Derived categories and homotopy categories There are two constructions that look quite similar to me: the derived category of an abelian category, and the homotopy category of a model category. Is there any explicit relationship between these two constructions? (This question is related to, and indeed the inspiration for, one of my previous questions.)
 A: Both give rise to derivators, and indeed thinking about homotopy theories as non-abelian derived categories is what led Grothendieck to introduce then (note that Heller and Franke independently came up with derivators, but I'm not sure they had the same motivation) 
A: Unfortunately, it isn't quite right to say that derived categories of abelian categories are a special case of model categories. Morally this might be true, but for a general abelian category there is no known model category structure on its (unbounded) category of chain complexes whose weak equivalences are the quasi-isomorphisms. There is such a model structure when the abelian category is a Grothendieck category; this is shown in 


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*Mark Hovey -- Model category structures on chain complexes of sheaves (2001).


Quillen originally gave the example of a model structure on the category of non-negatively bounded complexes of R-modules, but the case of unbounded complexes of R-modules seems not to have appeared in print until the publication of Hovey's book


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*Mark Hovey -- Model categories (1999).


As far as I'm aware, none of the standard references on model categories talk about unbounded derived categories of abelian categories---probably because in general they don't arise as the homotopy category of any known model structure on the category of chain complexes!
A: Yes. The former is a special case of the latter. There is a model category structure on the category of (say bounded) chain complexes of objects in your given abelian category. The weak equivalences are the quasi-isomorphisms, and the homotopy category is the derived category. 
In the case of R-modules, for a ring R, this is explained in detail in
this paper by Dwyer-Spalinski.
A: Some information may be found at nLab: homotopy category. Following the links there you also find information on all the other keywords mentioned above.
Urs Schreiber
A: I think you don't want any bounded condition. I don't see how the category of chain complexes with bounded cohomology could be a model category. It doesn't have all small colimits; just take longer and longer chain complexes with trivial differentials, and you get something with unbounded cohomology.
