Active questions tagged model-categories - MathOverflow most recent 30 from www.4124039.com 2019-08-18T05:59:32Z http://www.4124039.com/feeds/tag?tagnames=model-categories&amp;sort=newest http://www.creativecommons.org/licenses/by-sa/3.0/rdf http://www.4124039.com/q/337771 9 Non-small objects in categories Peter Bonart http://www.4124039.com/users/143968 2019-08-06T17:02:50Z 2019-08-16T09:10:29Z <p>An object <span class="math-container">$c$</span> in a category is called small, if there exists some regular cardinal <span class="math-container">$\kappa$</span> such that <span class="math-container">$Hom(c,-)$</span> preserves <span class="math-container">$\kappa$</span>-filtered colimits.</p> <p>Is there an example of a (locally small) category <span class="math-container">$C$</span> and an object <span class="math-container">$c$</span> of <span class="math-container">$C$</span>, such that <span class="math-container">$c$</span> is not small, i.e. such that <span class="math-container">$Hom(c,-)$</span> doesn't preserve all <span class="math-container">$\kappa$</span>-filtered colimits for any <span class="math-container">$\kappa$</span> whatsoever?</p> http://www.4124039.com/q/336292 4 Bousfield localization of a left proper accessible model category Philippe Gaucher http://www.4124039.com/users/24563 2019-07-17T06:24:47Z 2019-08-13T20:44:40Z <p>What is known about the Bousfield localization of a left proper <a href="https://ncatlab.org/nlab/show/accessible+model+category" rel="nofollow noreferrer">accessible model category</a> by a set of maps ? (I mean not combinatorial which is already known)</p> http://www.4124039.com/q/338198 4 Homotopy pullback of motivic weak equivalences Alexis http://www.4124039.com/users/144294 2019-08-12T14:16:27Z 2019-08-12T14:16:27Z <p>How to see that whether <span class="math-container">$\mathbb{A}_1$</span>-weak equivalence is closed under homotopy pullback? Let <span class="math-container">$L_{Nis}(Sm_S)$</span> be the Nisnevich localization, how to compute the homotopy pullback of maps <span class="math-container">$\mathbb{A}_1\times_S X\to X$</span> along itself? </p> http://www.4124039.com/q/338195 6 The model category structure on $\mathbf{TMon}$ Matt http://www.4124039.com/users/103150 2019-08-12T13:12:13Z 2019-08-12T13:24:59Z <p>I ask this question here since I asked it <a href="https://math.stackexchange.com/questions/3310262/the-model-category-structure-on-tmon-and-homotopy-colimits">here</a> on Math.SE, and got no answers after a week of a bounty offer.</p> <p>I am trying to understand the homotopy colimit of a diagram of topological monoids, and whether there is an explicit construction of this object (even in the case of simple pushout diagrams).</p> <p>Let <strong>TMon</strong> denote the category of well-pointed topological monoids which have the homotopy types of cell complexes.</p> <p><strong>TMon</strong> can be equipped with a model category structure where the fibrations and weak equivalences are the (Serre) fibrations and weak equivalences of the underlying topological spaces - this comes from section <span class="math-container">$3$</span> of the paper</p> <blockquote> <p>R. Schwänzl and R.M. Vogt, <em>The categories of <span class="math-container">$A_\infty$</span>- and <span class="math-container">$E_\infty$</span>-monoids and ring spaces as closed simplicial and topological model categories</em>, Arch. Math <strong>56</strong> (1991) pp 405–411, doi:<a href="https://doi.org/10.1007/BF01198229" rel="noreferrer">10.1007/BF01198229</a>.</p> </blockquote> <p>Cofibrations are the morphisms which have the appropriate lifting property.</p> <p>I wish to understand what the homotopy colimit of a diagram of topological monoids is. One way of approaching this is to take the colimit of the cofibrant replacement of the diagram in question. This involves (firstly) understanding cofibrations in <strong>TMon</strong>. </p> <p>I struggle with this. I really have no intuition for what a cofibration in this category is at all. This is the first thing preventing me from understanding homotopy colimits of diagrams in <strong>TMon</strong>.</p> <blockquote> <p><strong>Question:</strong> </p> <p>Are there constructions of the homotopy colimit in <strong>TMon</strong> in the case of simple diagrams? For example, if a diagram has morphisms which are all inclusions on the level of topological spaces? Or if the diagram is Reedy? Or under any other sufficiently nice assumptions?</p> </blockquote> http://www.4124039.com/q/338048 3 Proving a Kan-like condition for functors to model categories? Julian Chaidez http://www.4124039.com/users/123015 2019-08-09T23:28:12Z 2019-08-10T08:07:14Z <p>I've been trying to prove this version of the Kan condition for a project that I'm thinking about, and I'm pretty stuck. My experience asking questions about this stuff on MO in the past has been great, so I'm hoping the (higher) category theory people here can help me out!</p> <p><strong>Categories Of Functors:</strong> Let <span class="math-container">$\cal{C}$</span> be a (closed) model category and let <span class="math-container">$\mathcal{S}$</span> be a finite poset. </p> <p>The category of functors <span class="math-container">$[\mathcal{S},\mathcal{C}]$</span> can be imbued with the <em>Reedy model structure</em> (see <a href="https://ncatlab.org/nlab/show/Reedy+model+structure" rel="nofollow noreferrer">[1]</a>). A functor of posets <span class="math-container">$f:\mathcal{S} \to \mathcal{T}$</span> induces a pullback functor <span class="math-container">$$f^*:[\mathcal{T},\mathcal{C}] \to [\mathcal{S},\mathcal{C}]$$</span> This functor fits into a Quillen adjunction with the left Kan extension <span class="math-container">$f_!$</span> (see Barwick <a href="https://arxiv.org/pdf/0708.2832.pdf" rel="nofollow noreferrer">[2]</a>). <span class="math-container">$$ f_!:[\mathcal{S},\mathcal{C}] \leftrightarrow [\mathcal{T},\mathcal{C}]:f^* $$</span> Furthermore, if <span class="math-container">$\iota:\mathcal{S} \to \mathcal{T}$</span> is the inclusion of a sub-poset that is downward closed (i.e. <span class="math-container">$s \in \mathcal{S}$</span> and <span class="math-container">$s' \prec s$</span> implies <span class="math-container">$s' \in \mathcal{S}$</span>) then</p> <p><span class="math-container">$$ \iota^*:[\mathcal{T},\mathcal{C}] \to [\mathcal{S},\mathcal{C}] $$</span> preserves cofibrations.</p> <p>Maybe I should mention for clarity that we are viewing <span class="math-container">$\mathcal{S}$</span> as a directed category, i.e. a Reedy category where <span class="math-container">$\mathcal{S}_+ = \mathcal{S}$</span> and <span class="math-container">$\mathcal{S}_-$</span> is the trivial sub-category with every object. </p> <p>Let me fix some notation for a special sub-category of the functor category <span class="math-container">$[\mathcal{S},\mathcal{C}]$</span>. I haven't totally settled on the definition yet, the conditions are mostly coming from the situation I'm in.</p> <blockquote> <p><strong>Definition 1:</strong> The <em>category <span class="math-container">$\text{Ch}[\mathcal{S},\mathcal{C}]$</span> of <span class="math-container">$\mathcal{S}$</span>-chains in <span class="math-container">$\mathcal{C}$</span></em> is the full sub-category of <span class="math-container">$[\mathcal{S},\mathcal{C}]$</span> consisting of functors <span class="math-container">$x:\mathcal{S} \to \mathcal{C}$</span> such that</p> <ul> <li>(a) <span class="math-container">$x$</span> is a cofibrant diagram with respect to the Reedy model structure.</li> <li>(b) <span class="math-container">$x_S \to x_T$</span> is a quasi-isomorphism for each <span class="math-container">$S \to T$</span> in <span class="math-container">$\mathcal{S}$</span>.</li> </ul> </blockquote> <p>I may want to enhance the assumptions above as so, if it helps prove the result that I'm looking for (Proposition 1 below).</p> <ul> <li>(a') <span class="math-container">$x$</span> is cofibrant and fibrant with respect to the Reedy model structure.</li> <li>(b') <span class="math-container">$x_S \to x_T$</span> is a trivial cofibration for each <span class="math-container">$S \to T$</span> in <span class="math-container">$\mathcal{S}$</span>.