most recent 30 from http://mathoverflow.net 2013-05-19T13:52:50Z http://mathoverflow.net/feeds/question/39316 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/39316/model-category-with-formally-smooth-morphisms-as-fibrations Harry Gindi 2010-09-19T16:57:19Z 2010-09-19T18:41:17Z

<p>Let's view the category of algebraic spaces as a full subcategory of the category of "spaces" over the opposite category of commutative rings, that is, the category of sheaves on $CRing^{op}$ in the étale topology. Is there any sort of interesting model structure on this category, or a suitable enlargement of it (perhaps by looking at simplicial sheaves or by changing the topology), capturing the theory of formally smooth morphisms (as fibrations)? As a bonus, is there any way to describe the interesting spaces of this category, say algebraic spaces, as a category of fibrant or cofibrant objects? </p> <p>If there is some obvious failure that I'm overlooking, is there any way to rescue the idea? Is there any homotopical content in the definition of formally smooth morphisms by a lifting property?</p>

http://mathoverflow.net/questions/39316/model-category-with-formally-smooth-morphisms-as-fibrations/39320#39320 Peter Arndt 2010-09-19T18:41:17Z 2010-09-19T18:41:17Z

<p>This addresses just the last question "Is there any homotopical content...". It would belong into a comment but doesn't fit.</p> <p>Mathieu Anel shows in a <a href="http://arxiv.org/abs/0902.1130" rel="nofollow">very recommendable article</a> how two classes of maps in the opposite of a locally presentable category, one having the (left/right) lifting property w.r.t. the other, yield factorization systems - every map of the category factors as a map of the left class followed by a map of the right class. To a factorization system he associates a Grothendieck topology (you can just read the two pages addressing this, the article is very readable). Now a category of presheaves over a site is a model category with weak equivalences those morphisms which become isomorphisms after applying sheafification, cofibrations the monomorphisms and fibrations those with the lifting property. This already gives a homotopical content to a lifting system, but yet not as suggested in your question.</p> <p>Now if the topology on a site arose via a lifting system, I would guess that the fibrations of the model structure on presheaves can be described in terms of maps from the right class (I am thinking of something like: a map of sheaves is a fibration if every pullback of it into the realm of affines gives a map from the right class). Or maybe one can find a new model structure on presheaves with the same weak equivalences involving both the left and the right class from the site. I guess this is known to someone, but not to me...</p>