most recent 30 from http://mathoverflow.net 2013-05-19T07:23:42Z http://mathoverflow.net/feeds/question/372 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/372/derived-categories-and-homotopy-categories Kevin Lin 2009-10-12T19:15:48Z 2011-09-08T23:33:39Z

<p>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 <a href="http://mathematics.stackexchange.com/questions/141/model-category-structures-on-categories-of-complexes-in-abelian-categories" rel="nofollow">questions</a>.)</p>

http://mathoverflow.net/questions/372/derived-categories-and-homotopy-categories/373#373 Chris Schommer-Pries 2009-10-12T19:46:50Z 2009-10-12T19:46:50Z

<p>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. </p> <p>In the case of R-modules, for a ring R, this is explained in detail in <a href="http://hopf.math.purdue.edu/Dwyer-Spalinski/theories.pdf" rel="nofollow">this paper by Dwyer-Spalinski</a>.</p>

http://mathoverflow.net/questions/372/derived-categories-and-homotopy-categories/581#581 Urs Schreiber 2009-10-15T08:20:02Z 2009-10-15T08:20:02Z

<p>Some information may be found at <a href="http://ncatlab.org/nlab/show/homotopy+category" rel="nofollow">nLab: homotopy category</a>. Following the links there you also find information on all the other keywords mentioned above.</p> <p>Urs Schreiber</p>

http://mathoverflow.net/questions/372/derived-categories-and-homotopy-categories/604#604 AH 2009-10-15T15:51:53Z 2009-10-15T16:01:10Z

<p>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.</p>

http://mathoverflow.net/questions/372/derived-categories-and-homotopy-categories/74937#74937 Beren Sanders 2011-09-08T21:02:49Z 2011-09-08T21:02:49Z

<p>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 <em>is</em> such a model structure when the abelian category is a Grothendieck category; this is shown in </p> <ul> <li>Mark Hovey -- Model category structures on chain complexes of sheaves (2001).</li> </ul> <p>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</p> <ul> <li>Mark Hovey -- Model categories (1999).</li> </ul> <p>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 <em>don't</em> arise as the homotopy category of any known model structure on the category of chain complexes!</p>

http://mathoverflow.net/questions/372/derived-categories-and-homotopy-categories/74951#74951 David Roberts 2011-09-08T23:33:39Z 2011-09-08T23:33:39Z

<p>Both give rise to <a href="http://ncatlab.org/nlab/show/derivator" rel="nofollow">derivators</a>, 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) </p>