Questions tagged [model-categories]

Model categories are categories with three distinguished classes of morphisms: the weak equivalences, the fibrations and the cofibrations. They provide a natural setting for (homotopy-theory) in an arbitrary category, by mimicking the usual properties of (co)fibrations and weak equivalences in topological spaces.

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Proof of Proposition A.2.6.13 in Higher Topos Theory

I am reading Lurie's Higher Topos Theory and I need some help to understand a part of the proof of Proposition A.2.6.13. In the proposition, we are working with a locally presentable category $\...
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Iterated homotopy pullbacks

I would like a reference for the following fact, which I believe to be true. Consider the simplicial model category of Kan complexes with the Quillen model structure, and suppose given a commutative ...
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comparing homotopy colimits of "equivalent" diagrams

Motivation Let us work in a fixed model category $\mathcal{C}$. I'm interested in homotopy colimits with indexing categories $\mathcal{I}=\bullet \leftarrow\bullet\to \bullet $ and $\mathcal{I}=\...
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Why does tensor product satisfy pushout-product axiom?

Example 11.4 in this paper claims that the tensor product of chain complexes of bimodules (over not-necessarily-commutative rings) satisfies the pushout-product axiom (the first condition of a Quillen ...
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Reference for loop space objects

Okay so, hi! I would need to have some discussion related to loop space objects in a project of mine, but the only reference I can find for loop space objects is https://ncatlab.org/nlab/show/loop+...
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When does a left Quillen functor preserve weak equivalences?

I am interested in an answer to the following question: suppose we have a left Quillen functor $L: \mathcal{C} \rightarrow \mathcal{D}$ between symmetric monoidal model categories $\mathcal{C}$ and $\...
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CW complexes with cofibration morphisms

Let $CW$ be the category of CW complexes and $CW_{cof}$ be the wide subcategory whose objects are CW complexes and morphisms given by inclusions of subcomplexes (i.e. cofibrations in the model ...
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Dold-Kan correspondence, Model structures and Homology

I am fairly new to the concept of model categories, simplicial sets, etc. And so there is some questions, which may be obivuous, that I need to clarify. Consider the cateogry of simplicial abelian ...
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The class $l(F)$ of morphisms which have the left lifting property with respect to $F$ is stable under transfinite compositions.

I am reading Cisinski's Higher Categories and Homotopical Algebra and I am having trouble trying to verify some claims there. My background in category theory is not very solid. I would like some help ...
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Rotman's Algebraic Topology Lemma 9.11

This is the Lemma 9.11 of Rotman's "An Introduction to Algebraic Topology". The topic where I found this simple lemma of homological algebra is the Theorem of Acyclic Models. So we are ...
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When does the left adjoint of the base change functor between categories of algebras over operads preserve quasi-isomorphisms?

I have been thinking about the following question: given a morphism of coloured dg-operads $$\phi: P \longrightarrow Q$$ we derive a lax morphism between their respective monads $T= T_P \...
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Understanding isomorophism of the two sided bar constuction

I want to better understand the following canonical isomorphism: $$B(X,G,Y)\cong X\times_G B(G,G,Y)$$ Here we are looking in the category of simplicial sets or topological spaces. I thought about ...
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What is meant by the phrase "inclusion of categories induces an equivalence of categories"?

I am looking through Hovey's "Model categories" and the above mentioned phrase comes up several times. Here is a couple of examples: Does this phrase have a strict meaning? Is there a ...
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$f$ - isomorphism in a model category $\implies$ $f$ is fibration, cofibration and weak equivalence?

Why must an isomorphism in a model category be fibration, cofibration and weak equivalence simultaneously? Thank you.
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Is it customary to call a zig-zag of quasi-isomorphisms a weak homotopy equivalence? [closed]

Is it customary to call a zig-zag of quasi-isomorphisms a weak homotopy equivalence? $$ A_0 \leftarrow A_1 \rightarrow \dotsb \leftarrow A_{k-1}\rightarrow A_k $$ Could someone, please, elaborate on ...
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Natural transformation between sheaves in homotopy theory

Firstly a small disclaimer. I am not an expert in the theory of higher sheaves and their presentation in the model categories, so please feel free to correct all inaccuracies in the question itself! ...
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Homotopy Limit is the Limit in the Homotopy Category

