Left derivable monoidal criterion

Remark 20.1.4. In the model category literature, there exists a notion of symmetric monoidal model category. The underlying category with weak equivalences and cofibrations of such a symmetric monoidal model category is always left derivable. However, the condition of being left derivable is much weaker: we require only that tensoring preserve the relevant weak equivalences, with no general compatibility between tensor products and cofibrations.

A convenient sufficient criterion is the following. Assume that the monoidal unit is cofibrant, that the tensor product of two cofibrant objects is again cofibrant, and that for every cofibrant object \(Z\) the functor \(-\otimes Z\) sends weak equivalences between cofibrant objects to weak equivalences. Then every tensor-cofibrant object is again cofibrant, and an induction on the cardinality of \(J\) shows that the monoidal structure is left derivable in the sense of the definition.