</li> </ul> <p>I would also be happy to use the hypothesis that <span class="math-container">$x$</span> is cofibrant in the projective model structure, which is just object-wise cofibrance.</p> <p><strong>Categories Of Strata:</strong> One nice class of posets arises from simplicial complexes. </p> <blockquote> <p><strong>Definition 2:</strong> Let <span class="math-container">$X$</span> be a simplicial complex. The <em>category of strata</em> <span class="math-container">$X\mathcal{S}$</span> is the poset whose objects are simplices <span class="math-container">$S$</span> in <span class="math-container">$X$</span> and where there is a morphism <span class="math-container">$S \to T$</span> if <span class="math-container">$S$</span> is contained in <span class="math-container">$T$</span>.</p> </blockquote> <p>Clearly the map <span class="math-container">$X \to X\mathcal{S}$</span> is functorial. Any map of simplicial complexes <span class="math-container">$f:X \to Y$</span> induces a map of posets <span class="math-container">$f:X\mathcal{S} \to X\mathcal{T}$</span>.</p> <p><strong>Main Question:</strong> The result that I've been trying to prove is the following horn filling property.</p> <blockquote> <p><strong>Proposition 1 (?):</strong> Let <span class="math-container">$\iota:\Lambda^{n,k} \to \Delta^n$</span> denote the standard inclusion of the horn <span class="math-container">$\Lambda^{n,k}$</span> into the <span class="math-container">$n$</span>-simplex <span class="math-container">$\Delta^n$</span>, and let <span class="math-container">$\iota:\Lambda^{n,k}\mathcal{S} \to \Delta^n\mathcal{S}$</span> also denote the induced functor on strata categories. Then the corresponding pullback functor</p> <p><span class="math-container">$$\iota^*:\text{Ch}[\Delta^n\mathcal{S},\mathcal{C}] \to \text{Ch}[\Lambda^{n,k}\mathcal{S},\mathcal{C}]$$</span> admits a section, i.e. a functor <span class="math-container">$\sigma:\text{Ch}[\Lambda^{n,k}\mathcal{S},\mathcal{C}] \to \text{Ch}[\Delta^n\mathcal{S},\mathcal{C}]$</span> with <span class="math-container">$\iota \circ \sigma = \text{Id}$</span>.</p> </blockquote> <p>My main question is the following.</p> <blockquote> <p><strong>Question:</strong> Is Proposition 1 true? What about if I implement some of the possible modifications to Definition 1 suggested above?</p> </blockquote> <p><strong>Ideas For Proof:</strong> Here's a sketch of the proof that I had in mind. </p> <p>You can extend a functor <span class="math-container">$x \in \text{Ch}[\Lambda^{n,k}\mathcal{S},\mathcal{C}]$</span> to a functor that <span class="math-container">$\bar{x} \in [\Delta^n\mathcal{S},\mathcal{C}]$</span> by filling in the two faces <span class="math-container">$T_0,T_1$</span> of <span class="math-container">$\Delta^n$</span> that are missing in <span class="math-container">$\Lambda^{n,k}$</span> with the colimit <span class="math-container">$\text{colim}(x)$</span> and the inclusions <span class="math-container">$S \to T_i$</span> with the colimit maps <span class="math-container">$x_S \to x_{T_i}$</span>. A map <span class="math-container">$x \to y$</span> in <span class="math-container">$\text{Ch}[\Lambda^{n,k}\mathcal{S},\mathcal{C}]$</span> induces a map <span class="math-container">$\bar{x} \to \bar{y}$</span> in an obvious way, and this defines a functor <span class="math-container">$\sigma$</span> as in the Proposition. We need to show that <span class="math-container">$\bar{x}$</span> has properties (a) and (b) from Definition 1 .</p> <p>To show Property (b) from Definition 1, we note that since the nerve of <span class="math-container">$\Lambda^{n,k}\mathcal{S}$</span> is the barycentric sub-division of <span class="math-container">$\Lambda^{n,k}$</span> (thus contractible) and the maps <span class="math-container">$x_S \to x_T$</span> are quasi-isomorphisms, the map <span class="math-container">$x_S \to \text{colim}(x)$</span> is a quasi-isomorphism. </p> <p><strong>Property (a) is the issue:</strong> The colimit <span class="math-container">$\text{colim}(x)$</span> is cofibrant, because the the colimit is the left Kan extension of the pullback by <span class="math-container">$\Lambda^{n,k}\mathcal{S} \to *$</span> and this is a left Quillen adjoint. However, there seems to be no reason for the extended diagram to be cofibrant. You could try cofibrant replacement, but this will ruin the property that <span class="math-container">$\iota^*\bar{x} = x$</span>. </p> <p>I'm not sure if (a') and/or (b') help at all, and switching to the projective model structure (so, just assuming that <span class="math-container">$x$</span> is object-wise cofibrant) seems to ruin the property that the colimit will be cofibrant, which is bad. Anyway, this is where I'm stuck.</p> <p>One last remark is, if I just use the (pointwise) left Kan extension <span class="math-container">$\iota_!$</span> then the quasi-isomorphism property (b) of Definition 1 isn't satisfied in general, as far as I can tell.</p> <p><strong>Thanks:</strong> For reading the long question, and for any help or advice you might have!</p> http://www.4124039.com/q/299958 1 Euclidean model structure on multipointed $d$-spaces Philippe Gaucher http://www.4124039.com/users/24563 2018-05-11T13:53:26Z 2019-08-08T00:01:40Z <p>I use the notation of <a href="http://www.4124039.com/q/135738/24563">this question</a>. A non-decreasing continuous bijection from $[0,a]$ to $[0,b]$ where $a,b\geq 0$ are two real numbers is denoted by $[0,a] \cong^+ [0,b]$. If $\phi:[0,a]\to U$ and $\psi:[0,b]\to U$ are two continuous maps for some topological space $U$ with $\phi(a)=\psi(0)$, the map $\phi*\psi:[0,a+b]\to U$ is the composition of the paths, $\phi$ on $[0,a]$ and $\psi$ on $[a,a+b]$. <strong>All topological spaces are $\Delta$-generated. Therefore all following categories are locally presentable.</strong> </p> <p>A <strong>multipointed $d$-space</strong> $X$ is a variant of Marco Grandis' $d$-spaces. It consists of a topological space $|X|$, a subset $X^0$ (of states) of $|X|$ and a set of continuous maps (called execution paths) $\mathbb{P}^{top}X$ from $[0,1]$ to $|X|$ satisfying the following axioms: </p> <ol> <li>for any $\phi\in \mathbb{P}^{top}X$, $\phi(0)$ and $\phi(1)$ belong to $X^0$</li> <li>for any $\phi\in \mathbb{P}^{top}X$, a composite $[0,1] \cong^+ [0,1] \stackrel{\phi}\longrightarrow |X|$ belongs to $\mathbb{P}^{top}X$</li> <li>if $\phi$ and $\psi$ are two execution paths, all composites like $[0,1] \cong^+ [0,2] \stackrel{\phi*\psi}\longrightarrow |X|$ are execution paths.</li> </ol> <p>Tu summarize, a multipointed $d$-space has not only a distinguished set of continuous paths but also a distinguished set of points (the other points are intuitively not interesting). Unlike Grandis' notion, the constant paths are not necessarily execution paths. It is one of the role of the cofibrant replacement of the model category structure constructed in <a href="http://www.tac.mta.ca/tac/volumes/22/22/22-22abs.html" rel="nofollow noreferrer">Homotopical interpretation of globular complex by multipointed d-space</a> to remove from a multipointed $d$-space all points which do not belong to an execution path. The cofibrant replacement cleans up the underlying space by removing the useless topological structure.</p> <p>It turns out that the model structure constructed in <a href="http://www.tac.mta.ca/tac/volumes/22/22/22-22abs.html" rel="nofollow noreferrer">Homotopical interpretation of globular complex by multipointed d-space</a> is the left determined model category with respect to the set of generating cofibrations $\mathrm{Glob}(\mathbf{S}^{n-1}) \subset \mathrm{Glob}(\mathbf{D}^{n})$ for $n\geq 0$ and the map $\{0,1\} \to \{0\}$ identifying two points where $\mathbf{S}^{n-1}$ is the $(n-1)$-dimensional sphere, $\mathbf{D}^{n}$ the $n$-dimensional disk, and where $\mathrm{Glob}(Z)$ is the multipointed $d$-space whose definition is explained in the paper (I don't think that it is important to recall it in this post).