Active questions tagged model-categories - MathOverflowmost recent 30 from www.4124039.com2019-05-23T16:07:25Zhttp://www.4124039.com/feeds/tag?tagnames=model-categories&sort=newesthttp://www.creativecommons.org/licenses/by-sa/3.0/rdfhttp://www.4124039.com/q/3319042Which set of compact objects generates the subcategory of a compactly generated stable model category?Doelt_khttp://www.4124039.com/users/1096022019-05-19T00:07:30Z2019-05-19T08:12:12Z
<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/3307502Is the Hurewicz model category left proper?Dmitry Vaintrobhttp://www.4124039.com/users/71082019-05-05T00:35:56Z2019-05-05T01:09:34Z
<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/3305373Homotopy limit over a diagram of nullhomotopic mapsCharleshttp://www.4124039.com/users/1170882019-05-02T14:05:39Z2019-05-02T14:35:58Z
<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">$\dim V < \dim W$</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/3302995Is every dg-coalgebra the colimit of its finite dimensional dg-subcoalgebras?Victor TChttp://www.4124039.com/users/1123482019-04-29T17:24:05Z2019-04-30T11:45:02Z
<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/2999581Euclidean model structure on multipointed $d$-spacesPhilippe Gaucherhttp://www.4124039.com/users/245632018-05-11T13:53:26Z2019-04-10T00:01:15Z
<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/3275124Cotangent complex and its distinguished triangle- a generalisation?Meowhttp://www.4124039.com/users/1190122019-04-08T18:50:53Z2019-04-08T18:50:53Z
<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/3265966Model category structure on spectraTintinhttp://www.4124039.com/users/122042019-03-28T16:24:45Z2019-04-03T13:41:38Z
<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="nofollow 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="nofollow 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/3266478On model categories where every object is bifibrantSimon Henryhttp://www.4124039.com/users/221312019-03-29T08:53:01Z2019-04-02T08:52:57Z
<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/3268054Thomason fibrant replacement and nerve of a localizationMartin Franklandhttp://www.4124039.com/users/161092019-03-31T05:30:52Z2019-03-31T21:48:18Z
<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/3237682simplicial objects in a model categoryParishttp://www.4124039.com/users/1361282019-02-21T19:03:20Z2019-03-28T05:14:08Z
<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/32640012Example of non accessible model categoriesPhilippe Gaucherhttp://www.4124039.com/users/245632019-03-26T14:41:31Z2019-03-27T15:06:11Z
<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 Vop¨§nka's principle).</p>
http://www.4124039.com/q/25512513Is there an analog of Kan's $Ex^\infty$ functor for quasicategories?Tim Campionhttp://www.4124039.com/users/23622016-11-20T00:18:57Z2019-03-24T23:29:36Z
<p>Is there a fibrant replacement functor in the Joyal model structure which can be described non-recursively, like <span class="math-container">$Ex^\infty$</span> for the Quillen model structure? I believe another way to put this is to ask: is there a fibrant replacement functor in the Joyal model structure which is a right adjoint, or else a colimit of functors which are right adjoints?</p>
<p><strong>What I mean by "non-recursive"</strong></p>
<p><a href="https://arxiv.org/abs/0712.0724" rel="nofollow noreferrer">Garner's small object argument</a> certainly provides a functorial fibrant replacement which is simple enough to describe, but the description is still recursive. I think one could hope for something better.</p>
<p>Let me illustrate this in the setting of the Quillen model structure. In this setting Garner's construction also provides a fibrant replacement functor <span class="math-container">$G$</span>, but I think <a href="https://ncatlab.org/nlab/show/Kan+fibrant+replacement#_functor" rel="nofollow noreferrer">Kan's <span class="math-container">$Ex^\infty$</span> functor</a> is clearly "simpler" than <span class="math-container">$G$</span>. There is a ``closed-form" description of <span class="math-container">$Ex^\infty X$</span>: an <span class="math-container">$n$</span>-cell consists of a map <span class="math-container">$\mathrm{sd}^k\Delta^n \to X$</span> for some <span class="math-container">$k$</span>, where <span class="math-container">$\mathrm{sd}$</span> is the subdivision functor. Whereas <span class="math-container">$GX$</span> can seemingly only be described recursively: an <span class="math-container">$n$</span>-cell of <span class="math-container">$GX$</span> is the result of some sequence of horn-fillers being pasted in and possibly identified.</p>
<p>You might object that describing the <span class="math-container">$Ex^\infty$</span> functor does require <em>some</em> recursion, in that we need to consider iterates of the subdivision functor <span class="math-container">$\mathrm{sd}$</span>. I think the crucial distinction is that this recursion is independent of <span class="math-container">$X$</span> -- we only need to consider iterated subdivisions in a small set of universal cases -- the simplices <span class="math-container">$\Delta^n$</span>. Whereas to compute <span class="math-container">$GX$</span> we need to perform a recursion separately for each <span class="math-container">$X$</span> we consider.</p>
