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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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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65 views

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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1answer
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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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1answer
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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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1answer
52 views

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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1answer
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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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1answer
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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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1answer
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Discrete functor preserves serre fibrations

Consider $Top$ eqquiped with the Serre's struture 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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Model Structure, finite product and weak equivalence

Consider a model category $(M, Cof, Fib, WE)$ and $M$ is a category that admits finite products. Let $p : Z \to X$ and $q: Z \to Y$ be $WE$. Is it possible to show that $p \times q$ is a $WE$? where $...
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4answers
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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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1answer
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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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1answer
87 views

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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1answer
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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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1answer
60 views

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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1answer
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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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Model categories: $\text{Ho}$ and $\cal C_{cf}/\sim$

I have asked this question about model categories: Why $\text{Ho} \ \cal C$ is $\cal C_{cf}/\sim$ and not $\cal C/\sim$ and I got this answer: take for cofibratiobns Iso, weak equivalences all arrows. ...
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1answer
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Are these pointwise cofibrant cosimplicial objects cofibrant in the Reedy model structure?

Suppose I have a Quillen pair $F \dashv G$ with $F:\text{Psh}(\mathcal{C}\times{\Delta}) \to \mathcal{M},$ and consider also the category of cosimplicial objects in $\mathcal{M}$ denoted $\mathcal{M}^...
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Hovey book Model Categories

In his book Model Categories Hovey uses often the symbol $I$ as in the snippet below. But on the page 14 he says that $I$ is a set and on the page $30$ that $I$ is a set of class of maps in $\cal C.$ ...
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Fibrant Replacement Functor: its action on morphisms

I'm reading the below in Model Categories by Hovey. And before we go further, here is the definition of model category I am working with: From reading the answer from Fibrant replacement functor, I ...
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Induced map in model categories

In the snippet below I do not understand what is $$X_i\to X_i \coprod_{L_i X}L_i Y ,$$ i.e. how is it defined and why is it a cofibration.
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Cosimplicial resolution and fibrations

A cosimplicial resolution of a functor $\gamma: \mathcal{C}\to \mathcal{M}$ is given by a functor $\Gamma: \mathcal{C}\to \mathcal{M}^{\Delta}$ such that for every $X\in \mathcal{C},$ $\Gamma(X)$ is ...
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natural isomorphisms

In the snippet below (taken from Hovey's MC book), I would like to understand in some detail the lemma 6.1.2 (what it says) and the proof of it. Its too sketchy for me. I do not even know what is Map$...
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1answer
48 views

Fibrant replacement functor

In the snippet below (taken from Hovey's MC book) why is $$X\mapsto QX$$ but then the direction is reversed:$$QX\to X$$? Also, I would like to understand in that paragraph how $\alpha$ and $\beta$ are ...
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1answer
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What is an infinity category, really?

I'm interested in precisely what information an infinity category encodes. For example, consider the infinity category of spaces. I like to think about this as the homotopy category of spaces equipped ...
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1answer
46 views

Right Lifting Properties Against Relative Cell Complexes

So I've been studying these notes on homotopy theory. There is a proposition (2.10) which states that for any collection of morphisms $K \subset \mathrm{Mor}(C)$ the collection KProj of $K$-...
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Reference or counterexample: simplicial algebra isomorphism

I'm looking for either a reference, a quick proof, or a counter example to the following claim: Let $R_{\bullet}$ and $T_{\bullet}$ be simplicial $k$-algebras. Let $S_{\bullet}$ be the polynomial ...
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1answer
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Correspondence of left and right homotopies produces a homotopy of maps $A\times I \to X^I$ (Hovey, Lemma 6.1.5)

My question pertains to a specific piece of the proof of Lemma 6.1.5 in Hovey's Model Categories (page 160 in pdf, 150 by numbering). I'll briefly recall the definitions and set up my question first, ...
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0answers
62 views

Kan extension and singular cohomology

Let $Sm$ denote the category of smooth manifolds and $Top$ the category of topological spaces. Then we have a faithful functor $Sm\rightarrow Top$ which gives a functor $$Ho(Sm)\rightarrow Ho(Top).$$ ...
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Fibrantly generated model category

An important concept in the study of model categories is that of "cofibrantly generated model categories". These are nice because all morphisms can be obtained from a small subset of them ...

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