</p> <p>Now here is the question. I would be interested in considering the multipointed $d$-spaces $\vec{[0,1]^n}$ defined as follows</p> <ol> <li>The underlying space is the $n$-cube $[0,1]^n$</li> <li>The set of distinguished states is the set of vertices $\{0,1\}^n \subset [0,1]^n$</li> <li>The set of execution paths is generated by the continuous maps from $[0,1]$ to $[0,1]^n$ such that of course $0$ and $1$ are mapped to a point of $\{0,1\}^n$ and such that these maps are nondecreasing with respect to each axis of coordinates.</li> </ol> <p>The multipointed $d$-space $\partial\vec{[0,1]^n}$ is defined in the same way by removing the interior of the $n$-cube.</p> <p>Using Vopenka's principle and a result of Tholen and Rosicky, there exists a left determined model category structure with respect to the set of generating cofibrations $\partial\vec{[0,1]^n} \subset \vec{[0,1]^n}$ with $n\geq 0$ and $R:\{0,1\}\to \{0\}$. </p> <blockquote> <p>How is it possible to remove Vopenka's principle from this statement ?</p> </blockquote> <p>This question is probably too complicated for a post but if someone could give me a starting point, I would be very grateful. It is the reason why I ask the question anyway. Note: the presence of the map $\{0,1\}\to \{0\}$ in the set of generating cofibrations is not mandatory because I start considering in other parts of my work model structures where I remove this map from the set of generating cofibrations.</p> http://www.4124039.com/q/337556 2 Fibrant objects in Bousfield localization of homotopy pullback closure of Nisnevich hypercovers Nicky http://www.4124039.com/users/124163 2019-08-03T16:39:03Z 2019-08-06T22:59:58Z <p>Let <span class="math-container">$M$</span> be a model topos and <span class="math-container">$S$</span> a set of morphisms, there exists a set of morphism <span class="math-container">$\bar{S}$</span> which is generated by the <span class="math-container">$S$</span>-local equivalences which is closed under homotopy pullbacks in <span class="math-container">$M$</span>. Suppose that <span class="math-container">$M$</span> is left proper combinatorial, then the Bousfield localisation <span class="math-container">$M_{\bar{S}}$</span> is a model topos [Prop.6.2.1.2, HTT, Lurie]. </p> <p>Let <span class="math-container">$sPre(C)$</span> be the category of simplicial presheaves of an essentially small category <span class="math-container">$C$</span>, the fibrant objects of the Bousfield localisation with respect to the set of all hypercovers <span class="math-container">$S$</span> are precisely the <span class="math-container">$S$</span>-local objects, that is, the presheaves of Kan complexes satisfying descent for all hypercovers. Under suitable condition, this localisation is equivalent to the localisation of the set of all bounded hypercovers. And a presheaf satisfies descent for all bounded hypercover is equivalent to that it satisfies ?ech descent. Thus the fibrancy condition can be reformulated for ?ech descent which is a relatively concrete condition. </p> <p>In the case of Nisnevich descent, where <span class="math-container">$C$</span> is a site whose topology is defined by a complete bounded regular cd structure, the fibrancy condition can be stated for the distinguished square, that is, a simplicial presheaf <span class="math-container">$F$</span> is fibrant if and only if <span class="math-container">$F$</span> takes values in Kan complexes and it takes every elementary ditinguished square to a homotopy pullback. </p> <p>Let <span class="math-container">$\alpha$</span> be a Nisnevich distinguished square, <span class="math-container">$\require{AMScd}$</span> <span class="math-container">\begin{CD} W @&gt;&gt;&gt; Y \\ @VVV @VpVV\\ U @&gt;i&gt;&gt; X \end{CD}</span> and let <span class="math-container">$P(\alpha)\to X$</span> be the morphism from the pushout of the upper part <span class="math-container">$U\leftarrow W \rightarrow Y$</span> to <span class="math-container">$X$</span>. </p> <p><strong>Question</strong>: Consider the set of morphism <span class="math-container">$Nis=\{P(\alpha)\to X\}_{\alpha}$</span> and the localisation <span class="math-container">$sPre(C)_{\overline{Nis}}$</span> of <span class="math-container">$\overline{Nis}$</span>, are the fibrant objects of <span class="math-container">$sPre(C)_{\overline{Nis}}$</span> the same as <span class="math-container">$sPre(C)_{{Nis}}$</span>? </p> <p><strong>Further question after edit:</strong> Consider the category <span class="math-container">$sPre(C)_{\mathbb{A}_1,\overline{Nis}}$</span> which has a different model category structure than the motivic homotopy category <span class="math-container">$sPre(C)_{\mathbb{A}_1,{Nis}}$</span>, are the fibrant objects in <span class="math-container">$sPre(C)_{\mathbb{A}_1,\overline{Nis}}$</span> the same as that for <span class="math-container">$sPre(C)_{\mathbb{A}_1,{Nis}}$</span>?</p> <p>I slightly modified the question as now the question is not generally about hypercover as I thought. What I'm looking for are the fibrant objects in the model topos <span class="math-container">$sPre(C)_{\mathbb{A}_1,\overline{Nis}}$</span>. In the comments below, Pavlov mentioned the paper by <a href="https://arxiv.org/pdf/1704.08467.pdf" rel="nofollow noreferrer">Strunk and Raptis</a>, </p> <blockquote> <p>Let <span class="math-container">$\mathcal{C}$</span> be a small simplicial category. Then there is a bijective correspondence between Grothendieck topology <span class="math-container">$\bar{\tau}$</span> on <span class="math-container">$Ho(\mathcal{C})$</span> and homotopy left exact left Bousfield localization of <span class="math-container">$sPSh^{\Delta}(\mathcal{C})$</span> which are <span class="math-container">$t$</span>-complete. </p> </blockquote> <p>where a <span class="math-container">$\mathcal{U}$</span>-local model topos <span class="math-container">$sPSh^{\Delta}(Sm_S)_{\mathcal{U}Nis}$</span> is given together with its Grothendieck topology arising from Nisnevich topology on the ordinary category <span class="math-container">$Sm_S$</span>. <span class="math-container">$sPSh^{\Delta}(\mathcal{C})$</span> denotes the functor category of simplicial functors <span class="math-container">$\mathcal{C}^{op}\to \mathcal{sSet}$</span> for a small simplicial category <span class="math-container">$\mathcal{C}$</span>. A Grothendieck topology for a simplicial category <span class="math-container">$\mathcal{C}$</span> is by definition a Grothendieck topology on <span class="math-container">$Ho(\mathcal{C})$</span>. </p> <p>Given a Grothendieck topology above, the model structure of the localization can be described by hypercovers of this topology. By this theorem, there should be a Grothendieck topology for <span class="math-container">$Ho(Sm_k)$</span> that gives the localisation <span class="math-container">$(sPSh^{\Delta}(Sm_S)_{\mathcal{U}Nis})_{\overline{\mathcal{H}(Nis)}}\cong sPsh(Sm_k)_{\mathcal{A_1},\overline{Nis}}$</span> of <span class="math-container">$sPsh^{\Delta}(Sm_k)$</span>. The Grothendieck topology of the <span class="math-container">$\mathcal{U}$</span>-local model topos <span class="math-container">$sPSh^{\Delta}(Sm_S)_{\mathcal{U}Nis}$</span> arises from the Nisnevich topology of <span class="math-container">$Sm_k$</span> as ordinary category. So this topology isn't the one that gives the final localization.</p> <p><strong>How can one determine this Grothendieck topology hence find the fibrant objects of <span class="math-container">$sPre(C)_{\mathbb{A}_1,\overline{Nis}}$</span>?</strong> </p> http://www.4124039.com/q/277513 5 Maximal Cisinski model structure on simplicial sets Valery Isaev http://www.4124039.com/users/62782 2017-07-29T07:13:34Z 2019-08-02T00:36:46Z <p>This is a very simple question coming from the observation that every (pre)sheaf category has the maximal Cisinski model structure on it. This is the Cisinski model structure with the smallest class of weak equivalences possible. </p> <p>Now, it is natural to ask: what is the maximal Cisinski model structure on the most canonical category in this setting, namely the category of simplicial sets? Is it larger than the Joyal model structure? Assuming that the answer is "yes", can we give an explicit description of its weak equivalences and fibrant objects?