Questions tagged [model-categories]
A model category is a category equipped with notions of weak equivalences, fibrations and cofibrations allowing to run arguments similar to those of classical homotopy theory.
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How do you prove that the category of weak equivalences of sSet is accessible?
I am trying to prove that the category of simplicial sets is a combinatorial model category by using Proposition A.2.6.15 of Lurie's book, and this requires proving that the weak equivalences are ...
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60 views
Explaining the “free left fibration” functor for infinity categories
This is a cross-post from here
I am reading A. Mazel-Gee's paper "All about the Grothendieck construction". In that paper he explains that the left adjoint ${\mathrm{Cat}_{\infty}}_{/\mathcal{C}}\to \...
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1answer
173 views
Applications of model categories
I was wondering if someone could explain some of the concrete applications of model categories. My possibly naive understanding of the motivation is that one wants to mimic the category of ...
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1answer
45 views
Why does this construction give a weak factorization system in the category of span diagrams?
In Dwyer and Spalinski's classic paper Homotopy Theories and Model Categories, they describe homotopy pushouts by defining a model structure on the category of span diagrams in a given model category $...
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2answers
224 views
Model categories and chain complexes
I'm fairly new to thinking about homological algebra and chain complexes in their own right, i.e outside of isolated examples such as for constructing simplicial homology, or for computing $Ext$ ...
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65 views
Simple, explicit, functorial cylinder object in CDGA
In the model category of graded commutative dg-algebras CDGA over $\mathbb{Q}$ (with the projective model structure) there is a simple, functorial construction of a path object given by tensoring with ...
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82 views
What is the meaning of lifting property? and some question in $\infty$-category
When I learn model category, the important compute tool is the lifting property between $(\operatorname{Cof}, \operatorname{Fib} \cap W)$ and $(\operatorname{Cof} \cap W, \operatorname{Fib}) $, where $...
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1answer
50 views
Generating trivial cofibrations of Bousfield localization
Suppose $\mathfrak{M}$ is a left proper celluar model category and $S$ is a set of cofibrations in $\mathfrak{M}$. What are the generating trivial cofibrations of $L_S\mathfrak{M}$? Are they $J\cup S$,...
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What is the fibrant replacement of an Eilenberg-MacLane space in unstable motivic homotopy theory?
One can take an Eilenberg-MacLane space $K(A, n)$ for some abelian group $A$, and view it as (locally) constant simplicial sheaf on $Sch/k$, the category of schemes, or smooth schemes, over a field $k$...
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1answer
126 views
Split cofibrations up to quasi-isomorphism
$R$ a ring $(1\neq 0)$, $\mathbf{Perf}(R)$ is the category of perfect complexes (of right $R$-modules).
Suppose that $A_{\bullet}\rightarrow B_{\bullet}\rightarrow B_{\bullet}/A_{\bullet}$ a short ...
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65 views
A sectionwise fiber sequence is homotopy fiber sequence?
Let $\mathscr{C}$ be a site and $\mathsf{sPre}(\mathscr{C})$ the category of simplicial presheaves on $\mathscr{C}$ equipped with Jardine's local model structure. Let $E\to B$ is a sectionwise Kan ...
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Fibrant objects in $\mathbb{S}$-local model structure on $Top_*$
Let $\mathbb{S}$ be the sphere spectrum. We can localize the category of based spaces, $Top_*$ at a homology theory, and hence at $\mathbb{S}$.
Equipping $Top_*$ with the Quillen model structure (...
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2answers
182 views
Projective/injective object in functor category
Let $\mathcal{C}$ denote the functor category $Fun(\textbf{Man} , \textbf{Ab})$, where $\textbf{Man}$ and $\textbf{Ab}$ denote the category of smooth manifolds and abelian groups respectively. I want ...
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1answer
100 views
Conditions for certain inclusion functor to preserve internal homs
Suppose that $\mathcal{C}$ is a locally cartesian closed right proper Quillen model category for which all objects are fibrant. Let $x$ be an object of $\mathcal{C}$. Let $\mathbb{F}$ denote the class ...
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228 views
When is the model structure on functors correct, i.e. when does localization commute with taking functor categories?
Let $C$ be a small category and $M$ a model category. Then there are various "global" model structures (projective, injective, Reedy) on the category $Fun(C,M)$ of functors from $C$ to $M$, all with ...
