# 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### $k$-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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...