User boris chorny - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T10:50:10Z http://mathoverflow.net/feeds/user/30641 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/95452/uniqueness-of-f-localization/129193#129193 Answer by Boris Chorny for uniqueness of $f$-localization Boris Chorny 2013-04-30T08:51:40Z 2013-04-30T08:51:40Z <p>I think that in general the answer is `no', since your functor $F$ need not preserve $f$-equivalences. Take $f\colon \ast \to \ast$ to be the identity map or any other weak equivalence, then the $f$-local objects are the fibrant objects and $L_f$ is equivalent to the fibrant replacement is simplicial sets. Now take any functor $F$ which does not preserve weak equivalences and takes fibrant values. For example, $F(-) = \mathrm{fib}(\mathrm{hom}(A,-))$, where $\mathrm{fib}$ is a fibrant replacement functor and $A$ a non-trivial space. Then $L_f\circ F$ does not preserve weak equivalences (which are also $f$-equivalences) of non-fibrant simplicial sets.</p> <p>I think that the condition that you need to assume about $F$ is that $F$ preserves $f$-equivalences.</p> http://mathoverflow.net/questions/116702/the-quillen-model-structure-on-simplicial-sets-as-a-bousfield-localization/121566#121566 Answer by Boris Chorny for The Quillen model structure on simplicial sets as a Bousfield localization Boris Chorny 2013-02-12T07:59:42Z 2013-02-12T07:59:42Z <p>Such sequence of localizations/colocalizations does not exist, since <a href="http://www.tac.mta.ca/tac/reprints/articles/6/tr6.pdf" rel="nofollow">homotopy is not concrete</a>. </p> <p>Suppose, for contradiction, that such sequence is constructed, then we obtain a corresponding sequence of reflections/coreflections on the level of homotopy categories, producing a fully faithful embedding of $\mathrm{Ho}(sSet_{\mathrm{standard}})\rightarrow \mathrm{Ho}(sSet_{\mathrm{trivial}})\cong sSet$. Next, it is possible to construct a faithful functor $sSet\rightarrow Set$, say, by sending every simplicial set into the product of the sets of its simplices, obtaining a contradiction with Freyd's theorem. </p> http://mathoverflow.net/questions/111022/is-there-a-notion-of-a-model-category-which-admits-left-bousfield-localization/121403#121403 Answer by Boris Chorny for Is there a notion of a “model category which admits left Bousfield localization?” Boris Chorny 2013-02-10T16:20:03Z 2013-02-10T16:20:03Z <p>One relevant reference, though not answering any of the original questions, is a paper by George Raptis <a href="http://arxiv.org/abs/0907.2726" rel="nofollow">On the cofibrant generation of model categories</a>, where it is shown that under Vopenka's principle, every cofibrantly generated model category is Quillen equivalent to a combinatorial model category. Jiry Rosicky proves the converse direction in <a href="http://arxiv.org/abs/0905.0595" rel="nofollow">Are all cofibrantly generated model categories combinatorial?</a>, so this question is set-theoretical.</p> <p>An example for your second question may be found in the article by Emmanuel Dror Farjoun <a href="http://www.jstor.org/stable/2046572?origin=crossref" rel="nofollow">Homotopy theories for diagrams of spaces</a>, where a non-cofibrantly generated model category is constructed, which I learned to localize in my theses. It admits functorial localizations with respect to any set of maps and even with respect to some particularly nice classes of maps. I do not have a general theory of left Bousfield localizations for such categories, but it can be often constructed from the functorial localization by Bousfield-Friedlander technique. </p> <p>By the way, the localization with respect to a set of maps may not be expressed as a localization with respect to a single map in a general model category. See our paper with Carles Casacuberta <a href="http://arxiv.org/abs/math/0502329" rel="nofollow">The orthogonal subcategory problem in homotopy theory</a> for a simple counterexample.</p> http://mathoverflow.net/questions/116702/the-quillen-model-structure-on-simplicial-sets-as-a-bousfield-localization/121566#121566 Comment by Boris Chorny Boris Chorny 2013-02-14T14:32:16Z 2013-02-14T14:32:16Z You are welcome! I enjoyed thinking about your question. http://mathoverflow.net/questions/111022/is-there-a-notion-of-a-model-category-which-admits-left-bousfield-localization/121403#121403 Comment by Boris Chorny Boris Chorny 2013-02-11T13:36:35Z 2013-02-11T13:36:35Z Previous comment continued. Before we have a bunch of counterexamples of different nature we have no chance to classify model categories which admit Bousfield localizations. http://mathoverflow.net/questions/111022/is-there-a-notion-of-a-model-category-which-admits-left-bousfield-localization/121403#121403 Comment by Boris Chorny Boris Chorny 2013-02-11T13:33:48Z 2013-02-11T13:33:48Z @Fernando: Thanks! I held myself back for quite a long time, but the temptation to join had finally overcome. @David: I have a comment on your third question. I do not know many examples, where the localization does not exist. In fact, the only example that comes to mind (apparently, I learned it from Bill Dwyer) is the following: Consider the category of pointed simplicial sets with the trivial model structure, then the localization with respect to the class of all weak equivalences does not exist, since homotopy is not concrete.