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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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Mapping spaces for chain complexes

For any module category $M$, there is a mapping space $Map(X,Y)$ for two objects $X,Y$ of $M$ such that $\pi_0(Map(X,Y)) = Hom(X,Y)$ in $Ho(M)$. Chain complexes over a ring $R$ have a model structure ...
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When does a fibration $f:X\rightarrow Y$ in a model category admit a section?.

If we have a fibration $f:X\rightarrow Y$ in a model category $C$, where $Y$ is cofibrant and both $X, Y$ are fibrant. Does f admit a section (right inverse)?. If it does not work in general, ...
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CW complexes are the cofibrant objects in the Quillen model structure on Top?

If $J$ is a class of maps in a category, the $J$-cellular maps are by definition transfinite compositions of pushouts of coproducts of maps in $J$. Now if $J$ denotes the family of inclusions $S^{n-...
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Applying Quillen's small object argument to a specific morphism

I've been learning about model category theory and cofibrant generation of model structures, and I've come across Quillen's small object argument. In the informal paper I'm writing I'm choosing to ...
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Why are some maps monomorphisms?

Let $\mathbb{S}$ be an excellent model category (see https://ncatlab.org/nlab/show/excellent+model+category for the definition), why are all maps from the initial object of $\mathbb{S}$ monomorphisms? ...
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Mapping spaces for pro-objects in a simplicial model category

If $C$ is a simplicial model category (I have simplicial commutative rings in mind), then one can simplicially enrich pro-$C$ over sSet by taking $$\lim_j \mathrm{colim}_i \underline{\mathrm{Hom}}_C(...
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Factorisation into cofibration and trivial fibration

Suppose $M$ is a simplicial model category, $f: R \to T$ a morphism between fibrant objects in $M$. Is there a nice way to construct a factorization of $f$ into a cofibration followed by a trivial ...
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Definition of homotopy fiber

In definition 2.2 of nlab we define the homotopy fiber Let $f:X \rightarrow Y$ be a morphism in a model category $C$, its homotopy fiber $$hofib(f) \rightarrow X $$ is given by the morphism ...
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Examples of weak monoidal Quillen equivalences

Schwede-Shipley introduced the notion of weak monoidal Quillen equivalences between monoidal model categories in "Equivalences of monoidal model categories". Are there any examples of such Quillen ...
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Why Quillen equivalence is used to define “equivalent” model categories?

Why is Quillen equivalence used as the notion of equivalence of model categories? So given two model categories $C,D$, $$L \dashv R:C \rightarrow D$$ is called a Quillen adjunction if $L$ preserves ...
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Definition of reduced cylinder object in pointed category, model category

Definition 4.1 Let $C$ be a model category, with $C^{*/}$ its model structure on pointed objects. For $f:X \rightarrow Y$ a morphism between cofibrant objects, its reduced mapping cone is the object ...
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Grothendieck fibrations via right lifting property

The class of Grothendieck fibrations is a subclass of isofibrations, which constitute the class of fibrations in the canonical model structure on $Cat$. Is there any characterization of Grothendieck ...
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Perfect class of morphisms closed under retracts?

Suppose we are working in a presentable class of morphisms $\mathcal{C}$. A class of morphisms $P$ is said to be perfect if : 1) The class $P$ contains all isomorphisms. 2) The class $P$ ...
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(wrong) Proof of stability of homotopy equivalences under pullback

In studying the category Top localized by homotopies, I asked me this question: "Is homotopy equivalence stable by pullback (base change)?" I know that it is necessary to have a further condition (...
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Model category of all model categories

Is there some natural model category structure on the category of all small model categories and some specified functors among them (i.e. not necessarily all functors), say Quillen adjunctions? What ...
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Theorem 2.1.2.2 Higher Topos Theory

At the page 74 of HTT, there is the following theorem Let $S$ be a simplicial set, $\mathcal{C}$ a simplicial category, and $\phi: \mathfrak{C}[S] \rightarrow \mathcal{C}^{op}$ a simplicial functor....
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Right homotopy relation on the projective model structure of chain complexes

In Mark Hovey's book $\textit{Model Categories}$ it is said that in the projective model structure of chain complexes, the right homotopy relation is precisely the chain homotopy relation. However, I'...
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Homotopy category of Chain complex - isomoprhism = quasi isomoprhism?

