# Is $\text{DGA}^{-}$ a monoidal model category?

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?

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