Do simplicial objects in a Topos form a model category? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T09:22:32Z http://mathoverflow.net/feeds/question/80431 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/80431/do-simplicial-objects-in-a-topos-form-a-model-category Do simplicial objects in a Topos form a model category? 36min 2011-11-08T22:33:38Z 2011-11-09T01:06:58Z <p>Sometimes people say "If you don't like the word 'topos', just think the category of Sets", but I'm not sure to what extent this analogy holds. </p> <p>The real question here is, do simplicial object in a topos have the structure of a model category? I'm just not sure if you really need the geometric realization functor (and topological spaces) to define the usual model category structure on simplicial sets.</p> http://mathoverflow.net/questions/80431/do-simplicial-objects-in-a-topos-form-a-model-category/80441#80441 Answer by Peter Arndt for Do simplicial objects in a Topos form a model category? Peter Arndt 2011-11-09T01:00:55Z 2011-11-09T01:06:58Z <p>As has been established in the comments the answer for a general topos is no, while for a Grothendieck topos it is yes, by the work of Joyal. </p> <p>The general question of when one can transfer a model structure on a category based on Sets to an arbitrary Grothendieck topos is beautifully adressed in Tibor Beke's articles on "Sheafifiable Homotopy Model Categories", available <a href="http://faculty.uml.edu/tbeke/" rel="nofollow">here</a>. The examples include simplicial objects, cyclic objects and groupoid and category objects.</p> <p>The proofs really use the assumption that you are in a Grothendieck topos (e.g. he chooses a site defining the given topos and uses the existence of morphisms to the topos of sets, plus the existence of the necessary colimits to interpret infinitary geometric logic, possibly even accessibility somewhere) and I don't think his arguments can be saved for more general toposes. Anyway the papers are worth a look and contain several interesting remarks about the role of accessibility in establishing model structures, giving a good perspective on your original question.</p>