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Let $\text{DGA}^{-}$ denote the category of non-positively graded differential graded algebras with differentials of degree $+1$. It is well-known that $\text{DGA}^{-}$ has a model structure with

  1. Weak equivalences are quasi-isomorphisms;
  2. Fibrations are morphisms that are surjective on all negative degrees;
  3. Cofibrations are the morphisms having the left-lifting property with respect to acyclic fibrations.

It is also clear that $\text{DGA}^{-}$ has a tensor product which makes it a monoidal category.

My question is: Is the model structure compatible with the tensor product? More precisely, is $\text{DGA}^{-}$ a monoidal model category with the above model structure and tensor product?

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    $\begingroup$ Assuming the tensor product is the tensor product of complexes, $\mathrm{DGA}^-$ is not even closed monoidal. $\endgroup$
    – Pavel Safronov
    Commented Sep 26, 2018 at 8:32
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    $\begingroup$ See also mathoverflow.net/a/195194/6249 $\endgroup$
    – Bruno Stonek
    Commented Oct 17, 2018 at 11:18

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