<p>Well -- that's a bit of a lie. I believe <span class="math-container">$G$</span> commutes with filtered colimits, so we really only need to compute <span class="math-container">$GX$</span> on the small set of all finite simplicial sets <span class="math-container">$X$</span>, and then extend it formally. But it's already an undecidable problem to compute <span class="math-container">$GX$</span> for all finite <span class="math-container">$X$</span> because this includes the word problem for groups. So maybe the key distinction is that the recursive procedure involved in computing <span class="math-container">$Ex^\infty X$</span> is actually (easily!) decideable whereas the one involved in computing <span class="math-container">$G X$</span> is not.</p>
<p><strong>Why adjointness would help</strong></p>
<p>Suppose we have a fibrant replacement functor <span class="math-container">$R: \mathsf{sSet} \to \mathsf{sSet}$</span> which has a left adjoint <span class="math-container">$L: \mathsf{sSet} \to \mathsf{sSet}$</span>. Then we necessarily have <span class="math-container">$(RX)_n \cong \mathrm{Hom}(L\Delta^n,X)$</span>. So in order to compute <span class="math-container">$R$</span>, we need only compute <span class="math-container">$L\Delta^n$</span> for each <span class="math-container">$n$</span>. If <span class="math-container">$R$</span> is a colimit of functors <span class="math-container">$R_k$</span> with left adjoints <span class="math-container">$L_k$</span>, then we have <span class="math-container">$(RX)_n = \varinjlim Hom(L_k\Delta^n, X)$</span>, and again, we need only compute <span class="math-container">$L_k\Delta^n$</span> for each <span class="math-container">$k,n$</span>. I think this would be the kind of description I'm looking for (and totally analogous to the case of <span class="math-container">$Ex^\infty$</span>, which is the colimit of right adjoints to iterated subdivision <span class="math-container">$\mathrm{sd}^k$</span>).</p>
http://www.4124039.com/q/3261092Why is a homotopy limit of a cosimplicial space not the ordinary limit?Maanroofhttp://www.4124039.com/users/894982019-03-23T03:16:36Z2019-03-23T04:41:03Z
<p>I've been trying to compute a homotopy limit of a cosimplicial object <span class="math-container">$X: \Delta \to \mathscr{M}$</span>, where <span class="math-container">$\mathscr{M}$</span> is some simplicial model category, we may take it to be spaces. I'm wondering where I go wrong with the following argument.</p>
<p>Let <span class="math-container">$X'$</span> be a fibrant replacement of <span class="math-container">$X$</span> in the Reedy model category <span class="math-container">$\mathscr{M}^{\Delta}$</span>. Then we have maps <span class="math-container">$X \to X' \to *$</span>, where the first map is a trivial cofibration and the second a fibration. In particular, the maximal augmentations of <span class="math-container">$X,X'$</span> are isomorphic by (Hirschhorn, <em>Model Categories and their localizations</em>, Thm. 15.9.9). </p>
<p>Let <span class="math-container">$T$</span> be the category <span class="math-container">$[0] \rightrightarrows [1]$</span>. Then these maximal augmentations are nothing more than the limit on <span class="math-container">$T$</span>, and <span class="math-container">$T$</span> includes finally into <span class="math-container">$\Delta$</span>. Hence we would have
<span class="math-container">\begin{align}
\mathrm{holim}_{n \in \Delta} X \simeq \lim_{n\in \Delta} X' \cong \lim_{n \in T} X' \cong \lim_{n \in T} X \cong \lim_{n \in \Delta} X
\end{align}</span></p>
<p>The conclusion doesn't seem right to me. Could anybody point out where I went wrong?</p>
http://www.4124039.com/q/3260597What are the advantages of simplicial model categories over non-simplicial ones?Tim Campionhttp://www.4124039.com/users/23622019-03-22T15:05:51Z2019-03-22T17:47:50Z
<p>Of course, there are general results allowing one to replace a model category with a simplicial one. But suppose I want to stay in my original non-simplicial model category (say for some reason I'm a fan of quasicategories, for example). What do I miss out on?</p>
<p>I think the idea is supposed to be: the mapping spaces in a simplicial model category are functorial on the nose, rather than something which must be treated in an ad-hoc manner. But I'd like to better understand how this plays out.</p>
<p><strong>Questions:</strong></p>
<ol>
<li><p>What constructions / results are only available for simplicial model categories, and not for non-simplicial model categories?</p></li>
<li><p>What constructions / results were originally proven for simplicial model categories and only later adapted to the non-simplicial case, or else are easier in the simplicial case?</p></li>
<li><p>When adapting results to the non-simplicial case, are the issues mainly set-theoretical, or are there other issues?</p></li>
</ol>
<p>For example, there are general results on existence of Bousfield localization which require enrichment of the model structure; in order to get by without enrichment one needs to assume combinatorialness -- but this is basically a set-theoretical restriction.</p>
