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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There is a sometimes fully faithful functor from simplicial commutative algebras to differential graded algebras. What is this functor explicitly?

In Jacob Lurie's paper "Derived Algebraic Geometry," in 2.6 $E_{\infty}$-Ring Spectra and Simplicial Commutative Rings, page 25, there is the following claim: In general, we have functors $\...
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Is normalized chain complex functor the unique Quillen equivalence?

I don't have a grasp of model categories. I asked the question through Quillen equivalences in order to make it as general as possible. This might be too general to answer and/or might be above my ...
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25 views

Existence of limits and colimits in a pointed category. [duplicate]

I am reading Mark Hovey's model category theory. In the first chapter, on page $4$, we have a category $\mathcal C$ which has all small limits and colimits. He claims that the pointed category ${\...
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When is it enough to consider roofs in the derived category?

In the derived category $D^b(\mathcal{A})$ of an abelian category $\mathcal{A}$, obtained by taking the Verdier quotient wrt. all quasi-isomorphism, a morphism is given by a roof (or span) $ X\...
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Cofibrant objects are isomorphic to their cofibrant replacements

It is well-known that in a model category $M$ (assuming functorial factorizations as part of the axioms) we can replace every object with a cofibrant one, up to equivalence. Indeed, we can find a ...
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Why model categories in theory of stacks?

This question is bit related to my previous question about stacks. After understanding the definition of a stack (to be precise $(2,1)$-sheaf), now I am wondering about $\infty$-stacks. According to ...
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Understanding suspension in the model category of chain complexes.

In a model category $C$ and $X \in C$ we can define $\Sigma X$ as the homotopy pushout of $$* \leftarrow X \rightarrow * $$ letting $*$ denote the terminal object. The way I understand it is that if $...
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Description of the left adjoint to the forgetful functor from left fibrations to cocartesian fibrations

I am reading A. Mazel-Gee's paper "All about the Grothendieck construction". In that paper he explains that the left adjoint ${\mathrm{Cat}_{\infty}}_{/\mathcal{C}}\to \mathrm{coCFib}(\mathcal{C})$ to ...
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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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Weak equivalences in $\text{Top}$

Weak equivalences in $\text{Top}$ are the ones for which is $$\pi_n(f,x)$$ a bijection or more strictly group isomorphism ? See the definition $2.4.3.$ in the snippet below. I understand that $\pi_0(X,...
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Two apparently different definitions of a path object in the model category theory

I have a question about a path object in the context of model categories. For what Hovey says see the first snippet below. But here on page 9 the definition is different: Henry says (see the second ...
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Is $\mathcal{C} \times J_{m}$ a cylinder object in ${\bf Cat}$?

I am reading "A model structure on the category of small categories for coverings " https://arxiv.org/abs/0907.5339 and am interested in the question that "What is a cylinder object in ${\bf Cat}_{1}$?...
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Cofibrations in diagram category

Let $\mathcal{C}$ be a model category and $\mathcal{I}$ ba a small category. Then we have the projective model category structure on the diagram category $\mathcal{C}^\mathcal{I}$ where fibrations and ...
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What is the homotopy category of spaces (wrt homotopy equivalence)

I just realized I have always taken the homotopy category of spaces to be CGWH spaces inverted wrt weak homtopy equivalences. i.e. I am considering the homotopical category $(CGWH, whe)$, and $...
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Closed inclusion

I have a simple question in the context of (co)fibrations in the context of Model Categories: Why on the page $52$ in the snippet below $$g^{-1}(d)$$ must be a single point not in the image of $A$ ? ...
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Fibrations are thought of as epimorphisms

In the book More concise algebraic topology on the page 213 they write We think of fibrations as analogous to epimorphisms. BUT Hovey on the page 51 says $f$ is a fibration if it is in $J-inj$. My ...
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Mapping space in functor $\infty$-category as an end

In enriched category theory, we can endow functor categories with an enrichment such that the mapping object between two functors $F,G:\mathcal{C}\to \mathcal{D}$ is described as the end $\int_{c\in\...
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Brent's Method convergence criteria

