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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; see also the survey article Cotorsion pairs and model categories.

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!

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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.

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    $\begingroup$ Same for the motivating example in Hovey’s paper: the stable module category is triangulated and usually not abelian. $\endgroup$ – Dylan Wilson Jan 3 '18 at 13:59
  • $\begingroup$ Ok great, thanks! Sorry for the dumb question! $\endgroup$ – David White Jan 3 '18 at 14:01
  • $\begingroup$ Quick follow-up question: is the homotopy category of an abelian model category always triangulated at least? In other words, does abelian imply stable? $\endgroup$ – David White Jan 3 '18 at 14:02
  • $\begingroup$ @JeremyRickard, nevermind, I can answer my follow-up question. You only know the homotopy category is triangulated if the abelian model category is hereditary (I'm not sure if this is an "iff", and I doubt it). Without this condition you only know it's "pre-triangulated", as I learned from II.2.1.20 of Hanno Becker's PhD thesis. $\endgroup$ – David White Jan 3 '18 at 14:38

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