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Mar 3, 2017 at 16:01 comment added Bruno Stonek Nice example! One could say that the "monoid axiom", which holds for chain complexes, guarantees that the monoids get a model category structure, but it doesn't mean that the category of monoids itself has to be a monoidal model category (for monoidal model categories, the tensor product of two cofibrant objects is cofibrant).
Feb 11, 2015 at 19:35 comment added James thank you! I had myself convinced that things would have to be quite pathological for cofibrancy to fail, that's life.
Jan 29, 2015 at 14:43 history answered Fernando Muro CC BY-SA 3.0