I am using Brent's method to solve the BEM equations for a wind turbine model. I have run into a scenario where Brent's method has converged i.e., abs(m) is below set tolerance of 1e-8 but the value ...
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Construction of a corner of a diagram out of the homotopy pushout

Let \begin{array}{ccc} X & \xrightarrow{} & Y \\ \downarrow & & \downarrow\\ Z & \xrightarrow{} & W\\ \end{array} be a homotopy commutative diagram in a proper model catgeory $\...
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A question about homotopy pushout

Let \begin{array}{ccc} X & \xrightarrow{} &Y \\ \downarrow & & \downarrow \\ Z & \xrightarrow{} & W\end{array} be a commutative diagram in a proper model catgeory and $P$ be ...
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Questions about pushout in a suitable model category

Let us consider a diagram in any suitable( proper) model category as follows: Where $X' \to X,$ $Y' \to Y$ and $Z' \to Z$ are weak equivalences and P and P' are the pushout of the diagram $Z' \...
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A lifting problem in $\infty$-categories

Let $\mathcal{C}$ be an $\infty$-category and consider the outer horn inclusion $\Lambda[3]_3 \subset \Delta[3]$. Given a diagram $f:\Lambda[3]_3 \to \mathcal{C}$, if the image of $\Delta^{\{2,3\}}$ ...
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Basepoint in a category

What do they mean in the second paragraph by "takes $v$ to $w$" ?
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$\Omega X$-modules are functors from $X$

Let $X$ be a connected (nice) space, $x\in X$ and $\Omega X$ the loopspace at $x$. Then $\Omega X$ has an $E_1$-structure, and so we may consider left $\Omega X$-modules. There are a few ways to do ...
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Model categories, trivial fibrations and cofibrations l.l.p.

Defintion $\bf 1.1.2~$ Suppose $i:A\to B$ and $p:X\to Y$ are maps in a category $\mathcal C$. Then $i$ has the left lifting property with respect to $p$ and $p$ has the right lifting property with ...
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Model categories--Quillen adjunction Left and Right Quillen functor

In the book Model categories on the page 14, do they assume that $U$ in 3. (so that $(F,U,\phi)$ is a Quillen adjunction) is a RIGHT QUILLEN FUNCTOR or just Right adjoint ...
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Why two simplices become equal in the colimit

The following is an image of a proof found in Hovey's Model Categories, for which my question concerns just the second paragraph. I do not see the justification for the statement that we can find $\...
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Existence of certain factorization of simplicial map

The following is an image of a proof from Hovey's Model Categories: How exactly do we know that $s\restriction_{\partial{\Delta[n]}}$ factors through $X_n$? Since $\partial{\Delta[n]}$ has only ...
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Construction of diagram for given pushout

Let $\mathcal{C}$ be a proper model category and $P$ be the pushout of the diagram $Z \leftarrow X \rightarrow Y$ in $\mathcal{C}.$ Now consider $P' \in \mathcal{C}$ such that there is an weak ...
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cylinder object, weak equivalence

Let $C$ be a model category, $B$ an object. We take the coproduct of $B$, $(B\amalg B,i_0,i_1), i_0:B\rightarrow B\amalg B, i_1:B\rightarrow B\amalg B$ injections and then factor $(Id_B,Id_B):B\amalg ...
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Generalization of fully faithful functors between $sSet-$enriched categories

If we have two $sSet$ enriched categories $C$ and $D$ is there a name for a $sSet-$enriched functor $G:C \rightarrow D$ such that the map $C(x,y) \rightarrow D(Gx,Gy)$ is a weak equivalence of ...
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What fields of arithmetic/algebraic geometry benefit from infinity/derived techniques?

