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Suppose that $\mathcal{C}$ is a model category (with a fixed model structure). Are there any known examples where $\mathcal{C}$ is a (symmetric) monoidal model category in two different ways? I.e., can there exist tensor products $\otimes_1, \otimes_2\colon \mathcal{C}\times \mathcal{C}\to \mathcal{C}$ such that both $(\mathcal{C}, \otimes_1, A)$ and $(\mathcal{C}, \otimes_2, B)$ are symmetric monoidal model categories. Can this phenomenon even occur?

The canonical place to look for an answer to this would be Balchin's book A Handbook of Model Categories (specifically in the Die Kunstkammer) but I couldn't see anything relevant.

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    $\begingroup$ Yes, I know an example. If you are in Glasgow and interested in this kind of thing, you should email me. I'll be in Glasgow in June and happy to get a cup of coffee with you. Also, if you email me, I'll be happy to share the example I know. Nice to know others are interested in this kind of thing! $\endgroup$ May 14 at 13:12
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    $\begingroup$ How about chain complexes of modules over an elementary abelian $p$-group, and over a $p$-restricted Lie algebra with zero bracket and zero restriction map, and of the same rank? $\endgroup$ May 14 at 13:33
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    $\begingroup$ Take the model structure on $\mathrm{Set}$ with weak equivalences given by isomorphisms and (co)fibrations anything. Then surely, disjoint union and cartesian product give an example? $\endgroup$ May 14 at 14:29
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    $\begingroup$ @ConnorMalin Disjoint union doesn’t preserve colimits in each variable separately. For instance $X \amalg (-)$ doesn’t preserve the initial object. $\endgroup$
    – Tim Campion
    May 14 at 19:26

2 Answers 2

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Let $k$ be a field, and let $A$ be a finite-dimensional $k$-algebra. Suppose that $A$ admits the structure of a Hopf $k$-algebra. Then $A$ is quasi-Frobenius, so the category of $A$-modules admits a model structure where the cofibrations are the injective maps, the fibrations are the surjective maps, and the weak equivalences are the stable equivalences. This is in Hovey's Model Categories book, for example.

But now suppose $A$ admits two different Hopf $k$-algebra structures. Each one makes $Mod(A)$ into a symmetric monoidal model category.

For example, suppose $k$ has characteristic $2$, and let $A = k[x]/x^2$. If I recall correctly, you can make $A$ into a Hopf $k$-algebra by letting $x$ be primitive, or by letting the coproduct on $x$ be $x\otimes x + 1\otimes x + x\otimes 1$. These are really examples Dave Benson mentioned in a comment (although he suggested looking at the derived categories rather than the module categories themselves): the former Hopf algebra is the restricted enveloping algebra of a one-dimensional Lie $k$-algebra, while the latter Hopf algebra is isomorphic to the group algebra of the cyclic group $C_2$. The two are isomorphic as $k$-algebras but not isomorphic as Hopf $k$-algebras, so you get two different monoidal model structures on $Mod(A)$. The coproducts are both co-commutative, so these monoidal model structures are indeed symmetric monoidal.

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    $\begingroup$ This is nice and is roughly what I had in mind, but in the setting of the derived category as Dave said. $\endgroup$
    – JD1874
    May 14 at 19:29
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Let $\mathcal C$ be a cocomplete 1-category. Give it the model structure where every morphism is a cofibration and the weak equivalences are the isomorphisms. Then any (symmetric) monoidal closed structure on $\mathcal C$ makes it into a (symmetric) monoidal model category.

So you get an example any time you have a cocomplete 1-category with two (symmetric) monoidal closed structures. See e.g. here for some examples.

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