User philippe gaucher - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-26T05:33:25Z http://mathoverflow.net/feeds/user/24563 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/129074/cofibrant-replacements-of-a-given-object-in-a-combinatorial-model-category Cofibrant replacements of a given object in a combinatorial model category Philippe Gaucher 2013-04-29T09:00:08Z 2013-05-26T00:22:00Z <p>In a combinatorial model category, every $\lambda$-filtered colimit is a homotopy colimit for $\lambda$ regular big enough. So for $\lambda$ regular big enough, every $\lambda$-filtered colimit of a diagram of cofibrant replacements of a given object $X$ is a cofibrant replacement of $X$. Does the contrary hold, i.e. is the full subcategory of cofibrant replacements of a given object accessible ?</p> <p>EDIT : the class of cofibrations is accessible so for $\lambda$ regular big enough, a $\lambda$-filtered colimit of cofibrations $\varnothing \to X_i$ is a cofibration ; the class of weak equivalences is accessible so for $\lambda$- regular big enough, a $\lambda$-filtered colimit of weak equivalences $X_i\to X$ is a weak equivalence ; so for $\lambda$ regular big enough, a $\lambda$-filtered colimit of cofibrant replacements of $X$ is a cofibrant replacement of $X$.</p> http://mathoverflow.net/questions/129074/cofibrant-replacements-of-a-given-object-in-a-combinatorial-model-category/130244#130244 Answer by Philippe Gaucher for Cofibrant replacements of a given object in a combinatorial model category Philippe Gaucher 2013-05-10T10:08:24Z 2013-05-10T10:08:24Z <p>(sorry I have troubles with comments, I post here even if it is not an answer) I have a new information. In <a href="http://front.math.ucdavis.edu/1304.6974" rel="nofollow">On a fat small object argument</a>, it is proved that in a λ-combinatorial model category, every cofibrant object is a λ-filtered colimit of λ-presentable cofibrant objects, which is close to what I wanted.</p> http://mathoverflow.net/questions/128692/topological-question-about-right-lifting-property-and-the-evaluation-map Topological question about right-lifting property and the evaluation map Philippe Gaucher 2013-04-25T08:14:55Z 2013-04-25T12:51:48Z <p>Let $Z$ be a $\Delta$-generated space (a colimit of simplices -not sure that this hypothesis is important but it is the framework I am working in). The set of continuous maps $Z^{[0,1]}$ from $[0,1]$ to $Z$ is equipped with the <em>$\Delta$-kelleyfication</em> of the compact-open topology (the internal hom of the category, see below).</p> <p><strong>QUESTION :</strong> Suppose that the evaluation map at $0$ from $Z^{[0,1]}$ to $Z$ satisfies the right lifting property with respect to <em>any monomorphism</em> of $\Delta$-generated spaces (i.e. injections). Then I suspect that $Z$ must be discrete. Any counter-example ?</p> <p>Concerning $\Delta$-generated spaces, a short bibliography, just in case that it is important : 1. <em>Notes on Delta-generated spaces</em> by Dugger : <a href="http://math.uoregon.edu/~ddugger/delta.html" rel="nofollow">http://math.uoregon.edu/~ddugger/delta.html</a> 2. The proof that they assemble to a locally presentable category : <em>A convenient category for directed homotopy</em> by Fajstrup-Rosicky : <a href="http://www.tac.mta.ca/tac/volumes/21/1/21-01abs.html" rel="nofollow">http://www.tac.mta.ca/tac/volumes/21/1/21-01abs.html</a> 3. A survey of their properties : Section 2 of <em>Homotopical interpretation of globular complex by multipointed d-space</em> <a href="http://www.tac.mta.ca/tac/volumes/22/22/22-22abs.html" rel="nofollow">http://www.tac.mta.ca/tac/volumes/22/22/22-22abs.html</a></p> http://mathoverflow.net/questions/123864/about-the-cole-strom-model-category-structure-with-a-locally-presentable-category