http://www.4124039.com/q/3008798Left Bousfield localization without properness, what is known?Simon Henryhttp://www.4124039.com/users/221312018-05-23T08:34:49Z2019-03-19T16:19:36Z
<p>I'm interested in the existence of several example of left Bousfield localization of model categories that are not left proper (nor simplicial). I'm relatively convince that I can construct all those I need by hand, but that got me curious about what is known in general about existence of Left Bousfield localization of combinatorial model category at a set of maps:</p>
<ul>
<li><p>Is there any known example of a combinatorial model category with a set of maps such that the left Bousfield localization does not exist ? (Edit: this is answered <a href="http://www.4124039.com/questions/325383/counter-example-to-the-existence-of-left-bousfield-localization-of-combinatorial">there</a>)</p></li>
<li><p>Is there other kind of assumption under which we have a general theorem of existence ? For example I have the impression that putting condition on the set of maps we localize at (like localizing at a set of cofibrations between cofibration objects) might remove the need for left properness. </p></li>
<li><p>In <a href="http://www.maths.ed.ac.uk/~cbarwick/papers/complete.pdf" rel="nofollow noreferrer">On left and right model categories and left and right Bousfield localizations</a> C.Barwick claims (4.13) that if we only want to construct a left semi-model category then the left Bousfield localization always exists. But he does not prove it. What is the status of this claim ? is it proved somewhere ?</p></li>
</ul>
http://www.4124039.com/q/3256928$\Gamma$-sets vs $\Gamma$-spacesSimon Henryhttp://www.4124039.com/users/221312019-03-18T17:12:28Z2019-03-19T15:02:13Z
<p>I know that every <span class="math-container">$\Gamma$</span>-space is stably equivalent to a discrete <span class="math-container">$\Gamma$</span>-space, i.e. a <span class="math-container">$\Gamma$</span>-set.</p>
<p>For example, Pirashvili proves, as theorem 1.2 of <a href="https://link.springer.com/article/10.1007%2Fs002080000120" rel="nofollow noreferrer">Dold-Kan Type Theorem for <span class="math-container">$\Gamma$</span>-Groups</a>, that every <span class="math-container">$\Gamma$</span>-space is stably equivalent to a discrete <span class="math-container">$\Gamma$</span>-group (i.e. a group object in the category of <span class="math-container">$\Gamma$</span>-sets).</p>
<p>What I would like to know if this can be done in a "nice way," similar to how a quasi-category can be obtained from a Segal space. (For example, Pirashvili's argument is quite indirect and the its functoriality or universal properties are unclear). More precisely:</p>
<p>Consider the <a href="http://dodo.pdmi.ras.ru/~topology/books/bousfield-friedlander.pdf" rel="nofollow noreferrer">Bousfield-Friedlander</a> model structure on <span class="math-container">$\Gamma$</span>-spaces (here "space", will mean pointed simplicial sets) whose equivalences are the stable equivalences.</p>
<p>If <span class="math-container">$X$</span> is a simplicial set, denote by <span class="math-container">$X_0$</span> its set of vertices <span class="math-container">$X([0])$</span>. If <span class="math-container">$X$</span> is a <span class="math-container">$\Gamma$</span>-space, let <span class="math-container">$X_0$</span> be the application of this construction ``levelwise in <span class="math-container">$\Gamma$</span>''. i.e.:</p>
<p><span class="math-container">$$ X_0 : \Gamma^{op} \overset{X}{\rightarrow} sSet_* \overset{(\_)_0}{\rightarrow} Set_* $$</span></p>
<p><strong>Question:</strong> Given <span class="math-container">$X$</span> a <span class="math-container">$\Gamma$</span>-space fibrant in the Bousfield Friedlander model structure, is the natural morphism:</p>
<p><span class="math-container">$$ X_0 \rightarrow X$$</span></p>
<p>a (stable) weak equivalence ?</p>
<p>I believe this the right question to ask, but if there is something very similar with a positive answer (maybe "fibrant in the Bousfield-Friedlander model structure" is not quite the right condition?), please let me know!</p>
<p>Note that this exactly how the equivalence between complete Segal spaces (seen as simplicial-spaces satisfying a fibrancy condition) and quasi-categories works.</p>
http://www.4124039.com/q/32538312Counter-example to the existence of left Bousfield localization of combinatorial model categorySimon Henryhttp://www.4124039.com/users/221312019-03-13T21:08:03Z2019-03-13T22:00:23Z
<p>Is there any known example of a combinatorial model category <span class="math-container">$C$</span> together with a set of map <span class="math-container">$S$</span> such that the left Bousefield localization of <span class="math-container">$C$</span> at <span class="math-container">$S$</span> does not exists ?</p>
<p>It is well known to exists when <span class="math-container">$C$</span> is left proper, and it seems that it also always exists as a left semi-model structure, but I don't known if there is any concrete example where it is known to not be a Quillen model structure.</p>
<p>PS: I technically already asked this question <a href="http://www.4124039.com/questions/300879/left-bousfield-localization-without-properness-what-is-known">a year ago</a> but it was mixed with other related questions and this part was not answered, so I thought it was best to ask it again as a separate question.</p>