Let $Ho(Ch_Z)$ be the localization of the category of nonnegatively graded complexes of abelian groups, $Ch_Z$, wrt the quasi-isomorphisms. If two objects are isomorphic, in this localization, then ...
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What is the $\infty$-category associated to a model category?

It is often said that model categories are but a shadow of an $\infty$-category. It is also often said that model categories should give rise to an $\infty$-category via their homotopies. In fact, ...
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Bergner homotopy category of simplicially enriched caterories is cartesian closed

Let $Cat_{\Delta}$ be the model category of simplicially enriched categories with the Bergner model structure. In a paper I am reading, they state without proof that $Ho(Cat_{\Delta})$ of this model ...
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Subcomplexes of relative I-cell complexes

This is a problem concerning the definitions given by Hirschhorn in his book Model Categories and Their Localizations 10.6.7. Let $C $ be a cocomplete category and let $I $ be a set of maps in C. If $...
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connected components of homotopy pullbacks of nerves of categories

I am not sure if the question makes sense. Given three categories $\mathcal M_1$, $\mathcal M_2$ and $\mathcal M_3$ and a zig-zag $$\mathcal M_1\xrightarrow{f} \mathcal M_3\xleftarrow{g} \mathcal M_2$...
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map on connected components is injective

Consider the usual model structure on $SSet$ (which is proper). Let $X$ and $Y$ be two fibrant simplicial sets and $f:X\rightarrow Y$ a fibration. If for any two vertices $x,x'$ of $X$, the homotopy ...
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The fiber is a homotopy fiber

The problem comes from a paper of Bertrand Toën, Homotopical Algebraic Geometry II, Appendix A, Prop A.0.3. Let $M $ be a model category and $C $ be a full subcategory of the category of weak ...
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Loops Infinity of a spectrum

Let $\mathbf{X}$ be an (orthogonal) spectrum (can assume that it's an $\Omega$-spectrum if this helps give a positive answer) and give the category of orthogonal spectra the stable model structure. ...
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DG cofibrant replacement functor

This problem is one occurred in Bertrand Toën’s Lectures on DG-categories Prop 4.3.4. Let $M$ be a cofibrantly generated $C(k)-$model category. Then it is automatically a DG category and its ...
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Convenient categories in algebraic topology: their importance, and the role topology plays in their construction

Disclaimer. I have stated three questions but I felt that they are so related that they fit within a single post. Context. After reading Hatcher's Algebraic Topology I wanted to learn more about ...
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Weak factorization systems with right maps non closed under retracts

Is there a standard name for the following flavor of weak factorization system? I am interested in the data of two classes of maps $\mathfrak L$ and $\mathfrak R$ in a category $\mathcal C$ such that: ...
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geometric realization of a map is a strong deformation retract

A small category $\mathcal C$ having $O$ as its set of objects is called free if there exists a set $S$ of non-identity maps in $\mathcal C$ such that every non-identity map in $\mathcal C$ can ...
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Universal property of homotopy pullbacks

I am working in a model category $\mathcal C$. Given a fibration $p: Y \to B$ and a map $u : A\to B$ where $A$ and $B$ (and thus $Y$ also) are fibrant, it is know that the usual pullback $A\times_B Y$ ...
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functors preserve homotopy pushouts and homotopy sums

Let $\mathcal {M}$ and $\mathcal {N}$ be two cofibrantly generated model categories. Let $I$ be a small category. Then $\mathcal M^I$ has the projective model structure. The colimit functor is hence a ...
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one question about homotopy pushout

This question arises when I'm reading Jacob Lurie's Higher Topos Theory, p814. Suppose we are given a diagram $$A_0\leftarrow A\rightarrow A_1$$ in a model category $\mathcal C$. In general, the ...
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Underlying quasicategory of a model category through framings?