About the Cole-Ström model category structure with a locally presentable category Philippe Gaucher 2013-03-07T12:44:55Z 2013-03-07T19:26:34Z <p>In "Many homotopy categories are homotopy categories" (M. Cole / Topology and its Applications 153 (2006) 1084–1099), Cole generalizes the construction of the model category of Ström to any bicomplete category which is enriched, tensored and cotensored over compactly generated spaces and which satisfies an additional condition which is called <em>the cofibration hypothesis</em> (the mapping path object preserves colimits of towers of strong acyclic cofibrations). Now here are my questions : </p> <p>1) where is exactly used in the proof the fact that the topological spaces are weak Hausdorff (i.e. for any continuous map $f:K\to X$ with $K$ compact Hausdorff, $f(K)$ is closed in $X$) ? </p> <p>2) I ask the question since I would like to use this theorem for a locally presentable category which is enriched, tensored and cotensored over $\Delta$-generated spaces (i.e. colimits of simplices). Suppose that the weak Hausdorff condition above can be dropped, we then just have to check the cofibration hypothesis ; but the mapping path object turns out to be accessible so there is nothing to check. </p> <p>3) If the weak Hausdorff condition is necessary in Cole's theorem, do you think that it can be replaced in the context of $\Delta$-generated spaces by the following separation condition (I don't know its name) : a $\Delta$-generated space $X$ is $\Delta$-Hausdorff if for any continuous map $f:\Delta^n \to X$, $f(\Delta^n)$ is closed in $X$.</p> <p>Concerning $\Delta$-generated spaces, a short bibliography : </p> <ol> <li><em>Notes on Delta-generated spaces</em> by Dugger : <a href="http://math.uoregon.edu/~ddugger/delta.html" rel="nofollow">http://math.uoregon.edu/~ddugger/delta.html</a></li> <li>The proof that they assemble to a locally presentable category : <em>A convenient category for directed homotopy</em> by Fajstrup-Rosicky : <a href="http://www.tac.mta.ca/tac/volumes/21/1/21-01abs.html" rel="nofollow">http://www.tac.mta.ca/tac/volumes/21/1/21-01abs.html</a></li> <li>A survey of their properties : Section 2 of <em>Homotopical interpretation of globular complex by multipointed d-space</em> <a href="http://www.tac.mta.ca/tac/volumes/22/22/22-22abs.html" rel="nofollow">http://www.tac.mta.ca/tac/volumes/22/22/22-22abs.html</a></li> </ol> http://mathoverflow.net/questions/110781/about-the-n-cube About the $n$-cube Philippe Gaucher 2012-10-26T19:27:04Z 2012-10-26T19:55:28Z <p>$[n]$ is the set $\{0,1\}^n$ equipped with the product order $(\epsilon_1,\dots,\epsilon_n) \leq (\eta_1,\dots,\eta_n)$ if and only if $\forall i=1,\dots,n$, $\epsilon_i \leq \eta_i$. Let $$d((\epsilon_1,\dots,\epsilon_n),(\eta_1,\dots,\eta_n)) = \sum_{i=1}^{i=n}|\epsilon_i-\eta_i|.$$ A set map $f$ from $[m]$ to $[n]$ is adjacency-preserving if it is strictly increasing for the product order and if $d((\epsilon_1,\dots,\epsilon_m),(\eta_1,\dots,\eta_m)) = 1$ implies $d(f(\epsilon_1,\dots,\epsilon_m),f(\eta_1,\dots,\eta_m)) = 1$. Example : the adjacency-preserving maps from $[2]$ to itself are $(x_1,x_2)\mapsto (x_1,x_2)$, $(x_1,x_2)\mapsto (x_2,x_1)$, $(x_1,x_2)\mapsto (\min(x_1,x_2),\max(x_1,x_2))$ and $(x_1,x_2) \mapsto (\max(x_1,x_2),\min(x_1,x_2))$.</p> <p>Let $n\geq 3$. Let $f:[n]\to [n]$ be an adjacency-preserving map which commutes with all automorphisms of $[n]$ (the set of automorphisms of $[n]$ is in bijection with the permutation of the set $\{1,\dots,n\}$ by permuting the coordinates). Is $f$ necessarily the identity of $[n]$ ? For $n=2$, $(x_1,x_2)\mapsto (x_2,x_1)$ is a counter-example.