http://www.4124039.com/q/3252876Is the Thomason model structure the optimal realization of Grothendieck's vision?Tim Campionhttp://www.4124039.com/users/23622019-03-12T19:22:35Z2019-03-13T00:44:25Z
<p>In <em>Pursuing Stacks</em>, Grothendieck uses the category <span class="math-container">$Cat$</span> of small categories to model spaces. A recurring theme is the question of whether there is a Quillen model structure supporting this homotopy theory, i.e. with weak homotopy equivalences <span class="math-container">$\mathcal W_{min}$</span> -- those functors which become weak homotopy equivalences upon taking classifying spaces.</p>
<p>The Thomason model structure is indeed a model structure on <span class="math-container">$Cat$</span> with exactly these weak equivalences. But it's a little strange. The natural idea is to transfer the model structure from <span class="math-container">$sSet$</span> along the nerve / realization adjunction <span class="math-container">$c \dashv N$</span>, but this doesn't quite work so instead one transfers along <span class="math-container">$c \, sd^2 \dashv Ex^2 N$</span> where <span class="math-container">$sd \dashv Ex$</span> is the barycentric subdivision adjunction.</p>
<p>But Grothendieck envisioned a much more systematic connection between presheaf categories and <span class="math-container">$Cat$</span>. For a small category <span class="math-container">$A$</span>, he viewed the "category of elements" functor <span class="math-container">$Elts_A : Psh(A) \to Cat$</span> and its right adjoint as the fundamental way to endow <span class="math-container">$Psh(A)$</span> with a notion of weak equivalence. Indeed, he showed (see Cisinski's <em>Les Prefaisceaux comme modeles des types d'homotopie</em> Section 4.2) that if <span class="math-container">$A$</span> is a <a href="https://ncatlab.org/nlab/show/test+category" rel="noreferrer">test category</a>, then there is a model structure on <span class="math-container">$Psh(A)$</span> with cofibrations the monomorphisms and weak equivalences given by <span class="math-container">$Elts_A^{-1}(\mathcal W_{min})$</span>, which models the homotopy theory of spaces. For example, the Kan-Quillen model structure on <span class="math-container">$sSet$</span> is of this form.</p>
<p><strong>Questions:</strong> Let <span class="math-container">$A$</span> be a test category.</p>
<ol>
<li><p>Is there a model structure on <span class="math-container">$Cat$</span> induced projectively along <span class="math-container">$Elts_A$</span> from the Grothendieck model structure on <span class="math-container">$Psh(A)$</span>?</p></li>
<li><p>Is there a model structure on <span class="math-container">$Cat$</span> such that the Grothendieck model structure on <span class="math-container">$Psh(A)$</span> is induced injectively along <span class="math-container">$Elts_A$</span> from this model structure?</p></li>
<li><p>If the answer to both (1) and (2) is "yes", then is this perhaps one and the same model structure, possibly even independent of <span class="math-container">$A$</span>?</p></li>
</ol>
<p>Note that <span class="math-container">$Elts_A$</span> is the "canonical test functor". The categorical realization functor <span class="math-container">$c: sSet \to Cat$</span> is another test functor, but needs to be "improved" to <span class="math-container">$c sd^2$</span> before a model structure can be transferred. So one might also ask for which test functors such a model structure can be transferred, and apparently the answer is "not all of them". The hope is that things work out in the canonical case of <span class="math-container">$Elts_A$</span>.</p>
http://www.4124039.com/q/3251583Definition A.3.1.5 of Higher Topos TheoryFrank Konghttp://www.4124039.com/users/1237462019-03-11T08:40:48Z2019-03-11T09:15:26Z
<p>I am now reading the book Higher Topos Theory. In A.3.1.5, it gives the definition
of a <span class="math-container">$\mathbf{S}$</span>-enriched model category, where <span class="math-container">$\mathbf{S}$</span> is a monoidal model category. But in the book model structures are introduced only on <span class="math-container">$\mathsf{Set}$</span>-enriched categories. </p>
<blockquote>
<p>So, what does a model structure on a <span class="math-container">$\mathbf{S}$</span>-enriched category mean? </p>
</blockquote>
<p>Is it supposed to be that <span class="math-container">$\mathbf{S}$</span> obtains a forgetful functor to <span class="math-container">$\mathsf{Set}$</span>, and the model structure is defined on the category with respect to the <span class="math-container">$\mathsf{Set}$</span> enrichment, or is it something else?</p>
http://www.4124039.com/q/3250937For which categories of spectra is there an explicit description of the fibrant objects via lifting properties?Tim Campionhttp://www.4124039.com/users/23622019-03-10T16:32:28Z2019-03-10T22:00:05Z
<p>How explicit are the model structures for various categories of spectra?</p>
<p>Naive, symmetric and orthogonal spectra are obtained via left Bousfield localization of model structures with explicit generating (acyclic) cofibrations, so we have explicit generating cofibrations for them.</p>