I am trying to understand the homotopy limit. This question naturally appears to my mind. Let $I$ be a small category and $\mathcal{X}$ is an $I$-diagram of simplicial sets. There is a functor from ...
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Nisnevich local model structure is a left Bousfield localisation at hypercovers

I am reading $\mathbb{A}^1$-homotopy theory. I found the following statement in an article but can't prove it neither finding any reference to prove it. Please help. Thanks in advance. Let $\Delta^{op}...
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Nisnevich local model structure is a Bousfield localisation of global projective model structure

Suppose $\Delta^{op}Psh(Sm/k)$ is the category of simplicial presheaves on the category $Sm/k$ of finite type smooth schemes over $k$, endowed with Nisnevich topology. $\Delta^{op}Psh(Sm/k)$ has two ...
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Simpliciality of projective model structure on simplicial cofibrantly generated model categories

In Model Categories and Their Localizations, Definition 11.7.2, for $M$ a simplicial cofibrantly generated model category, $C$ a small category, Hirschhorn gives the following simplicial structure on $...
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Is the composition of hom-spaces entirely determined by the tensor in a simplicial model category?

Let $\mathcal{C}$ be a simplicial model category, as defined in https://ncatlab.org/nlab/show/simplicial+model+category. We can prove that the $\mathrm{Hom}$ functor is determined by the tensor, ...
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To check factorization in axiom in cofibration category, it suffices to check factorization of codiagonal map.

Say we have category $C$ with class of morphisms weak equivalence and cofibration. And say we have that (Acylic) cofibrations are closed under pushouts, composition. And that initial object is in $C$ ...
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Global Fibrations are Local Fibration in Local Injecive model Structure

Suppose $T$ is a small site with enough points and $\Delta^{op}Shv(T)$ is the category of simplicial sheaves on $T$ endowed with local injective model structure. Here a map $f: \mathcal{X} \to \...
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Splitting of square diagram in chain complexes

Let us consider a square in chain complexes over a field $k$ \begin{array}{ccc}A & \xrightarrow{f} & B \\ \downarrow{g} & & \downarrow{h} \\ C & \xrightarrow{k} & D\end{array} ...
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Are weak equivalences characterized by the homotopy fibers in an arbitrary model category?

Given a model category $\mathcal{C}$ we can define homotopy pullbacks like the loop space $$\begin{array}{ccc} \Omega_x(X) & \rightarrow & *\\ \downarrow & & \;\;\downarrow_x\\ * & ...
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Long exact sequence of homotopy groups groups in simplicial sets - reference request

I believe it is well-known that for a based map $f:X\to Y$ of simplicial sets (possibly with some extra hypotheses on $X$ and $Y$), there is a long exact sequence $$ \ldots \to \pi_n(F)\to \pi_n(X) \...
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A Formal Proof for the Strom Model Structure

Recall that the Strom-Model Structure is the model structure on compactly generated weakly Hausdorff spaces given by Hurewicz-cofibrations (having the homotopy extension property), Hurewicz-fibrations ...
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What is the motivation behind the Factorization Systems of Model Categories?

I am currently trying to get a grasp of higher category theory, being promised to get a nice framework to do homotopy theory (which I currently understand to be a theory of dealing with categories ...
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What is the use of homotopy pushouts in Lurie's HTT 2.1.4.10?

I have trouble understanding the usage of homotopy pushouts in the proof of proposition 2.1.4.10 in Lurie's Higher Topos Theory. My trouble is the way he shows that for a map $j\colon S\to S'$, the ...
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Homotopy (co)limits and model category structure on functor category

Let $C$ be a model category, $I$ a small category and $C^{I}$ the functor category. I was reading about homotopy (co)limits, and they define them the following way. First give a model category ...
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Chain homotopy as left homotopy and right homotopy

According to Hovey's Model categories (around Theorem 2.3.11), a chain homotopy can equivalently be described as a right homotopy wrt to the standard model structure of the category of chain complexes....
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Trying to unify techniques used to compute derived functors

I have noticed a pattern when dealing with different types of derived functors. Let $\mathcal A$ be an abelian category with enough injectives. If $F: \mathcal A \rightarrow \mathcal A$ is a left ...
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Group Structure on Homotopy Classes of Homotopies

Given a cylinder object $\nabla : X \coprod X \xrightarrow{(i_0,i_1)} Cyl(X) \xrightarrow{p} X$ for a fibrant-cofibrant object $X$ and given homotopies $a,b: Cyl(X) \rightarrow Y$ with $Y$ fibrant ...
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Limits in the localization of a category of fibrant objects