Consider a model category $\mathcal M$. Because it is a category with weak equivalences, we can use the following construction to obtain the "underlying quasicategory" of $\mathcal M$: taking the ...
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Comparison of model structures on $sMod_{B}$

Let $B$ be a commutative ring with unity. By Theorem 4 from Chapter 2 Section 4 of 'Homotopical Algebra' we can impose a model structure on $sMod_{B}$ by declaring a map of simplicial $B$-modules $f:...
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the Verlinde formula

The Verlinde formula writes the fusion coefficient in terms of S matrix. My question is that for fusion category without braiding, is there a similar formula which gives the fusion coefficient in ...
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Why is the flat-cotorsion pair actually a cotorsion pair?

Fix a ringed space $(X,\mathcal{O})$ and denote by $\mathcal{F}$ the class of flat modules. A module $C$ is called cotorsion if $Ext^1(F,C)=0$ for all flat modules $F$. I want to show that a module $...
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Small question on the Kan-Quillen model structure on simplicial sets

I would like to know two things about the Kan Quillen model structure on simplicial sets: Firstly, let $|-|\dashv S$ be the geomtric realization, simplicial complex functor, and let $\eta_X$ be its ...
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Two equivalent definitions of weak equivalences of simplicial sets

In and André Joyal and Myles Tierney their Notes on Simplicial Homotopy, two different definitions of a weak equivalence are given, which they say are equivalent (p. 59). I, however, only see why the ...
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Map induced by fibrations is a fibration

I have the following problem: given a model category $\mathcal{C}$ and an indexing set $I$ I have objects $X$ and $Y_i$ for every $i \in I$ with a fibration $f_i \colon X \rightarrow Y_i$. Now using ...
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Homotopy limits commute

I'm looking for a reference for the fact that homotopy limits commute. That is $\mathrm{holim}_U \mathrm{holim}_W E = \mathrm{holim}_W\mathrm{holim}_U E $. Preferably looking for a result that ...
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The right homotopy class and cofibrant replacement

Let $f:X\rightarrow Y$ be a map in a model category. Let us denote $Qf$ "the" map it induces on the cofibrant replacements of $X$ and $Y$. How can I show that if $Y$ is fibrant, the right homotopy ...
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A model category structure on chain complexes

The wikipedia claims that there is a model category structure of the category of arbitrary chain-complexes of R-modules which is defined by: weak equivalences are chain homotopy equivalences of chain-...
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Set theoretical problems in the construction of the homotopy category

I have often been told that model categories are nice because they give a framework where the construction of the homotopy category does not incur in set-theoretical problems, but I have never been ...
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model structure on the arrow category

I wish to show that given a model structure on a category $C$ one can define a model structure on $C^2$, where $2$ denotes the category with two objects and only one non-identity morphism. I define ...
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Trivial cofibration is a deformation retract?

I read somewhere that a trivial cofibration $w:E_1\to E_2$ (in $\mathcal sSet$ where cofibrations are monomorphisms) means that $E_1$ is a deformation retract of $E_2$. Why is that? I see that I can ...
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Factorization axiom for model categories

One of the axioms for a model structure on a category $C$ is that any morphism $f$ of $C$ can be factored as $f = pj$, where $p$ is a fibration, $j$ is a cofibration, and we can choose either one to ...
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How can I understand maps out of a limit or into a colimit?

If $L$ is a limit, we know that maps $A\to L$ are characterized by cones from $A$. Similarly if $C$ is a colimit, maps $C\to B$ are characterized by cocones into $B$. Is there a general way to ...
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Are all isomorphisms contained in the class of weak equivalences?

Suppose $\mathscr{C}$ is a model category, defined as in the book Simplicial homotopy theory by Goerss and Jardine, chapter $2$. Is it true that every isomorphism in $\mathscr{C}$ is a weak ...
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Left Homotopy is an Equivalence Relation

I was reading Mark Hovey's Model Categories and I am confused about the following proof for left homotopy being an equivalence relation Firstly, how does $B$ being cofibrant imply that $t$ is a weak ...
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How far are functors valued in Ho(Cat) from pseudofunctors?

Basically my question is: Does the Grothendieck construction make sense for (some?) functors valued in $\mathrm{Ho}\,(\mathbf{Cat})$? By $\mathrm{Ho}\,(\mathbf{Cat})$, I mean the category whose ...