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.)
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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. |
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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
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
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! |
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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 |
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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) |
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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. |
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