in \Sigma _3$ sends $a= (1,0)$ to $b = (0,1)$ and the element $(123) \in \Sigma _3$ sends $a$ to $b$ and $b$ to $c=a+b$ which equips $N$ with a $\Sigma _3$-action. As this $\Sigma _3$-action has no fixed points, the object $N$ is $f$-locally trivial.

However the object $N \otimes _{\mathbb {F}_2} N$ has a $\Sigma _3$-invariant element $a \otimes a + b \otimes b + c \otimes c$. As such, $L_ f R \textbf{-Mod}_{\mathrm{st}}$ is not a monoidal model category as required.

We can now look at one such condition of when left Bousfield localization does preserve structure that we care about.

Definition 5.4.3. Let $\mathcal{C}$ be a monoidal model category, then we say that a cofibrant object $X \in \mathcal{C}$ is flat if $f \otimes X$ is a weak equivalence whenever $f$ is a weak equivalence.

Proposition 5.4.4 ([Theorem 4.6, Whi18]). Let $\mathcal{C}$ be a cofibrantly generated monoidal model category in which all cofibrant objects are flat, then a left Bousfield localization $L_ S \mathcal{C}$ is a monoidal Bousfield localization if and only if every morphism of the form $f \otimes \mathrm{id}_ K$ is $S$-local where $f \in S$ and $K$ cofibrant.

Moreover, If the domains of the generating cofibrations $I$ happen to be cofibrant then it suffices to only check this condition where $K$ is in the set of domains and codomains of $I$.

5.5 References for Lecture 5

[Hov99] Mark Hovey. Model categories, volume 63 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI,1999.

[Hov14] Mark Hovey. Smith ideals of structured ring spectra. arXiv:1401.2850, 2014.

[Whi18] D. White. Monoidal Bousfield localizations and algebras over operads. arXiv:1404.5197, 2018.

[WY19] D. White and Donald Yau. Arrow categories of monoidal model categories. Math, Scand., 125(2):185-198, 2019.