<p>I'm thinking it's too much to ask for explicit generating acyclic cofibrations, but it would be nice to at least have a pseudo-generating set -- i.e. an explicit set of generating acyclic cofibrations, lifting against which characterizes the fibrant objects and fibrations between fibrant objects.</p>
<p><strong>Questions:</strong> Let <span class="math-container">$\mathcal C$</span> be a model category modeling spectra, (e.g. naive, <a href="https://ncatlab.org/nlab/show/Model+categories+of+diagram+spectra" rel="noreferrer">symmetric</a>, <a href="https://ncatlab.org/nlab/show/model+structure+on+orthogonal+spectra" rel="noreferrer">orthogonal</a>, <a href="https://www.math.uchicago.edu/~may/PAPERS/mmLMSDec30.pdf" rel="noreferrer">EKMM</a>, <a href="https://ncatlab.org/nlab/show/combinatorial+spectrum#ModelCategoryStructure" rel="noreferrer">combinatorial</a>...)</p>
<ol>
<li><p>Are explicit generating cofibrations available for <span class="math-container">$\mathcal C$</span>?</p></li>
<li><p>How about explicit generating acyclic cofibrations?</p></li>
<li><p>If not (2), how about an explicit pseudo-generating set of acyclic cofibrations?</p></li>
<li><p>If not (3), is there at least an explicit description of the fibrant objects via lifting properties?</p></li>
</ol>
http://www.4124039.com/q/2289666What structure of a monoidal simplicial model category is preserved by taking the opposite category?Jonathan Beardsleyhttp://www.4124039.com/users/115462016-01-21T19:13:44Z2019-03-08T15:05:52Z
<p>Suppose we have $(M,\otimes,1)$, a monoidal simplicial model category. Then we can consider the opposite model category $M^{op}$ with the opposite model structure (fibrations become cofibrations, etc.). This category is also still a monoidal category (the opposite of a monoidal category is canonically a monoidal category). How do all of these possible adjectives interact? Is it still closed? Does the push-out product axiom still hold? It seems like we must lose some things, for instance, if we wanted the axiom "for any cofibrant object $X$ there is an equivalence $1\otimes X\simeq X$" this can't in general be true anymore, can it, since the cofibrant objects are completely different now? </p>
<p>In general it seems clear that things break down, but I'm interested in precisely what breaks down. Do we still have a simplicial model category, but we're just missing some of the monoidal model category axioms? I'm also okay with dumping the simplicial requirement. </p>
http://www.4124039.com/q/3249153Monoidalness of a model category can be checked on generatorsTim Campionhttp://www.4124039.com/users/23622019-03-08T05:43:37Z2019-03-08T12:30:16Z
<p>If <span class="math-container">$C$</span> is a cofibrantly generated model category which is also monoidal biclosed, then to check that <span class="math-container">$C$</span> is a monoidal model category, it suffices to check that the Leibniz product of generating cofibrations is a cofibration, that the Leibniz product of a generating cofibration with a generating acyclic cofibration is an acyclic cofibration, and that the unit axiom is satisfied.</p>
<p>The proof is a little involved, so I'd like to have a source to cite this from. Where can I find it in the literature?</p>
<p>I'm happy to assume that the unit is cofibrant.</p>
http://www.4124039.com/q/3248088Localization, model categories, right transfernumhttp://www.4124039.com/users/1366832019-03-06T20:57:17Z2019-03-07T18:22:11Z
<p>Suppose that we have a locally presentable category <span class="math-container">$M$</span> and <span class="math-container">$N$</span> is a locally presentable full subcategory of <span class="math-container">$M$</span>. Both categories are complete and cocomplete. Lets suppose that we have an adjunction <span class="math-container">$F:M\leftrightarrow N:i$</span> where <span class="math-container">$i$</span> is the embedding functor and <span class="math-container">$(i \circ F)^{2}=i \circ F=L$</span>. If <span class="math-container">$N$</span> is a combinatorial model category, could we define a model structure on <span class="math-container">$M$</span> such that <span class="math-container">$a\rightarrow b$</span> is a weak equivalence (cofibration) if and only if <span class="math-container">$F(a)\rightarrow F(b)$</span> is a weak equivalence (cofibration) in <span class="math-container">$N$</span>? </p>
<p><strong>Edit:</strong> In case it is not true in general, we assume one more condition, there exists a combinatorial left proper model structure on <span class="math-container">$M$</span> such that <span class="math-container">$F:M\leftrightarrow N:i$</span> is a Quillen adjunction. </p>
http://www.4124039.com/q/1203959Monoidal model category structure on a functor category.Geoffroy Horelhttp://www.4124039.com/users/107072013-01-31T10:13:08Z2019-03-07T08:41:32Z
<p>Let $A$ be a small simplicial category. The category $Fun(A,s\mathrm{Set})$ of simplicial functors from $A$ to simplicial sets can be given the projective model structure in which fibration and weak equivalences are objectwise.</p>
<p>Now assume further that $A$ is a symmetric monoidal category with respect to some binary operarion $\circ: A\times A\to A$. We can define the Day (or convolution) tensor product on $Fun(A,s\mathrm{Set})$ by the following coend:
$$F\otimes G(a)=A(-\circ -,a)\otimes_{A\times A}F(-)\times G(-)$$</p>
<p>My question is:</p>
<p>Is it true that the projective model structure is a monoidal model category in the sense that it satisfies the pushout-product axiom and if so is there a place where this is written down ?</p>
http://www.4124039.com/q/30370811Is there an "injective version" of the Bergner model structure?Tim Campionhttp://www.4124039.com/users/23622018-06-26T19:53:09Z2019-03-04T17:09:48Z
<p>The Bergner model structure on $sCat$ (simplicially enriched categories) has a "projective" flavor: fibrations are "levelwise" while cofibrations satisfy stringent relative "freeness" conditions.</p>
<p>This seems to be something inherent to modeling $\infty$-categories using strictly-associative composition. Similarly, Barwick and Kan's model structure on <a href="https://arxiv.org/abs/1011.1691" rel="nofollow noreferrer">relative categories</a> and Horel's model structure on <a href="https://arxiv.org/abs/1403.6873" rel="nofollow noreferrer">simplicially-internal categories</a> are induced projectively, and not very many objects are cofibrant.</p>
<p>Is this really a trade-off we're stuck with?</p>
<p><strong>Question 1:</strong> Is there a model structure $M$ on $sCat$ other than the Bergner model structure such that the identity functor is a left Quillen equivalence $sCat_M \to sCat_{Bergner}$? More generally, is there a model $M$ of the homotopy theory of $\infty$-categories whose underlying category is $sCat$ which doesn't have fewer cofibrant objects than $sCat_{Bergner}$?</p>
<p><strong>Question 2:</strong> Is there such an $M$ with a reasonable supply of cofibrant objects? Say, such that every ordinary category is cofibrant? Or perhaps at least such that every ordinary Reedy category is cofibrant?</p>
<p><strong>Question 3:</strong> As above, but for relative categories or simplicially-internal categories?</p>
<p>Two data points:</p>
<ul>
<li><p>If $C$ is a category and $M$ is a model category, then $M^C$ with levelwise weak equivalences has the correct homotopy theory. But from thinking about the Bergner model structure, you might think you'd have to consider the category of simplicial functors $C' \to M'$ where $C'$ is the standard resolution of $C$ and $M'$ is the simplicial localization of $M$. The fact that you don't have to do this hints at some alternate model category -type structure where $C$ is already cofibrant and $M$ is fibrant -- maybe this would be a model structure on relative categories.</p></li>
<li><p>A similar phenomenon happens with operads, if I recall correctly (which are also <em>strict</em> monoids for the Kelly tensor product on symmetric sequences) -- the usual model structure (Berger-Moerdijk, I believe) is much more stringent in its cofibrancy requirements than what is needed in practice.</p></li>
<li><p>As hinted at in the comments, it's a good idea to ask whether the existing "projective" model structures $M$ are left proper. For </p>
<ul>
<li><p>Left properness of a model structure depends only on the weak equivalences (saying that the co-base change Quillen adjunctions between co-slice categories are Quillen equivalences).</p></li>
<li><p>A model structure with all objects cofibrant is left proper.</p></li>
</ul>
<p>Well, it turns out that the Bergner, Lack, Barwick-Kan, and Horel model structures are all left proper! (I'm not sure about the Berger-Moerdijk model structure.) So this leaves open the possibility of model structures with the same weak equivalences and all objects cofibrant.</p></li>
</ul>
http://www.4124039.com/q/3245982Simplicial models for mapping spaces of filtered mapsJoao Faria Martinshttp://www.4124039.com/users/990882019-03-04T11:57:13Z2019-03-04T11:57:13Z
<p>Let <span class="math-container">$S$</span> and <span class="math-container">$K$</span> be simplicial sets with <span class="math-container">$K$</span> Kan. Given simplicial sets <span class="math-container">$S$</span> and <span class="math-container">$K$</span>, we let <span class="math-container">$SIMP(S,K)$</span> be the internal hom in the category of simplicial sets. Hence <span class="math-container">$SIMP(S,K)$</span> is Kan.</p>
<p>Suppose that <span class="math-container">${\cal S}=\{S_0 \subseteq S_1 \subseteq S_2 \subseteq S_3 \dots\}$</span> and <span class="math-container">${\cal K}=\{K_0 \subseteq K_1 \subseteq K_2 \subseteq K_3 \dots\}$</span> are fitrations of <span class="math-container">$S$</span> and <span class="math-container">$K$</span>, by sub-simplicial sets, where each <span class="math-container">$K_i$</span> is Kan. We can consider a filtered version <span class="math-container">$SIMP({\cal S},{\cal K})$</span> of <span class="math-container">$SIMP(S,K)$</span>. The <span class="math-container">$n$</span>-simplices are given by filtered simplical maps <span class="math-container">${\cal S} \times \Delta(n) \to {\cal K}$</span>. (The case of fitrations of length two is treated in May's book "Simplical Objects in Algebraic Topology", page 17.)</p>
<p>It <span class="math-container">$X$</span> and <span class="math-container">$Y$</span> are topological spaces, we use <span class="math-container">$TOP(X,Y)$</span> to denote the space of functions <span class="math-container">$X \to Y$</span>, with the k-ification of the compact open topology.