</p> <p>PS : the definitions are in this paper <a href="http://www.pps.univ-paris-diderot.fr/~gaucher/symcub.pdf" rel="nofollow">http://www.pps.univ-paris-diderot.fr/~gaucher/symcub.pdf</a> (published paper here <a href="http://dx.doi.org/10.1016/j.tcs.2009.11.013" rel="nofollow">http://dx.doi.org/10.1016/j.tcs.2009.11.013</a> if you have access);</p> http://mathoverflow.net/questions/19325/most-striking-applications-of-category-theory/103370#103370 Answer by Philippe Gaucher for Most striking applications of category theory? Philippe Gaucher 2012-07-28T09:05:19Z 2012-07-28T09:05:19Z <p>The recent developments in homotopical algebra (after 1990) would not be possible without the use of category theory, and more precisely the theory of locally presentable and accessible categories. I am talking about the theory of combinatorial model categories (model categories such that the underlying category is locally presentable). </p> http://mathoverflow.net/questions/103322/bibliographical-reference-needed-characterizing-the-weak-equivalences-of-a-model Bibliographical reference needed (characterizing the weak equivalences of a model category) Philippe Gaucher 2012-07-27T15:16:19Z 2012-07-27T21:12:54Z <p>I need a bibliographical reference for this fact: let $\mathcal{M}$ be a model category such that all objects are cofibrant; then the class of weak equivalences is the class of maps f such that $\mathcal{M}(f,T)/\simeq$ is a bijection for any fibrant object $T$ where $\simeq$ is the homotopy relation. I would prefer a reference in Hirschhorn's book (I have it but I cannot find where it is proved). Thanks in advance.</p> http://mathoverflow.net/questions/100153/left-determined-model-structure-on-delta-generated-topological-spaces Left determined model structure on delta-generated topological spaces Philippe Gaucher 2012-06-20T17:18:49Z 2012-07-06T18:02:09Z <p>Consider the class of cofibrations of the Quillen model structure, restricted to delta-generated topological spaces (the full subcategory of topological spaces generated by the colimits of simplices). Under Vopenka's principle, a left determined model structure w.r.t. a given class of cofibrations always exists if the underlying category is locally presentable. Since delta-generated spaces are locally presentable, how could this left determined model structure w.r.t. Quillen cofibrations look like ? Any idea ?</p> http://mathoverflow.net/questions/100153/left-determined-model-structure-on-delta-generated-topological-spaces/100846#100846 Answer by Philippe Gaucher for Left determined model structure on delta-generated topological spaces Philippe Gaucher 2012-06-28T07:06:43Z 2012-06-28T07:06:43Z <p>I can answer my question now... Not only the Quillen model structure on $\Delta$-generated spaces is left determined, but also the hypothesis $\Delta$-generated can be removed. The left determined model structure exists by Marc Olschok's PhD. The Quillen model structure has the same class of cofibrations and more weak equivalences. So the left determined model structure has more fibrant objects, that is all topological spaces. So the left determined model structure and the Quillen model structure have the same class of cofibrations and the same class of fibrant objects (all topological spaces). Therefore they are equal.