</p> http://www.4124039.com/q/282510 4 Filtered colimit of fibrations user95222 http://www.4124039.com/users/0 2017-10-02T09:30:01Z 2019-07-30T10:41:08Z <p>In a model category $\mathcal{C}$, is the filtered colimit of fibrations, resp. trivial fibrations, a fibration, resp. trivial fibration?</p> <p>Thm. 1.2.3.5 in Toen-Vezzosi's "Homotopical algebraic geometry, II" (<a href="https://arxiv.org/pdf/math/0404373.pdf" rel="noreferrer">https://arxiv.org/pdf/math/0404373.pdf</a>) seems to give a criterion, but it points to the wrong reference, as nothing of the sort is in Hovey's book "Model Categories".</p> http://www.4124039.com/q/337098 5 When do zigzags of weak equivalences detect isomorphisms in the localization? Valery Isaev http://www.4124039.com/users/62782 2019-07-27T18:21:21Z 2019-07-27T18:21:21Z <p>The usual way to prove that two model categories are equivalent is to construct a zigzag of Quillen equivalences between them, but is it always possible? We can ask a more general question.</p> <blockquote> <p><strong>Question:</strong> Suppose that objects <span class="math-container">$X$</span> and <span class="math-container">$Y$</span> of a <a href="https://ncatlab.org/nlab/show/category+with+weak+equivalences" rel="noreferrer">category with weak equivalences</a> <span class="math-container">$(\mathcal{C},\mathcal{W})$</span> are isomorphic in its localization. Under which conditions on <span class="math-container">$(\mathcal{C},\mathcal{W})$</span> is there a zigzag of weak equivalences between <span class="math-container">$X$</span> and <span class="math-container">$Y$</span>?</p> </blockquote> <p>This is true if <span class="math-container">$(\mathcal{C},\mathcal{W})$</span> is a model category, but is there a weaker condition? It is not true in general. The counterexample is the category <span class="math-container">$A \to B \to C \to D$</span> in which <span class="math-container">$A \to C$</span> and <span class="math-container">$B \to D$</span> are weak equivalences. It is also not enough to assume that <span class="math-container">$(\mathcal{C},\mathcal{W})$</span> satisfies 2-out-of-6. The counterexample is the simplest category with a retract of a weak equivalence. So, here are some natural conditions on <span class="math-container">$(\mathcal{C},\mathcal{W})$</span> which might be sufficient:</p> <ul> <li><span class="math-container">$(\mathcal{C},\mathcal{W})$</span> satisfies 2-out-of-6 and is closed under weak retracts (a weak retract is like an ordinary retract, but identity maps are replaced with weak equivalences).</li> <li><span class="math-container">$\mathcal{W}$</span> is <a href="https://ncatlab.org/nlab/show/two-out-of-six+property#saturation" rel="noreferrer">saturated</a> (this implies the previous condition).</li> <li><span class="math-container">$(\mathcal{C},\mathcal{W})$</span> has the structure of a category of fibrant objects.</li> <li><span class="math-container">$\mathcal{C}$</span> is the category of (small) model categories and Quillen adjunctions between them and <span class="math-container">$\mathcal{W}$</span> is the class of Quillen equivalences.</li> </ul> http://www.4124039.com/q/336735 3 Strøm model structure on nonnegatively graded chain complexes Najib Idrissi http://www.4124039.com/users/36146 2019-07-22T14:06:58Z 2019-07-22T16:27:20Z <p>Let <span class="math-container">$\newcommand{\Ch}{\mathsf{Ch}}\Ch_{\ge 0}(R)$</span> be the category of <span class="math-container">$\mathbb{N}$</span>-graded chain complexes over some ring <span class="math-container">$R$</span>, and <span class="math-container">$\Ch(R)$</span> the category of <span class="math-container">$\mathbb{Z}$</span>-graded chain complexes.</p> <p>The standard ("Quillen") projective model structure on <span class="math-container">$\Ch_{\ge 0}(R)$</span> has quasi-isomorphisms for weak equivalence, monomorphisms with projective cokernel as cofibrations, and epimorphisms in positive degrees as fibrations. There is also an injective model structure with weak equivalence, monomorphisms in positive degrees, and epimorphisms with injective kernel.</p> <p>On <span class="math-container">$\Ch(R)$</span>, there are similar injective/projective model structures, but one drops the "positive degrees" where relevant. There is also a different, "Strøm" model structure on <span class="math-container">$\Ch(R)$</span>, which is worked out in Chapter 18 of <em>More Concise Algebraic Topology</em> (May–Ponto), see e.g. <a href="http://www.4124039.com/q/52508/36146">this MO question</a>. It has homotopy equivalences as weak equivalences, split monomorphisms as cofibrations, and split epimorphisms as fibrations.</p> <p>Is there an analogue of this "Strøm" model structure for nonnegatively-graded chain complexes? [Wild guess] Perhaps two of them, one with "split epis in positive degrees &amp; split monos" and one with "split monos in positive degrees and split epis"?</p> http://www.4124039.com/q/247953 8 When does the natural simplicial enrichment of the category of cdgas compute the derived mapping space? David Carchedi http://www.4124039.com/users/4528 2016-08-21T16:25:37Z 2019-07-19T08:45:43Z <p>Let $CDGA$ be the category of commutative differential graded algebras over a field $k$ of characteristic zero. Denote by $\Omega\left(\Delta^n\right)$ the cdga of algebraic differential forms on the $n$-simplex. There is a natural simplicial enrichment of CDGA given by $Map_n\left(A,B\right)=Hom(A,B \otimes \Omega\left(\Delta^n\right))$ which computes the correct derived mapping space (for the projective model structure) whenever $A$ is quasi-free (i.e. free with a possibly non-trivial differential), or more generally, when $A$ is projectively cofibrant. The main ingredient in showing this is showing that for any CDGA $A,$ if $A \to C$ is a quasi-free $A$-algebra, then the induced map $$Map(C,B) \to Map(A,B)$$ is a trivial Kan fibration. (This is true more generally for any trivial cofibration with respect to the projective model structure, but these are all retracts of quasi-free maps.) My question is, are there more general maps other than trivial cofibrations which get sent to trivial Kan fibrations by $Map(blank,B)$? For instance, is this more generally true for any map $A \to C$ which as a map of graded algebras is of the form $A \to A \otimes K,$ with $K$ acyclic (but not necessarily free)?</p> http://www.4124039.com/q/336116 3 The k-ification of the compact-open topology for weak Hausdorff compactly generated spaces Joao Faria Martins http://www.4124039.com/users/99088 2019-07-14T15:14:17Z 2019-07-15T16:51:37Z <p>Let CGWH be the category of weak Hausdorff compactly generated spaces; see e.g.</p> <p>N.P. Strickland. THE CATEGORY OF CGWH SPACES: Preprint available from <a href="https://neil-strickland.staff.shef.ac.uk/courses/homotopy/cgwh.pdf" rel="nofollow noreferrer">https://neil-strickland.staff.shef.ac.uk/courses/homotopy/cgwh.pdf</a>. 2009</p> <p>or</p> <p>Tammo tom Dieck. Algebraic topology. Zurich: European Mathematical Society (EMS), 2008. (7.9)</p> <p>CGWH is well known to be a cartesian closed category (references above prove this). However, there appears to be two different ways to define the topology on the space TOP(X,Y) of continuous maps. Strickland topologises TOP(X,Y) with the k-ification of the topology with sub-basis <span class="math-container">$$\{f \colon X \to Y \,\, |\,\, f(u(K))\subset A\},$$</span> where <span class="math-container">$u\colon K \to X$</span> is a continuous map, where <span class="math-container">$K$</span> is compact Hausdorff, and <span class="math-container">$A\subset Y$</span> is open. In the CGWH case, and by using Lemma 1.4(b) of Strickland's paper, this coincides with the definition in May's "A concise course in Algebraic topology" chapter 5, where TOP(X,Y) is topologised with the k-ification of the topology with sub-basis: <span class="math-container">$$\{f \colon X \to Y \,\,|\,\, f(K)\subset A\},$$</span> where <span class="math-container">$K$</span> is a compact Hausdorff subset of <span class="math-container">$X$</span>, and <span class="math-container">$A\subset Y$</span> is open.