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124 views
$k$-linear $\infty$ stable categories and dg categories
This question is related to this question, where I asked about the relation between the derived category of a fiber product $Y \times_Z W$ and the push out of the diagram of derived categories one ...
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Derivators and fibred $\infty$-categories
In his Cohomological methods in intersection theory, Cisinski writes:
"[...] note however that, by Balzin’s work [Bal19, Theorem 2], it is clear that one can go back and forth between the language of ...
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128 views
Almost transferred model structures
Let $F : \mathcal{C} \leftrightarrows \mathcal{D} : U$ be a Quillen adjunction between cofibrantly generated model categories. The model structure on $\mathcal{D}$ is called transferred if $U$ ...
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103 views
1-connected infinity groupoids, groupoids and 1-connected spaces
I am exploring a bit the world of groupoids. What I have in mind is that infinity groupoids correspond to spaces. So my first question is the following:
Consider the model category $\infty-Grpd$ of ...
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Generalization of familiar theorem about singular homology to general model category
I have two questions, the first one is just wether the following statement is true or not? Is there a reference for this?
The second question is maybe related, I don't know. But anyway, given $U:\...
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4answers
404 views
Existence of homotopy limits and colimits in model categories
I am not an expert, thus I apologize if my question is very naive. Let $\mathsf{M}$ be a model category (I do not assume any functoriality on the factorization),
Q1. Is there a reference where it ...
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1answer
349 views
Category of spaces/sheaves
Consider the following category $\mathcal C$:
An object of $\mathcal C$ is a pair $(X,\mathcal F)$ where $X$ is a space and $\mathcal F$ is a sheaf on $X$.
A morphism $(X,\mathcal F)\to(Y,\mathcal G)$...
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1answer
205 views
Does any accessible model category come from an algebraic model category?
I read in nLab : Every cofibrantly generated model category structure can be lifted to that of an algebraic model category. It is not clear whether or not this is true for any accessible model ...
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A characterization of local objects in model categories
Let $\mathcal{M}$ be a model category and $S$ a set of cofibrations between cofibrant objects. Then every $S$-local object has the right lifting property with respect to $S$. The converse does not ...
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Clarification on definition of closed $\mathcal{C}$-module for a category $\mathcal{C}$
Hovey introduces the notion of a closed monoidal structure and a closed monoidal functor. Then he goes on to say that this naturally gives rise to the notion of closed modules over a closed monoidal ...
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The homotopy category of the category of enriched categories
We know that if $\mathcal C$ is a combinatorial monoidal model category such that all objects are cofibrant and the class of weak equivalences is stable under filtered colimits, then $\mathsf{Cat}_{\...
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1answer
180 views
Is there Jeff Smith's theorem for left semi-model structures?
Jeff Smith's theorem gives a simple criterion for the existence of a combinatorial model category. Is there a similar theorem for combinatorial left semi-model categories? I see two problems that ...
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1answer
105 views
Cofibrant simplicial categories
Reading about simplicial categories, and in particular the model structure in sCat, I found various sources, among which I can mention this one or section 16.2 in Riehl's book “Categorical homotopy ...
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1answer
148 views
Transfer model structures, reflective subcategories and Bousfield localizations
The topic of Bousfield localizations has a lot of literature which has on most of the occasion some tameness assumption on the presentability of the model category. Recently I have been trying to ...
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1answer
171 views
Model structure for fiberwise Bousfield localization
I think the following should be in the literature but couldn't find it.
Recall that around the 1970's Bousfield described the $R$-localization $EX$ of any space $X$, for $R$ a fixed ring. The ...
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1answer
131 views
Relative Ext of Avramov-Martsinkovsky as a derived Hom
Avramov-Martsinkovsky (http://mathserver.neu.edu/~martsinkovsky/Relative.pdf) have defined an exotic version of Ext between two modules over (for simplicity) Gorenstein rings. The basic idea of their ...
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Is there a model category describing shape theory?
Is there a nice model structure on some category of topological spaces compatible with shape theory? In particular, weak equivalences should induce isomorphisms on sheaf cohomology.
As an example, ...
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181 views
Is the simplicial nerve a localization?
Given a simplicial category $\mathcal{C}_{\ast}$ (if necessary, you may assume it's fibrant), denote as $\mathcal{C}$ its underlying ordinary category, and as $\mathcal{W}$ the class of all ...