We also have a simplicial mapping space <span class="math-container">$TOP_{Simp}(X,Y)$</span>, which is essentially the singular complex <span class="math-container">$Sing(TOP(X,Y))$</span> of <span class="math-container">$TOP(X,Y)$</span>. </p>
<p>If <span class="math-container">${\cal X}=\{X_0 \subseteq X_1 \subseteq X_2\dots\}$</span> and <span class="math-container">${\cal Y}=\{Y_0 \subseteq Y_1\subseteq Y_e \dots\}$</span> are fitrations of <span class="math-container">$X$</span> and <span class="math-container">$Y$</span> by subspaces, we let <span class="math-container">$TOP({\cal X}, {\cal Y})$</span> be the space of filtered functions <span class="math-container">$\cal X \to \cal Y$</span>, with the <span class="math-container">$k$</span>-ification of compact open topology.</p>
<p>We have a weak homotopy equivalence <span class="math-container">$|Sing(X)| \to X$</span> given any topological space <span class="math-container">$X$</span>. There exists a well known weak homotopy equivalence <span class="math-container">$|SIMP(S,K)| \to TOP(|S|,|K|)$</span>. It is essentially derived from the fact that <span class="math-container">$K$</span> is a strong deformation retract of the singular complex <span class="math-container">$Sing(|K|)$</span> if <span class="math-container">$K$</span> is Kan. This weak homotopy equivalence is the composition of the obvious maps:
<span class="math-container">$$ |SIMP(S,K)| \to |SIMP(S,Sing|K|)| \stackrel{\cong}{\to} |TOP_{Simp}(|S|,|K|)|\to TOP(|S|,|K|). $$</span></p>
<p>The question is the following: do we have a filtered version of the weak homotopy equivalence <span class="math-container">$|SIMP(S,K)| \to TOP(|S|,|K|)$</span>? For instance, does it exist a weak homotopy equivalence <span class="math-container">$|SIMP({\cal S}, {\cal K})| \to TOP(|\cal S|,|\cal K|) $</span>, where <span class="math-container">$(|\cal S|$</span> and <span class="math-container">$|\cal K|)$</span> are filtered by the geometric realisations of the <span class="math-container">$S_i$</span> and the <span class="math-container">$K_i$</span>, respectively.</p>
http://www.4124039.com/q/1203825Reedy model structures on oplax limitsMike Shulmanhttp://www.4124039.com/users/492013-01-31T04:42:28Z2019-02-27T19:41:06Z
<p>Suppose $R$ is a category and $F:R\to Cat$ is a functor (or pseudofunctor). The <em>oplax limit</em> of $F$ is the category whose objects consist of an object $x_r \in F(r)$ for all $r$ together with a morphism $x_s \to F(d)(x_{r})$ for all morphisms $d:r\to s$ in $R$, satisfying obvious compatibility conditions.</p>
<p>I have some vague memory of reading a paper in which, given a functor as above in which $R$ is a Reedy category, each category $F(r)$ is a model category, and probably some other conditions, a "Reedy-type" model structure on the oplax limit (or perhaps some related category) was constructed. However, I have been totally unable to find this paper again; the closest I can find is <a href="http://arxiv.org/abs/1010.0717" rel="nofollow">this paper</a> which considers "injective-type" model structures on lax limits. Can anyone point me to the paper I am thinking of?</p>
<p>(I am not interested in seeing proofs or "it seems like this should work" arguments written out in the answers. I only want the reference.)</p>
http://www.4124039.com/q/3242773Simplicial models for fibrations between mapping spacesJoao Faria Martinshttp://www.4124039.com/users/990882019-02-27T18:22:51Z2019-02-27T19:35:14Z
<p>Let <span class="math-container">$S,T$</span> and <span class="math-container">$K$</span> be simplicial sets with <span class="math-container">$K$</span> Kan. Given simplicial sets <span class="math-container">$S$</span> and <span class="math-container">$K$</span>, we let <span class="math-container">$SIMP(S,K)$</span> be the internal hom in the category of simplicial sets. We let <span class="math-container">$|S|$</span> denote the geometric realisation of a simplicial set <span class="math-container">$S$</span>. </p>
<p>Given <span class="math-container">$X$</span> and <span class="math-container">$Y$</span>, (compactly generated) topological spaces, we use <span class="math-container">$TOP(X,Y)$</span> to denote the function space, with the k-ification of the compact open topology. </p>
<p>We also have a simplicial mapping space <span class="math-container">$TOP_{Simp}(X,Y)$</span>, which is essentially the singular complex <span class="math-container">$Sing(TOP(X,Y))$</span> of <span class="math-container">$TOP(X,Y)$</span>. </p>
<p>We have a weak homotopy equivalence <span class="math-container">$|Sing(X)| \to X$</span> given any topological space <span class="math-container">$X$</span>.</p>
<p>There exists a well known weak homotopy equivalence <span class="math-container">$|SIMP(S,K)| \to TOP(|S|,|K|)$</span>. It is essentially derived from the fact that <span class="math-container">$K$</span> is a strong deformation retract of the singular complex <span class="math-container">$Sing(|K|)$</span> if <span class="math-container">$K$</span> is Kan. This weak homotopy is the composition of the obvious maps:
<span class="math-container">$$ |SIMP(S,K)| \to |SIMP(S,Sing|K|)| \stackrel{\cong}{\to} |TOP_{Simp}(|S|,|K|)|\to TOP(|S|,|K|). $$</span></p>
<p>Suppose that we have a cofibration (meaning inclusion) <span class="math-container">$i\colon T \to S$</span> of simplicial sets. We have an induced fibration <span class="math-container">$i^*\colon SIMP(S,K) \to SIMP(T,K)$</span> of simplicial sets, hence a Serre fibration <span class="math-container">$|i^*|\colon |SIMP(S,K)| \to |SIMP(T,K)|$</span>. </p>