</p> <p>However, tom Dieck topologises <span class="math-container">$TOP(X,Y)$</span> with the k-ification of the topology with sub-basis <span class="math-container">$$\{f \colon X \to Y \,\,|\,\, f(K)\subset A\},$$</span> where <span class="math-container">$K$</span> is a compact (not necessarily Hausdorff, it seems) subset of <span class="math-container">$X$</span>, and <span class="math-container">$A$</span> is open. This is also the "official" function space topology in appendix A1 of Rudolf Fritsch and Renzo A. Piccinini. Cellular structures in topology, volume 19 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 1990.</p> <p>These two topologies in TOP(X,Y) are clearly the same if <span class="math-container">$X$</span> is Hausdorff. Do they coincide in general?</p> <p>EDIT: As Ivan Yudin mentioned below, given that we have the adjunction between <span class="math-container">$(\_)\times Y$</span> and <span class="math-container">$C(Y,\_)$</span>, where <span class="math-container">$X \times Y$</span> is topologised with the k-ification of the product topology, it then follows, assuming all references (and my interpretation of them) are correct that given CGWH spaces <span class="math-container">$X$</span> and <span class="math-container">$Y$</span> the two topologies on <span class="math-container">$TOP(X,Y)$</span>, which k-ify the topologies with sub-basis <span class="math-container">$\{f \colon X \to Y \,\,|\,\, f(K)\subset A\}$</span> <span class="math-container">$A\subset Y$</span> open, and <span class="math-container">$K \subset X$</span> compact, and with sub-basis <span class="math-container">$\{f \colon X \to Y \,\,|\,\, f(K)\subset A\}$</span> <span class="math-container">$A\subset Y$</span> open, and <span class="math-container">$K \subset X$</span> compact and Hausdorff coincide.</p> http://www.4124039.com/q/335742 6 Homotopy pullbacks and pushouts in stable model categories stblhtpy http://www.4124039.com/users/142851 2019-07-08T21:28:30Z 2019-07-11T14:41:15Z <p>There are lots of similar questions that have been answered on this topic (particularly <a href="http://www.4124039.com/questions/135462/homotopy-limit-colimit-diagrams-in-stable-model-categories">Homotopy limit-colimit diagrams in stable model categories</a>), but I have a specific question that I do not believe has been answered. I suspect there is a simple answer that I'm just not seeing, so apologize in advance if this is the case. Given a commutative square <span class="math-container">$\require{AMScd}$</span> <span class="math-container">\begin{CD} A @&gt;{}&gt;&gt; X\\ @V{f}VV @V{g}VV \\ B @&gt;{}&gt;&gt; Y \end{CD}</span> we say it is a <strong>homotopy pullback square</strong> if the canonical map <span class="math-container">$A \to holim(B \to Y \leftarrow X)$</span> is a weak equivalence and that it is a <strong>homotopy pushout square</strong> if the canonical map <span class="math-container">$hocolim(B \leftarrow A \rightarrow X) \to Y$</span> is a weak equivalence. </p> <p>I'm hoping to understand the proof that in a stable category, the two notions coincide. As pointed out in the reference above, in the 2007 version of Hovey's ``Model Categories,'' an argument is given in Remark 7.1.12 that goes roughly as follows. One can show that the square above is a homotopy pullback square if and only if <span class="math-container">$$ hofib(f) \overset{\sim}{\to} hofib(g) $$</span> is a weak equivalence. Similarly, it is a homotopy pushout square if and only if <span class="math-container">$$ hocofib(f) \overset{\sim}{\to} hocofib(g) $$</span> is a weak equivalence. One then concludes the result by claiming that <span class="math-container">$$ \Sigma hofib(f) \simeq hocofib(f) \text{ and } \Sigma hofib(g) \simeq hocofib(g) $$</span> This is the part on which I am stuck. Hovey claims that it follows from Thereom 7.1.11 (which is the same in both the 1999 and 2007 version) but I do not see this. It seems to me (and here is where I must be going wrong) that Hovey is claiming that the fiber in the homotopy category is the homotopy fiber -- because from what I understand, it is the homotopy category which is triangulated (to which 7.1.11 would apply), not the original stable model category. Since the fiber in the homotopy category is <em>not</em> (usually) the homotopy fiber, I must be misinterpreting what Hovey is saying.</p> <p>So my question is: how do we know that the suspension of the homotopy fiber is weakly equivalent to the homotopy cofiber (or, equivalently, that loops of the homotopy cofiber is weakly equivalent to the homotopy fiber). Am I misunderstanding what Hovey means by "fiber sequence?" Are the distinguished triangles actually <em>homotopy</em> cofiber sequences in the original model category? As is probably clear by now, I do not know much about triangulated categories. Any help/reference would be appreciated!</p> http://www.4124039.com/q/285988 5 Stable Dold-Kan correspondence and symmetric group actions iron feliks http://www.4124039.com/users/110510 2017-11-13T19:54:48Z 2019-06-23T17:01:39Z <p>There exists a Quillen equivalence between <span class="math-container">$HRModSpectra$</span> (model category of ring spectra over Eilenberg-MacLane spectra <span class="math-container">$EM(R)$</span>, where <span class="math-container">$R$</span> is a commutative ring, with stable model structure) and <span class="math-container">$Ch$</span> (model category of unbounded chain complexes of <span class="math-container">$R$</span>-modules).</p> <p>I was wondering what the Quillen functors are that give the above Quillen equivalence.</p> <p>One can start with an unbounded chain complex <span class="math-container">$X$</span> and apply the Dold-Kan functor <span class="math-container">$\Gamma$</span> to chain complex <span class="math-container">$X_{\geq 0}$</span> to get a simplicial abelian group <span class="math-container">$\Gamma(X_{\geq 0})$</span>, and then consider <span class="math-container">$\Gamma(X[-n]_{\geq0})$</span> (shifting <span class="math-container">$X$</span> to the left by n places, truncating and then applying <span class="math-container">$\Gamma$</span>). This way one gets an <span class="math-container">$\Omega$</span>-spectrum <span class="math-container">$Y$</span> = {<span class="math-container">${Y_{0}, Y_{1},...}$</span>}, with <span class="math-container">$Y_{n} = \Gamma(X[-n]_{\geq0})$</span>. </p> <p>How does then one proceed to prove that <span class="math-container">$Y$</span> is a symmetric spectrum? For that one needs an action of symmetric group <span class="math-container">$S_{n}$</span> on <span class="math-container">$Y_{n}$</span>. Now each <span class="math-container">$Y_{i}$</span> is in fact as a simplicial set equivalent to <span class="math-container">$\prod K(\pi_{k}(Y_{n}), k)$</span> and <span class="math-container">$K(\pi_{n}(Y_{n}), n)$</span> has an <span class="math-container">$S_{n}$</span> action, though I'm not sure if this action will satisfy the compatibility conditions that are required of a symmetric spectrum. </p> http://www.4124039.com/q/332942 5 Compact objects in the $\infty$-category presented by a simplicial model category Saal Hardali http://www.4124039.com/users/22810 2019-05-31T11:29:47Z 2019-06-17T19:49:01Z <p>Let <span class="math-container">$\mathsf{M}$</span> be a <strong>simplicial model category</strong> presenting an <span class="math-container">$\infty$</span>-category <span class="math-container">$\mathcal{M}$</span>. I'm interested in a general statement relating compact objects in <span class="math-container">$\mathcal{M}$</span> (in the <span class="math-container">$\infty$</span>-categorical sense) with the compact objects in <span class="math-container">$\mathsf{M}$</span>. Here's roughly what I expect to be true but if i'm missing some assumptions or if some are redundant feel free to phrase the correct statement as an answer.