A model structure on an abelian category $A$ is called an *abelian model structure* if the cofibrations are precisely the monomorphisms with cofibrant cokernel, and if the fibrations are precisely the epimorphisms with fibrant kernel. This terminology was introduced by Mark Hovey in [Cotorsion pairs, model category structures, and representation theory](https://link.springer.com/article/10.1007/s00209-002-0431-9); see also the survey article [Cotorsion pairs and model categories](http://homepages.math.uic.edu/~bshipley/hovey.pdf).


In his book, Hovey introduced *monoidal model categories* and proved that their homotopy categories are also monoidal. Similarly, a model category is *stable* if and only if it's homotopy category is triangulated. I always assumed from the terminology that the condition about (co)fibrations was there to guarantee that the homotopy category of an abelian model category would be an abelian category, but now I can't find a reference for this, and I'm not even sure if it's true. I'm trying to branch out a bit into homological algebra and representation theory, and abelian model categories are new to me. Any help would be much appreciated!



Is the homotopy category of an abelian model category abelian?
No. The projective model structure on chain complexes of modules over a ring is an abelian model category, and the homotopy category is the derived category, which is never abelian unless the ring is semisimple.