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1answer
128 views
Calculus of Functors and Model categories
In Calculus of functors and model categories II Biedermann and Rondigs claim in Corollary 6.18 that the $n$-homogeneous model structure on $\mathrm{Fun}(\mathcal{C}, \mathcal{D})$ is stable if $\...
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1answer
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What came of the problems posed in Hovey's book chapter 8
In his book "Model Categories", Hovey sets out to write a self-contained introduction to model categories. The final chapter briefly discusses some questions which stayed unresolved.
I have been ...
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4answers
851 views
How should I think about presentable $\infty$-categories?
Let me start out with a confession. I have never cared much for set-theoretic size issues, for they seem not to cause much trouble in my day-to-day mathematical life. Despite that, I have always been ...
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1answer
291 views
Correspondence between classes of model categories and classes of $\infty$-categories
We know by Karol Szumiło's thesis (https://arxiv.org/pdf/1411.0303.pdf) that there is an equivalence between the two fibration categories of cofibration categories on one side and cocomplete $\infty$-...
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Model structures on groupoids
Let me preface by saying that I'm very inexperienced with model categories. I was thinking about the following example, and I'm now wondering whether it fits into the theory of model stuctures:
...
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Is simplicial localization part of a Quillen equivalence between relative categories and simplicial categories?
There are many models for $\infty$-categories. One of them is relative categories – AKA categories with weak equivalences – which have a model structure due to Barwick–Kan. Another one is simplicial ...
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1answer
270 views
Where is Brown's lemma from?
In homotopy theory, Kenneth Brown's lemma states that if a functor send acyclic cofibrations between cofibrant objects to weak equivalences, then it send all weak equivalences between cofibrant objets ...
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Based loops objects in model categories
If I have a pointed model category then for I can define based loops objects as homotopy pullbacks:
$\require{AMScd}$
\begin{CD}
\Omega X @>>> *\\
@V V V @VV V\\
* @>>>...
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Do dg schemes have derived points?
Working over a base field $k$ of characteristic $0$, say $K$ is a field (over $k$) and $X$ is a ("nice" if necessary) dg scheme in the sense of Toen-Vezzosi and others, and say $X^0$ is the reduced ...
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Homotopy fibre sequence and left Bousfield localization
Let $\mathcal{M}$ be a pointed model category and $\mathcal{C}$ a class of maps in $\mathcal{M}$ for which the left Bousfield localization ${\rm L}_{\mathcal{C}}\mathcal{M}$ exists (see Hirschhorn, ...
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1answer
141 views
Are constant $\infty$-sheaves constant on connected components?
Let $C$ be an $\infty$-category endowed with a Grothendieck topology $J$ and consider the $\infty$-topos $\infty\text{Sh}(C, J)$. There is a
natural geometric morphism to $\infty\text{Grpd}$ whose ...
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1answer
178 views
Does a homotopy sheaf functor commute with group completion
Let $\text{sPre}(C)$ be the category of simplicial presheaves with Grothendieck topology $\tau$. Let $\pi_n^{\tau}$ be the $\tau$-homotopy sheaves defined as follow
Does $\pi_n^{\tau}$ commute with ...
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Does every span have a homotopy pushout?
Suppose we have an arbitrary span in an arbitrary model category
$$ B \leftarrow A \to C $$
Is it always possible to complete it to a homotopy pushout square?
$$ \require{AMScd} \begin{CD}
A @>>&...
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1answer
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Proof that $Sing^IX$ is $I$-invariant for an interval object in a site by “simplicial decomposition”
I just saw a proof showing that $Sing^{A^1}X$ is $A^1$ invariant where they use an algebraic topological analogue of ``simplcial decomposition'' defined as follow:
The argument works by showing that ...
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1answer
91 views
Is transfinite composition of weak equivalences of simplicial presheaves a weak equivalence?
In a left Bousfield localization of the projective model structure on the category of simplicial presheaves, what is the condition that transfinite composition preserves weak equivalences? How about ...
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Examples of non-proper model structure
I have recently been thinking about left and right semi-model categories and in which case they can be promoted to Quillen model structure, and I have come to the conclusion that that absolutely all ...
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When is the localization of all hypercovers equivalent to that of Čech covers?
In Theorem A.6 of Dugger's paper, it is shown that a few localizations are equivalent to the localization of Čech covers.
The Nisnevich localization
at all hypercovers is equivalent to the ...