</p> <p>Suppose further that <span class="math-container">$\mathsf{M}$</span> satisfies that <strong>weak equivalences between fibrant objects are stable under filtered colimits</strong>. Then is the following true </p> <blockquote> <ol> <li><p>Let <span class="math-container">$X$</span> be a <strong>compact cofibrant object</strong> in <span class="math-container">$\mathsf{M}$</span>. Is <span class="math-container">$X$</span> compact as an object in <span class="math-container">$\mathcal{M}$</span>? </p></li> <li><p>Is every compact object in <span class="math-container">$\mathcal{M}$</span> a retract of (the image in <span class="math-container">$\mathcal{M}$</span>) of some compact object in <span class="math-container">$\mathsf{M}$</span>?</p></li> </ol> </blockquote> http://www.4124039.com/q/333996 1 Existence of tensor product of infinity operads Andrea Marino http://www.4124039.com/users/140013 2019-06-14T12:13:05Z 2019-06-14T13:18:03Z <p>I am trying to show, or find a reference, for the following fact:</p> <p>"Given O,P two infinity operads [in the sense of Lurie, HA, Definition 2.1.1.10], there always exist a tensor product".</p> <p>In other words, this means the following. One can consider an infinity operad as a simplicial set over <span class="math-container">$K=N(Fin_*)$</span> with some marked edges, namely the inert maps. We call the (<span class="math-container">$\infty$</span>) category of pairs <span class="math-container">$(X,\Gamma)$</span> where <span class="math-container">$X$</span> is a simplicial set and <span class="math-container">$\Gamma$</span> is a subset of edges that contains degenerate edges a <em>marked simplicial set</em>, denoted by <span class="math-container">$sSet^+_{/K}$</span>.</p> <p>Then, by the very definition, a map of infinity operads is a morphism as marked simplicial sets. Note that the product of two infinity operads, as marked simplicial set, is not necessarily an infinity operad. Indeed, one can show that there exist a model category on <span class="math-container">$sSet^+_{/K}$</span> such that infinity operads are exactly the fibrant objects, and every object is cofibrant.</p> <p>Define bifunctors from <span class="math-container">$O,P$</span> to <span class="math-container">$Q$</span> (everything being an infinity operad) as <span class="math-container">$$Map_{sSet^+_{/K}}(O\times P, Q) $$</span> We say that <span class="math-container">$\alpha: O \times P \to Q$</span> exhibit Q as the tensor product of O,P if, for any other ifninity operad R, the composition map</p> <p><span class="math-container">$$ \alpha^*: Map_{sSet^+_{/K}}(Q,R) \to Map_{sSet^+_{/K}}(O \times P,R) $$</span> is an equivalence of sSets. </p> <p>In literature is often said that it is enough to take a fibrant replacement of <span class="math-container">$O\times P$</span> in the category of marked sSets over K. I am trying to show, and here there are my attempts:</p> <ol> <li>Note that <span class="math-container">$\alpha^*$</span> is surjective in every degree. Indeed the diagram determined by <span class="math-container">$$O \times P \times (\Delta^n)^{\#} \to R$$</span> <span class="math-container">$$ O \times P \times (\Delta^n)^{\#} \to Q \times (\Delta^n)^{\#}$$</span> <span class="math-container">$$Q\times (\Delta^n)^{\#} \to 0,R \to 0$$</span></li> </ol> <p>admits a lifting, because <span class="math-container">$R \to 0$</span> is a fibration and <span class="math-container">$O \times P \to Q$</span> is an acyclic cofibration. Recall that <span class="math-container">$(\Delta^n)^{\#}$</span> is the (left) canonical marked simplicial set associated to the n-cell, with just degenerate edges marked. This is the cosimplicial object we use for maps in degree n.</p> <ol start="2"> <li>For every marked S, the sequence</li> </ol> <p><span class="math-container">$$ S \times (\Delta^n)^{\#} \coprod S \times (\Delta^n)^{\#} \to S \times (\Delta^{n+1})^{\#} \to S \times (\Delta^n)^{\#} $$</span></p> <p>exhibits <span class="math-container">$ S \times (\Delta^{n+1})^{\#}$</span> as a canonical cylinder object of <span class="math-container">$S \times (\Delta^n)^{\#}$</span> (canonical meaning that the last is a fibration). This is true because it is at the level of <span class="math-container">$\Delta^n$</span>, and multiplying by <span class="math-container">$Q$</span> preserve at least cofibrations (~injectives) and equivalences. </p> <ol start="3"> <li><p>I would hope that, with some being fibrant argument, one can show that every left homotopy can be done via tha cylinder object above.</p></li> <li><p>As the map <span class="math-container">$O\times P \to Q$</span> is a weak equivalence, both are cofibrant, and <span class="math-container">$R$</span> is fibrant, we have that <span class="math-container">$\alpha^*$</span> is an equivalence between <span class="math-container">$Map(O \times P, R)_n/ homotopy$</span> and <span class="math-container">$Map(Q, R)_n /homotopy$</span>.</p></li> </ol> <p>Even if everything goes in the good direction, it seems that these facts point at <span class="math-container">$\alpha_*$</span> being a <em>weak</em> equivalence, and not an equivalence. In other words, it hints at the fact that it may be a simplicial model category (see <a href="https://ncatlab.org/nlab/show/simplicial+model+category" rel="nofollow noreferrer">https://ncatlab.org/nlab/show/simplicial+model+category</a>, definition 2.1, point 3). But how the hell showing that fibers are contractible and not weakly contractible? </p> http://www.4124039.com/q/327512 7 Why does the cotangent complex really have a distinguished triangle? Meow http://www.4124039.com/users/119012 2019-04-08T18:50:53Z 2019-06-12T21:14:38Z <p>Associated to any ring maps <span class="math-container">$A\to B\to C$</span> there is the distinguished triangle <span class="math-container">$$\mathbf{L}_{B/A}\otimes^L_BC\ \longrightarrow \ \mathbf{L}_{C/A} \ \longrightarrow \ \mathbf{L}_{C/B} \ \stackrel{+1}{\longrightarrow} \ $$</span> in <span class="math-container">$D(C)$</span>. The cotangent complex <span class="math-container">$\mathbf{L}_{C/A}$</span> is (the value at <span class="math-container">$C$</span> of) the Quillen left derived functor of <span class="math-container">$$C\otimes_-\Omega^1_{-/A} \ : \ \text{simplicial }A\text{ rings over }C\ \longrightarrow \ \text{simplicial }C\text{ modules}$$</span> Note that <span class="math-container">$D(C)$</span> is the homotopy category of simplicial <span class="math-container">$C$</span> modules, by Dold-Kan.</p> <blockquote> <p>Is there a deeper reason for the triangle, or is it something very special about <span class="math-container">$\mathbf{L}$</span>? i.e. can we say anything about when a derived Quillen functor (say between stable model categories) admits a long exact sequence like this?</p> </blockquote> <p>The question must be fairly subtle because already the result fails if we replace rings with general schemes in the above, except under certain conditions on the maps <span class="math-container">$X\to Y\to Z$</span>.</p> http://www.4124039.com/q/333398 3 On cofibrations of simplicially enriched categories F.Abellan http://www.4124039.com/users/141150 2019-06-06T13:28:53Z 2019-06-06T13:28:53Z <p>Let <span class="math-container">$\mathbb{C}$</span> be an strict 2-category and denote by <span class="math-container">$C$</span> is underlying 1-category viewed as as a 2-category only having identity 2-cells.</p> <p>We have a canonical inclusion functor ,</p> <p><span class="math-container">$$i: C \longrightarrow \mathbb{C}.$$</span></p> <p>The usual nerve functor,</p> <p><span class="math-container">$$N: \mathsf{Cat} \longrightarrow \mathsf{Set}_{\Delta},$$</span> preserves limits therefore it induces another functor,</p> <p><span class="math-container">$$N_{2}: 2\text{-}\mathsf{Cat} \longrightarrow \mathsf{Cat}_{\Delta}$$</span> obtained by applying nerve to each Hom-category, where <span class="math-container">$\mathsf{Cat}_{\Delta}$</span> is the category of simplicially enriched categories.</p> <p>Let us consider <span class="math-container">$\mathsf{Cat}_{\Delta}$</span> endowed with the Joyal-enriched model structure. I would like to know if then <span class="math-container">$N(i)$</span> would be a cofibration of simplicially enriched categories.