# 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.)

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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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Do you know a reference for the general case? –  Kevin H. Lin Oct 12 '09 at 19:51
I think this can be found in the standard texts "Model Categories" by Mark Hovey or "Model Categories and Their Localizations" by Philip Hirschhorn. It is also probably in Quillen's original papers. I would look at Quillen's stuff first. –  Chris Schommer-Pries Oct 12 '09 at 20:06
I wasn't able to find it in Hovey's book. I'll try Hirschhorn's book, and Quillen's stuff. Thanks! –  Kevin H. Lin Oct 12 '09 at 20:10
Weak equivalences may be quasi-isomorphisms, but what are fibrations and cofibrations, or at least the fibrant and cofibrant objects? Since this question does not seem to have a good answer in general, what you say is unlikely to be true. –  Leonid Positselski Dec 2 '09 at 16:56
This is an exercise in Gelfand-Manin, right? Section V.2, maybe? –  Aaron Bergman Sep 8 '11 at 23:10

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

• 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

• 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!

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@Beren, in those cases when there is no known model category structure on the category of unbounded complexes on an abelian category, the derived category is not know to exist either. –  Fernando Muro Sep 9 '11 at 7:49
@Fernando: Thanks for pointing that out. Is there a feeling among experts that all derived categories of abelian categories which exist should have models (even if we aren't currently aware of such models)? Secondly, are there any abelian categories for which we have proofs that their derived categories don't exist? –  Beren Sanders Sep 9 '11 at 12:45
@Beren: 1) As far as I remember, I don't know any abelian category with a derived category which does not come from a sort of model category. 2) Look at Freyd's Abelian Categories Chapter 6, Exercise A, pp. 131–132. –  Fernando Muro Sep 10 '11 at 8:07

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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Bounded can mean that they are universally bounded (ie, consider only chain complexes that are positively graded). –  Eric Wofsey Oct 15 '09 at 17:29
This is what most people call "concentrated in positive degree". I've never heard anyone refer to it as being bounded. –  user332 Oct 16 '09 at 16:07
@Rex, I think that Quillen model categories are not required to be cocomplete. –  Fernando Muro Sep 9 '11 at 7:51
The definition I am working with is the one here: en.wikipedia.org/wiki/Model_category –  user332 Sep 9 '11 at 8:22
Of course, if you adapt the definition of the notion somebody else is asking about, you'll probably always be true, correct and precise! ;-) –  Fernando Muro Sep 10 '11 at 8:11