Small model categories? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T08:34:28Z http://mathoverflow.net/feeds/question/108739 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/108739/small-model-categories Small model categories? Shlomi A 2012-10-03T19:22:24Z 2012-10-03T21:50:34Z <p>All the standard examples for model categories are large categories. Is it possible to have a small model category? Are there any interesting examples?</p> <p>EDIT:</p> <p>Since a complete small category is a preorder (proposition V.2.3 in MacLane's Categories), I'd be glad to compromise the limit axioms to be as in Quillen's original definition, demanding only finite limits and colimits. In particular, I don't consider a trivial model structure to be interesting.</p> http://mathoverflow.net/questions/108739/small-model-categories/108746#108746 Answer by Todd Trimble for Small model categories? Todd Trimble 2012-10-03T20:14:16Z 2012-10-03T20:14:16Z <p>Any complete preorder where all isomorphisms are weak equivalences and all morphisms are both fibrations and cofibrations is an example. <a href="http://ncatlab.org/nlab/show/model+category#examples_52" rel="nofollow">Reference.</a> </p> http://mathoverflow.net/questions/108739/small-model-categories/108752#108752 Answer by Zhen Lin for Small model categories? Zhen Lin 2012-10-03T21:40:27Z 2012-10-03T21:40:27Z <p>One of Quillen's original examples was the category of chain complexes of finitely-generated modules over a ring – this is obviously equivalent to a small category, and of course, one has to use Quillen's original definition which only required limits and colimits for <em>finite</em> diagrams, rather than the usual definition nowadays which demands limits and colimits for all <em>small</em> diagram.</p>