</p> <p>If this is not true in general I would be very interested in knowing if there are some conditions on <span class="math-container">$\mathbb{C}$</span> that ensure that the induced map will be a cofibration.</p> http://www.4124039.com/q/332759 2 Homotopy colimits of simplicial objects Edoardo Lanari http://www.4124039.com/users/57280 2019-05-29T10:23:23Z 2019-05-29T10:23:23Z <p>Given a simplicial combinatorial model category <span class="math-container">$\mathcal{M}$</span> and a simplicial diagram <span class="math-container">$F\colon \Delta^{\mathrm{op}} \rightarrow \mathcal{M}$</span>, is there a nice (i.e. explicitely computable) way of expressing its homotopy colimit? For instance, if one could give a cofibrant replacement <span class="math-container">$\widetilde{F}$</span> of <span class="math-container">$F$</span> (with respect to the projective model structure) which is quite explicit at least in dimension 0 and 1, by which I mean the value on the restriction <span class="math-container">$\Delta^{\mathrm{op}}_{\leq 1}$</span>, then the coequalizer of <span class="math-container">$\widetilde{F}_1\stackrel{\longrightarrow}{\longrightarrow}\widetilde{F}_0$</span> would do the job.</p> http://www.4124039.com/q/332597 4 Right adjoint preserving (trivial) cofibrations between cofibrant objects Philippe Gaucher http://www.4124039.com/users/24563 2019-05-27T14:09:23Z 2019-05-27T14:09:23Z <p>Consider a right Quillen adjoint which is not a categorical left adjoint which takes (trivial resp.) cofibrations between cofibrant objects to (trivial resp.) cofibrations between cofibrant objects.</p> <blockquote> <p>Question: does this situation have a name ? Is it studied abstractly somewhere ? </p> </blockquote> <p>I would like to have other examples like mine to help me to separate what is a general fact from what is particular to the objects I am studying.</p> http://www.4124039.com/q/331904 1 Which set of compact objects generates the subcategory of a compactly generated stable model category? Doelt_k http://www.4124039.com/users/109602 2019-05-19T00:07:30Z 2019-05-19T08:12:12Z <p>I couldn't find any info on what set of compact objects generates the following subcategory:</p> <blockquote> <p>Let <span class="math-container">$k$</span> be a field of positive characteristic and let <span class="math-container">$G$</span> be either a finite group or a finite group scheme over <span class="math-container">$k$</span>. Then let <span class="math-container">$\mathrm{stab}(k[G])$</span> be a stable model category consiting of finitely generated <span class="math-container">$k[G]$</span>-modules (modulo the projectives). It is a subcategory of <span class="math-container">$\mathrm{Stab}(k[G])$</span> - the stable model category of all <span class="math-container">$k[G]$</span>-modules (modulo the projectives). For <span class="math-container">$\mathrm{Stab}(k[G])$</span>, it is known that compact objects are precisely finitely-generated modules and the simple modules generate <span class="math-container">$\mathrm{Stab}(k[G])$</span>.</p> </blockquote> <p>More generally, let <span class="math-container">$\mathcal{K}$</span> be a compactly generated stable model category with a set <span class="math-container">$\mathcal{C}$</span> of compact objects and a set <span class="math-container">$\mathcal{G}$</span> of compact generators. What are the compact generators of the subcategory of <span class="math-container">$\mathcal{K}$</span> spanned by objects in <span class="math-container">$\mathcal{C}$</span>? I realize that this may be unknown, so I'd appreciate any particular examples (I'm more interested in cases where the orignal stable model category has a set of compact generators rather than a single one).</p> <p>For the definitions I use, I refer to <a href="http://homepages.math.uic.edu/~bshipley/classTopFinal.pdf" rel="nofollow noreferrer">Schwede-Shipley</a>. Of course, substituting the term "stable model category" for "triangulated category" across my question makes no difference.</p> http://www.4124039.com/q/330750 2 Is the Hurewicz model category left proper? Dmitry Vaintrob http://www.4124039.com/users/7108 2019-05-05T00:35:56Z 2019-05-05T01:09:34Z <p>A model structure is left proper if the pushout of a weak equivalence along a cofibration is a weak equivalence. In the Hurewicz (or Strom) model structure on the category of topological spaces, weak equivalences are homotopy equivalences and cofibrations are defined in terms of a lifting property for Hurewicz fibrations. Is this model structure left proper?</p> http://www.4124039.com/q/330537 4 Homotopy limit over a diagram of nullhomotopic maps Charles http://www.4124039.com/users/117088 2019-05-02T14:05:39Z 2019-05-02T14:35:58Z <p>Let <span class="math-container">$I$</span> be a <span class="math-container">$\mathrm{Top}_*$</span>-enriched poset and <span class="math-container">$X: I \to \mathrm{Top}_*$</span>, and consider the homotopy limit <span class="math-container">$$ \underset{i \in I}{\mathrm{holim}}X(i), $$</span> where the maps <span class="math-container">$X(i) \to X(j)$</span> are nullhomotopic for <span class="math-container">$i \leq j$</span> and <span class="math-container">$X(i) \to X(j)$</span> a weak homotopy equivalence whenever <span class="math-container">$i \cong j$</span>. </p> <p>Can we conclude that <span class="math-container">$\underset{i \in I}{\mathrm{holim}}X(i)$</span> is trivial?</p> <h2>An example</h2> <p>The example I have in mind is whenever <span class="math-container">$I$</span> is the poset of non-zero subspaces of <span class="math-container">$\mathbb{R}^n$</span> (topologized as a disjoint union of Grassmannians), and <span class="math-container">$X$</span> is the sphere functor <span class="math-container">$\mathbb{S}: V \mapsto S^V$</span>. Then when <span class="math-container">$\dim V &lt; \dim W$</span> the map <span class="math-container">$S^V \to S^W$</span> is null homotopic and when <span class="math-container">$\dim V = \dim W$</span> the map <span class="math-container">$S^V \to S^W$</span> is a weak homotopy equivalence. </p> <p>I'd like to conclude that the homotopy limit, <span class="math-container">$$ \underset{0 \neq U \subseteq \mathbb{R}^n}{\mathrm{holim}}S^U $$</span> is weakly contractible. </p> http://www.4124039.com/q/330299 5 Is every dg-coalgebra the colimit of its finite dimensional dg-subcoalgebras? Victor TC http://www.4124039.com/users/112348 2019-04-29T17:24:05Z 2019-04-30T11:45:02Z <p>I saw this result in <a href="https://ncatlab.org/nlab/files/GetzlerGoerss99.pdf" rel="nofollow noreferrer"><em>A Model Category Structure for Differential Graded Coalgebras</em></a> by Getzler-Goerss, but when the coalgebra is non-negatively graded, is this property also satisfied when the dg coalgebra is <span class="math-container">$\mathbb{Z}$</span>-graded?.</p> <p>Thanks.</p> http://www.4124039.com/q/326596 7 Model category structure on spectra Tintin http://www.4124039.com/users/12204 2019-03-28T16:24:45Z 2019-04-03T13:41:38Z <p>I have a concrete question for the algebraic category of spectra, but if there is an answer for its topological analogue I would be interested in it.</p> <p>Let <span class="math-container">$S$</span> be a finite dimensional <a href="https://en.wikipedia.org/wiki/Noetherian_scheme" rel="noreferrer">Noetherian scheme</a> and <span class="math-container">$\mathbf{Spt}(S)$</span> the category of spectra over <span class="math-container">$S$</span>. After inverting <span class="math-container">$\mathbb{A}^1$</span>-stable equivalences we obtain <a href="https://ncatlab.org/nlab/show/motivic+homotopy+theory" rel="noreferrer">Voevodsky's stable homotopy category</a> <span class="math-container">$\mathbf{SH}(S)$</span>. My question is:</p> <blockquote> <p>Is there a model structure on <span class="math-container">$\mathbf{Spt}(S)$</span>, having <span class="math-container">$\mathbf{SH}(S)$</span> as homotopy category, such that every object is fibrant? If so, could you provide a reference?