<p>We also have a cofibration <span class="math-container">$|i|\colon |T| \to |S|$</span>, thus a (Hurewicz, hence Serre) fibration <span class="math-container">$|i|^*\colon TOP(|S|,|K|) \to TOP(|T|,|K|)$</span>. (We just restrict a map <span class="math-container">$|S| \to |K|$</span> to <span class="math-container">$|T|\subset |S|$</span>.)</p>
<p>It is also clear that we have a commutative diagram:
<span class="math-container">$\require{AMScd}$</span>
<span class="math-container">\begin{CD}
|SIMP(S,K)| @>>> TOP(|S|,|K|)\\
@V |i^*| V V @VV |i|^* V\\
|SIMP(T,K)| @>>> TOP(|T|,|K|)\\
\end{CD}</span></p>
<p>The question I would like to ask is the following. Does <span class="math-container">$i^*\colon SIMP(S,K) \to SIMP(T,K)$</span> give a 'faithful' simplicial model of the fibration <span class="math-container">$|i|^*\colon TOP(|S|,|K|) \to TOP(|T|,|K|)$</span> (whatever this means). For instance (this would suffice), does the homotopy long exact sequence of <span class="math-container">$|i|^*$</span> coincide with that of <span class="math-container">$|i^*|$</span>?</p>
http://www.4124039.com/q/3236816Simplicial localization of the cofibrant-fibrant objectsReid Bartonhttp://www.4124039.com/users/1266672019-02-20T22:47:23Z2019-02-22T10:19:51Z
<p>Let <span class="math-container">$M$</span> be a model category. I don't assume that <span class="math-container">$M$</span> has functorial factorizations or that <span class="math-container">$M$</span> is simplicial. Write <span class="math-container">$M^{c}$</span> (respectively, <span class="math-container">$M^{cf}$</span>) for the full subcategory of <span class="math-container">$M$</span> on the cofibrant objects (respectively, the cofibrant and fibrant objects).</p>
<p>Does the inclusion <span class="math-container">$M^{cf} \to M^c$</span> induce a Dwyer-Kan equivalence on the simplicial localizations at the class of weak equivalences?</p>
<p>Assuming the answer is "yes", does anyone know of a reference? I couldn't find this in the original papers of Dwyer and Kan for general <span class="math-container">$M$</span>, only when <span class="math-container">$M$</span> has functorial factorizations.</p>
http://www.4124039.com/q/1047764Need M combinatorial for existence of injective model structure on $M^G$?David Whitehttp://www.4124039.com/users/115402012-08-15T17:00:04Z2019-02-13T11:54:46Z
<p>I'm doing some work with model categories and operads, and to check a certain hypothesis I've had to learn a bit of equivariant homotopy theory. Let $M$ be a model category and $G$ be a finite group. We can assume $M$ is cofibrantly generated and left proper, but I'm trying to avoid assuming $M$ is combinatorial. If necessary we can assume $M$ is cellular, but right now I don't see how that can help. It has turned out to be useful to know that $M^G$ has a model structure where the cofibrations are $G$-equivariant maps which are cofibrations in $M$. This is simply the <a href="http://ncatlab.org/nlab/show/model+structure+on+functors" rel="nofollow"><em>injective</em> model structure</a> on $M^G$, where a map $f$ is a weak equivalences or cofibrations if the underlying map in $M$ is such.</p>
<p>I tend to think of $M^G$ as a special case of a diagram category $M^I$ (where $I$ is small), because I feel like I have a decent grasp on diagram categories. In that setting, I believe one must know that $M$ is combinatorial in order to know the injective model structure exists. However, I don't have a good reference for this other than having seen it mentioned without proof or reference in a number of papers.</p>
<blockquote>
<p>(1) Can anyone provide a reference which proves the injective model structure on $M^I$ exists? I'd like to see where the hypothesis that $M$ is combinatorial gets used. I'd also like the reference to prove the injective model structure is cofibrantly generated.</p>
</blockquote>
<p>It's worth noting that Proposition A.2.8.2 in Lurie's <em>Higher Topos Theory</em> proves existence of the injective model structure, but I'm not satisfied with that for a couple of reasons. First, the proof is very complicated because Lurie wants it to hold in the setting where $M$ and $I$ are enriched over an excellent model category. My ideal reference would be a simpler proof holding in the non-enriched setting, preferably the first place the injective model structure was defined. Second (and related), because everything in this appendix is about combinatorial model categories, I can't help but wonder if there's a proof which relies on that hypothesis less. Finally, it's almost impossible for me to get my hands on the generating (trivial) cofibrations from that proof. Lurie relies on the very-complicated Lemma A.3.3.3 to get generating cofibrations and on Proposition A.2.6.8, which says basically that if you're in a category which is almost combinatorial (missing only generating trivial cofibrations) then you can get the generating trivial cofibrations for free from the generating cofibrations.</p>
<p>In the special case where $I$ is a group $G$, I can't seem to find anything on the injective model structure. Most of the work I can find on equivariant homotopy theory uses the projective model structure instead of the injective (and this one is known to exist if $M$ is cofibrantly generated). I imagine that with so much structure on $I$ and with so much theory which has been developed out there for equivariant homotopy theory, one should be able to come up with a much better proof in this setting than the one in HTT.</p>
<blockquote>
<p>(2) Is the hypothesis that $M$ is combinatorial still necessary to prove existence of the injective model structure on $M^G$? In what ways is this model structure nicer than $M^I$ for a generic $I$?</p>
</blockquote>
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