</p> http://mathoverflow.net/questions/100158/negative-objects-in-categories/100160#100160 Answer by Philippe Gaucher for Negative objects in categories Philippe Gaucher 2012-06-20T17:53:19Z 2012-06-20T17:53:19Z <p>With Yoneda ? For every object $X$, $Mor(a\oplus (-a),X)$ is a singleton since $a \oplus (-a)$ is initial. And $Mor(a\oplus (-a),X) \cong Mor(a,X) \times Mor((-a),X)$ is a singleton as well. So $Mor(a,X) \cong Mor((-a),X) \cong Mor(0,X)$. Hence $a \cong (-a) \cong 0$. </p> http://mathoverflow.net/questions/129074/cofibrant-replacements-of-a-given-object-in-a-combinatorial-model-category/129177#129177 Comment by Philippe Gaucher Philippe Gaucher 2013-04-30T19:00:42Z 2013-04-30T19:00:42Z It is yes if the category of cofibrations is accessible... I have no more information. http://mathoverflow.net/questions/129074/cofibrant-replacements-of-a-given-object-in-a-combinatorial-model-category/129177#129177 Comment by Philippe Gaucher Philippe Gaucher 2013-04-30T17:13:53Z 2013-04-30T17:13:53Z I believed that &quot;On combinatorial model categories&quot; by J. Rosicky was the answer (<a href="http://www.math.muni.cz/~rosicky/papers/comb2.pdf" rel="nofollow">math.muni.cz/~rosicky/papers/comb2.pdf</a>), and page 8 (top of the page), one can read that $cof(S)$ is not always accessible, $S$ being a set ! On the contrary, $inj(S)$ is always accessible (Proposition 3.3 of the same paper) as a small injectivity class... http://mathoverflow.net/questions/129074/cofibrant-replacements-of-a-given-object-in-a-combinatorial-model-category/129177#129177 Comment by Philippe Gaucher Philippe Gaucher 2013-04-30T16:01:44Z 2013-04-30T16:01:44Z I cannot edit my comment so I rephrase my question. Take a &quot;very good&quot; cofibrant replacement $f.g$ of $X$ ($f$ cofibration and $g$ trivial fibration). Why does it come from a factorization by a functor $T$ constructed using the small object argument ? http://mathoverflow.net/questions/129074/cofibrant-replacements-of-a-given-object-in-a-combinatorial-model-category/129177#129177 Comment by Philippe Gaucher Philippe Gaucher 2013-04-30T15:56:37Z 2013-04-30T15:56:37Z Indeed, the class of accessible categories is closed under lax limits, not under limits. I do understand that your homotopy pullback proves that the class of composables maps (f,g) with $f.g=\varnothing \to X$ is accessible. But the intersection with the image of the functor $T$ does not answer the question. How do you choose $T$ ? Where does it come from ? http://mathoverflow.net/questions/129074/cofibrant-replacements-of-a-given-object-in-a-combinatorial-model-category/129177#129177 Comment by Philippe Gaucher Philippe Gaucher 2013-04-30T06:34:33Z 2013-04-30T06:34:33Z There is a similar (but not sure that it is simpler) problem. In the category of $\Delta$-generated spaces, is the class of cofibrant contractible spaces accessible ? My intuition tells me &quot;yes&quot;. http://mathoverflow.net/questions/129074/cofibrant-replacements-of-a-given-object-in-a-combinatorial-model-category/129177#129177 Comment by Philippe Gaucher Philippe Gaucher 2013-04-30T06:12:48Z 2013-04-30T06:12:48Z I don't understand your proof. Do you mean the map $\mathcal{C}^{[2]}\to \mathcal{C}^{[1]}$ which takes the composite ? And even with that, I still do not understand. By cofibrant replacement of $X$, I mean a pair $(X,f:Y\to X)$ where $Y$ is cofibrant and $f$ is a weak equivalence (I am also interested in the more restricted definition $f$ trivial fibration). And why use homotopy limits ? I believe (but I may be wrong) that the class of accessible categories, unlike locally presentable ones, is closed under a lot of operations like limits. http://mathoverflow.net/questions/129074/cofibrant-replacements-of-a-given-object-in-a-combinatorial-model-category Comment by Philippe Gaucher Philippe Gaucher 2013-04-29T10:39:22Z 2013-04-29T10:39:22Z I added the reason in my question. http://mathoverflow.net/questions/123864/about-the-cole-strom-model-category-structure-with-a-locally-presentable-category/123869#123869 Comment by Philippe Gaucher Philippe Gaucher 