Suppose we have category $\mathcal{C}$ which has the structure of a category of fibrant objects, and suppose we have a functor $F:I\to \mathcal{C}$ with a limit $\lim F$ in $\mathcal{C}$. If we have ...
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Model categories: Any two solutions to a lifting problem are homotopic

I am reading the paper "Model Categories and Simplicial Methods" by Goerss and Schemmerhorn and I am struggling to prove Lemma 2.11, which states the following. Suppose that there is a ...
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Discrete functor preserves serre fibrations

Consider $Top$ equipped with the Serre's structure and $Set$ with the trivial model structure (All maps fibrations and cofibrations, bijections weak equivalences) and the functor $D: Set \to Top$ that ...
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Basic Homotopy Question

I am starting to read the book "Rational Homotopy Theory" by Yves Felix, Stephen Halperin, J.-C. Thomas and I have a quick question about the very beginning (which only concerns basic ...
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What does Hinich mean by "homotopy" and "contractible"?

In "Homological Algebra of Homotopy Algebras", Hinich talks about homotopies and contractible (cochain) complexes. More specifically, I want to look at the following two points: §4.3, see ...
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Injectivity or projectivity criterion in modules and model categories

Here in Hovey's book on Model Categories in as opposed to errata there is already injective in the lemma 2.2.8. The errata says injective and not projective. Also I wonder what is $A$ in $A\to Q$ ...
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Different definitions of suspension in a pointed model category

Let $\mathcal{C}$ be a pointed model category (i.e. $\mathcal{C}$ has the structure of a model category, and in the category $\mathcal{C}$, the initial and terminal object $*$ are isomorphic). I ...
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Example of right homotopy not transitive in Kan-Quillen model structure

I am learning about abstract homotopy theory and know that when $Y$ is fibrant then right homotopy is an equivalence relation on the set of maps from $X$ to $Y$. When $Y$ isn’t, then transitivity ...
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4 answers
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Are weak homotopy equivalences always homotopic?

I was wondering whether two weak equivalences between spaces $X$ and $Y$, are homotopic, or not. If yes, a proof / reference would be very welcome. If not, a counterexample would be interesting! More ...
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What is a Homotopy of Maps in the Sense of Quillen-Model-Structure

I am fairly new to model categories, so apologies if this question has an obvious solution or if it doesn't even make sense... Naive homotopy in $\mathsf{Top}$ (using the unit interval $I$ and maps $X\...
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A path object for a non-negatively graded chain complex

In Mark Hovey's Model categories (chapter 2), a path object of a chain complex is presented. For a chain complex $X$, a chain complex $P$ defined by $P_n := X_n \oplus X_n \oplus X_{n+1}$ is a path ...
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Orthogonal factorization system in a model category

Are there any model category $\langle\mathcal{M},\mathfrak{C},\mathfrak{F},\mathfrak{W}\rangle$ whose factorization systems $\langle\mathfrak{C}\cap\mathfrak{W},\mathfrak{F}\rangle$ and $\langle\...
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Doubt in the proof of Lemma 8.5 (chapter II) in Goerss & Jardine

In Lemma 8.5 (chapter II) of Goerss & Jardine's book, we are given the following pushout square in a category of cofibrant objects: $$\begin{array}{cc} A & \xrightarrow {u} & B \\ \...
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Why are acyclic fibrations called so?

This question refers to acyclic fibrations in model categories. Why is the word "acyclic" being used here? Is there a reason for it, at least in a particular case such as algebraic topology ...
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Intuitions for constructing fibrant/cofibrant objects for a homotopy theory.

I have heard that doing homotopy theory in a category requires only weak equivalences since we can formally localise the category. The class of fibrations and cofibrations are auxiliary constructs ...
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Canonical model structure on a 2-category

It surprised me that the homotopy category of the canonical model structure on $\text{Cat}$ is its familiar quotient category $\text{Cat}/\sim$ whose morphisms are functors modulo natural ...
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1 answer
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Does strict group completion preserve weak equivalences?

Let $i : \mathbf{Grp}\to \mathbf{Mon}$ denote the forgetful functor from groups to monoids. It has a left adjoint, $(-)^{gp}$, which one could call group completion. We have an induced functor $\...
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