</p> </blockquote> <p>For example, does the obvious candidate, given by the class of <span class="math-container">$\mathbb{A}^1$</span>-stable equivalences as weak equivalences, surjective morphisms as fibrations, and cofibrations defined via the left lifting property, define a model structure on <span class="math-container">$\mathbf{Spt}(S)$</span>? </p> http://www.4124039.com/q/326647 8 On model categories where every object is bifibrant Simon Henry http://www.4124039.com/users/22131 2019-03-29T08:53:01Z 2019-04-02T08:52:57Z <p>Most model structures we use either have that every object is fibrant or that every object is cofibrant, and we have <a href="https://arxiv.org/abs/1003.1342" rel="nofollow noreferrer">various</a> <a href="https://arxiv.org/abs/1403.5303" rel="nofollow noreferrer">general</a> <a href="https://arxiv.org/pdf/math/0007070.pdf" rel="nofollow noreferrer">constructions</a> that allow (under some assumption) to go from one situation to the other.</p> <p>But there are very few examples of model categories where every object is both fibrant and cofibrant ("bifibrant").</p> <p>The only example I know is when one starts with a strict 2-category with strict 2-limits and 2-colimits there are <a href="https://ncatlab.org/nlab/show/2-trivial+model+structure" rel="nofollow noreferrer">model structures</a> on its underlying 1-category where every object is bifibrant and whose Dwyer-Kan localization is equivalent to the <span class="math-container">$2$</span>-category itself (where one drop non-invertible 2-cells). So this typically applies to the canonical model category on Cat or on Groupoids. I'm not even sure it can be used to model things like the <span class="math-container">$2$</span>-category of categories with finite limits.</p> <p>I don't believe there are that many other examples. But I have never seen any obstruction for this. So:</p> <p>Is there any example of a model category where every object is bifibrant whose localization is not a <span class="math-container">$2$</span>-category?</p> <p>Is every presentable <span class="math-container">$\infty$</span>-category represented by a model category where every object is bifibrant? If (as I expect) this is not the case, can we give an explicit 'obstruction' or an example to show it isn't the case?</p> <p><strong>Edit :</strong> The first question has been completely answered, but I havn't accepted answer so far because I'm still hoping to get an answer (positive or negative) to the second question.</p> http://www.4124039.com/q/326805 4 Thomason fibrant replacement and nerve of a localization Martin Frankland http://www.4124039.com/users/16109 2019-03-31T05:30:52Z 2019-03-31T21:48:18Z <p>The <a href="https://ncatlab.org/nlab/show/Thomason+model+structure" rel="nofollow noreferrer">Thomason model structure</a> on the category <span class="math-container">$\mathrm{Cat}$</span> of small categories is transferred along the right adjoint of the adjunction <span class="math-container">$$\tau_1 \circ \mathrm{Sd}^2 \colon s\mathrm{Set} \rightleftarrows \mathrm{Cat} \colon \mathrm{Ex}^2 \circ N,$$</span> where <span class="math-container">$\tau_1 \colon s\mathrm{Set} \to \mathrm{Cat}$</span> denotes the fundamental category functor, left adjoint to the nerve <span class="math-container">$N$</span>. The functor <span class="math-container">$\mathrm{Sd} \colon s\mathrm{Set} \to s\mathrm{Set}$</span> denotes the barycentric subdivision, and <span class="math-container">$\mathrm{Ex}$</span> is its right adjoint.</p> <p>A functor <span class="math-container">$F \colon \mathcal{C} \to \mathcal{D}$</span> is a Thomason weak equivalence if and only if it induces a weak equivalence on nerves <span class="math-container">$NF \colon N\mathcal{C} \to N\mathcal{D}$</span>. The adjunction displayed above is a Quillen equivalence.</p> <blockquote> <p><strong>Question 1.</strong> Is the fibrant replacement <span class="math-container">$\mathcal{C} \to \mathcal{C}'$</span> in the Thomason model structure a localization? Here I mean localization in the <span class="math-container">$1$</span>-categorical sense, i.e., a functor <span class="math-container">$\mathcal{C} \to \mathcal{C}[S^{-1}]$</span> that inverts a set of maps <span class="math-container">$S$</span> in <span class="math-container">$\mathcal{C}$</span>.</p> </blockquote> <p>My hunch is that the answer is no in general, but I'd be interested in situations where the answer is yes.</p> <p>I've looked at Thomason's original paper [1], this paper by Meier and Ozornova on Thomason-fibrant categories [2], and <a href="https://arxiv.org/abs/1603.05448" rel="nofollow noreferrer">this paper</a> by Bruckner and Pegel on Thomason-cofibrant categories.</p> <hr> <p>A related topic is what a localization does to the nerve, in particular, when does it preserve the homotopy type. </p> <p>In Proposition 3.7 of [3], Dwyer and Kan show that if a category is a free product <span class="math-container">$\mathcal{C} = \mathcal{D} \ast \mathcal{W}$</span>, where <span class="math-container">$\mathcal{W}$</span> is a <em>free</em> category, then the localization <span class="math-container">$\mathcal{C} \to \mathcal{C}[\mathcal{W}^{-1}]$</span> induces a weak equivalence <span class="math-container">$$N\mathcal{C} \to N(\mathcal{C}[\mathcal{W}^{-1}])$$</span> on nerves. Technically, their statement is happening in <span class="math-container">$O$</span>-categories, with a fixed set of objects <span class="math-container">$O$</span>.</p> <blockquote> <p><strong>Question 2.</strong> Are there other conditions on the category <span class="math-container">$\mathcal{C}$</span> and the set of maps <span class="math-container">$S$</span> under which the localization <span class="math-container">$\mathcal{C} \to \mathcal{C}[S^{-1}]$</span> induces a weak equivalence <span class="math-container">$N\mathcal{C} \to N(\mathcal{C}[S^{-1}])$</span> upon applying the nerve?</p> </blockquote> <hr> <p>[1] <em>Thomason, R. W.</em>, <a href="http://www.numdam.org/item?id=CTGDC_1980__21_3_305_0" rel="nofollow noreferrer"><strong>Cat as a closed model category</strong></a>, Cah. Topol. Géom. Différ. 21, 305-324 (1980). <a href="https://zbmath.org/?q=an:0473.18012" rel="nofollow noreferrer">ZBL0473.18012</a>.</p> <p>[2] <em>Meier, Lennart; Ozornova, Viktoriya</em>, <a href="http://dx.doi.org/10.4310/HHA.2015.v17.n2.a5" rel="nofollow noreferrer"><strong>Fibrancy of partial model categories</strong></a>, Homology Homotopy Appl. 17, No. 2, 53-80 (2015). <a href="https://zbmath.org/?q=an:1332.18019" rel="nofollow noreferrer">ZBL1332.18019</a>.</p> <p>[3] <em>Dwyer, W. G.; Kan, D. M.</em>, <a href="http://dx.doi.org/10.1016/0022-4049(80)90049-3" rel="nofollow noreferrer"><strong>Simplicial localizations of categories</strong></a>, J. Pure Appl. Algebra 17, 267-284 (1980). <a href="https://zbmath.org/?q=an:0485.18012" rel="nofollow noreferrer">ZBL0485.18012</a>.</p> http://www.4124039.com/q/323768 2 simplicial objects in a model category Paris http://www.4124039.com/users/136128 2019-02-21T19:03:20Z 2019-03-28T05:14:08Z <p>Suppose that we have a (combinatorial if necessary) model category <span class="math-container">$M$</span>, and let <span class="math-container">$F:\Delta^{op}\rightarrow M$</span> a simplicial object in <span class="math-container">$M$</span>, such that for any natural number <span class="math-container">$n$</span>, <span class="math-container">$F([n])$</span> is a fibrant object in <span class="math-container">$M$</span>. We define a new object <span class="math-container">$X= colim_{n} F([n]) $</span>. Is it true that <span class="math-container">$X$</span> is a fibrant object ?</p> http://www.4124039.com/q/326400 13 Example of non accessible model categories Philippe Gaucher http://www.4124039.com/users/24563 2019-03-26T14:41:31Z 2019-03-27T15:06:11Z <p>By curiosity, I would like to see an example of a model category with the underlying category locally presentable which is not accessible <a href="https://ncatlab.org/nlab/show/accessible+model+category" rel="noreferrer">in this sense</a> (and just in case: even by using Vopěnka's principle).</p> 特码生肖图
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