I'd like to get a better picture of how infinity/derived techniques become more important in algebraic/arithmetic geometry. I'd therefore like to know: What questions/subfields of algebraic/...
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Neighborhood deformation retracts vs cofibrations

I got really confused over the different notions of neighborhood deformation retracts and cofibrations one can find in various sources on algebraic topology and alike, so I would really appreciate, if ...
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Homotopy cofinality of $\Delta^{op}$ in $\Delta^{op}\times \Delta^{op}$

There is the usual diagonal inclusion $i:\Delta^{op}\to\Delta^{op}\times \Delta^{op}$ which is easily seen to be cofinal in the $1$-categorical sense, and so one can compute colimits on $\Delta^{op}\...
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Pullback against a fibration invariant under homotopic maps

So I am currently trying to piece together little bits of knowledge I have acquired about fibrations in various context when I came across this question. If $p : E \rightarrow B$ is a Serre fibration ...
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Proof of homotopy invariance of homology : any way to make it better?

Suppose you want to prove that homotopic maps induce the same morphisms in singular homology. One way to do that is the following : you have your homotopy $X\times I \to Y$, apply $Sing$ to it to get $...
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Reedy model structure, simplicial sets and model categories

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 one is much more open, it's basically what restricitions do we ...
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Why is this the definition of Quillen bifunctor?

This the definition of a Quillen bifunctor from nlab. Let $C,D,E$ be model categories. $$ \otimes: C \times D \rightarrow E$$ is a Quillen bifunctor if it preserves colimits in each variable and ...
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How can I show that a specific object is cofibrant in $s\mathbf{Alg}_{R}$?

Set-up: I will use the notation $\underline{S}$ to denote the constant simplicial commutative ring which is $S$ in all degrees and has face and degeneracy maps the identity. Let $R_{\bullet}$ be a ...
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Exercise Sheets of a course based on Hovey's Model Categories book

I am not entirely sure this type of question belongs here, but I'll try. I am preparing for an exam based on Hovey's book 'Model Categories'. The book is light on exercises, so I have been wondering ...
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Is the proof of Lemma A.2.3.1 in HTT correct?

I am reading the proof of proposition A.2.3.1 in Lurie's HTT, the proposition can be stated as the following: Let $\mathscr{C}$ be a a model category, $A$ and $B$ be cofibrant objects, and $X$ be a ...
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How to compute a derived tensor product?

Let $A_*$ and $B_*$ be simplicial algebras over a simplicial commutative ring $R_*$. I would like to understand how one explicitly computes the derived tensor product $A_* \otimes^L_{R_*} B_*$. More ...
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A $\mathbb N$-indexed cofibrant replacement in projective model structure.

Let $M$ be a model category, and suppose the projective model structure on $M^{\mathbb N_{\ge 0}}$. Then The diagram $$A_0 \rightarrowtail A_1 \rightarrowtail \cdots$$ with cofibrations between ...
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Retract of a chain complex.

Let $A_*$ and $B_*$ be projective chain complexes which are zero in dimensions $i=-1$ and down. Now suppose there exist maps $A_* \xrightarrow i B_* \xrightarrow r A_*$ such that $r \circ i = \text{id}...
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What's the difference between an anodyne map and a trivial monomorphism?

This sounds silly but they seem the same to me and I can't find any reference confirming or denying this. The argument in my head goes like this: In the standard model structure on simplicial sets, ...
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Reference for homotopy colimits

In the first paragraph of the following paper by Heuts and Moerdijk they speak of A well known construction of homotopy colimits $$h_{!}:sSet^A \rightarrow sSet/NA$$ $A$ is small category. $sSet$...
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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 $\infty Sh(C, J)$. There is a natural geometric morphism to $\infty \text{Grpd}$ whose left ...
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HTT, 2.1.1.3, Lurie

On Lurie's Higher Topos Theory, Prop. 2.1.1.3, Let $F:C \rightarrow D$ be a functor between categories. Then $C$ is cofibered in groupoids over $D$ if and only if the induced map $N(F):N(C)\...
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Transferring model structure along adjunction

I want to use the transfer principal for cofibrantly generated model category, but it’s difficult to check the conditions. Given an adjunction such that $F:C\to D$ is a left adjoint of $G: D \to C$, ...
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Localizations of infinity categories and fibrant replacement

In Higher Topos Theory by J. Lurie, Def. 5.2.7.2, a localization functor between $\infty$-categories is defined as a functor having a fully faithful right adjoint. Now, given any $\infty$-category $\...

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