# 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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### 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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### 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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### 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 ...