2013-03-07T15:22:06Z 2013-03-07T15:22:06Z The paper you mention even gives the answer: top of page 24 : &quot;We should remark that any locally presentable topologically bicomplete category also satisfies our hypothesis&quot;. And as far as I can understand the paper, any convenient category of topological spaces is fine, by convenient it is meant cartesian closed and containing enough topological spaces (e.g. CW-complexes). http://mathoverflow.net/questions/116702/the-quillen-model-structure-on-simplicial-sets-as-a-bousfield-localization Comment by Philippe Gaucher Philippe Gaucher 2012-12-18T15:49:37Z 2012-12-18T15:49:37Z Here is the beginning of an idea: by right-Bousfield localizing by all trivial fibrations, you will reduce the class of cofibrations. So the trivial fibrations must be interpreted as colocal equivalences. So I suggest first a right Bousfield localization by the set of simplices, and then if the new model category has exactly the monomorphisms as cofibrations (?), it should be &quot;between&quot; the minimal model structure and the usual model structure by Cisinski's result (so it should be left proper), then a left Bousfield localization by the accessible class of weak equivalences could work. http://mathoverflow.net/questions/103322/bibliographical-reference-needed-characterizing-the-weak-equivalences-of-a-model Comment by Philippe Gaucher Philippe Gaucher 2012-07-31T04:14:50Z 2012-07-31T04:14:50Z It is a shortcut for denoting the map $\mathcal{M}(Y,T)/\simeq \rightarrow \mathcal{M}(X,T)/\simeq$ if $f:X\rightarrow Y$. http://mathoverflow.net/questions/100153/left-determined-model-structure-on-delta-generated-topological-spaces/100846#100846 Comment by Philippe Gaucher Philippe Gaucher 2012-06-28T21:53:06Z 2012-06-28T21:53:06Z Actually, there is a slightly shorter argument: Marc Olschok already proves in his PhD that, in this case, all objects of the left determined one are fibrant. So the left determined one has the same cofibration and the same fibrant objects than Quillen's, so the two model structures are equal. http://mathoverflow.net/questions/100153/left-determined-model-structure-on-delta-generated-topological-spaces/100846#100846 Comment by Philippe Gaucher Philippe Gaucher 2012-06-28T21:48:21Z 2012-06-28T21:48:21Z I mean: every trivial cofibration of the left determined one is a trivial cofibration of Quillen's, so dually, every fibrant of the Quillen's is fibrant in the left determined one, i.e. all spaces are fibrant in the left determined one. BTW, the same argument also proves that with Vopenka's principle, every combinatorial model category such that all objects are fibrant is left determined, fact that I had never noticed before asking this question. http://mathoverflow.net/questions/100153/left-determined-model-structure-on-delta-generated-topological-spaces Comment by Philippe Gaucher Philippe Gaucher 2012-06-22T18:00:44Z 2012-06-22T18:00:44Z Maybe I should say that I asked this question because I am trying to understand what the homotopy exchange property means for topological spaces. This notion is introduced by Marc Olschok in his PhD to generalize Lafont-Metayer-Worytkiewicz's construction of the folk model structures on globular $\omega$-categories. Marc told me that the homotopy exchange property is satisfied by the cylinder of topological spaces. So unless I am missing something, that implies that the Quillen model structure on delta-generated spaces is left determined, which is very weird. http://mathoverflow.net/questions/100158/negative-objects-in-categories/100160#100160 Comment by Philippe Gaucher Philippe Gaucher 2012-06-20T18:05:20Z 2012-06-20T18:05:20Z Yes, the only hypothesis is $a \oplus (-a)$ initial.