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

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small limits in the pointed category of a category with all small limits

In Hovey, Model Categories, he says that given a category $\mathcal{C}$ with a small limits and colimits, if we form the category of pointed objects ,i.e. the category with objects as morphisms ...
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1answer
28 views

definition of a model category

In Quillen's book ,,Homotopical Algebra" he defines a model category as a category with three types of arrows satyfying some axioms. I have problems with understanding two of them. These axioms says ...
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41 views

Homotopy and chain homotopy determine each other

In continuation of this previous question, I'm having problems with the following proposition from Kamps and Porter's Abstract Homotopy and Simple Homotopy Theory: Proposition (3.7). [page 210] ...
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1answer
118 views

Constructing model category from given category

Given a model category $\mathcal{M}$, Goerss and Hopkins constructed a subcategory (see Structured Ring Spectra, p. 160) $\mathbf{E}$ of $\mathcal{M}$ such that: If $X\in\mathbf{E}$ and $Y$ is ...
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Doubt about proof of factorization $f=pi$, where $i$ is acyclic cofibration and $p$ is fibration

I try to understand a proof in More Concise Algebraic Topology: Localization, completions and model categories by May & Ponto (pdf). The proof is on page 262, and it is for the statement Any ...
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38 views

Direct way to show that 2-out-of-6 holds for weak equivalences in a model category?

In a model category, when weak equivalences are inverted, nothing else gets inverted. It follows that weak equivalences satsify 2-out-of-6. But the first sentence takes some work to show. Is there a ...
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1answer
33 views

Chain complexes as a model category?

I'm reading a paper called Model Categories and Simplicial Methods by Paul Goerss and Kristen Schemmerhorn, and they show that cofibrations have lifting property with respect to acyclic fibrations for ...
2
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1answer
101 views

Functorial cofibrant replacement does not have to be fibration?

I'm new to model category theory, and I find myself confused about the different meanings of cofibrant replacement in literature. The usual definition is that we assign to every object $X$ in our ...
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1answer
44 views

are equivalences in an $(\infty,1)$-category preserved under colimits

Let $C$ be an $(\infty,1)$-category (e.g. quasicategory) having all small colimits. If $f_i:x_i \to y_i$ are equivalences in $C$ indexed by a small set $I$, is $f:=\mathrm{colim}_{i \in I} f_i$ an ...
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1answer
53 views

Strong (trivial) cofibration in Lurie's HTT

in Lurie's book HTT in annexe A, proof of Proposition A.2.8.2 page 824, he mentions that a map is a "strong (trivial) cofibration" but I didn't succeed to find the definition of this notion that seems ...
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1answer
73 views

Hurewicz model structure and cofibrantly generated model categories

Is it an open problem if $\mathbf{TOP}$ with Hurewicz (Strøm) model structure is cofibrantly generated?
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1answer
64 views

Derived categories as homotopy categories of model categories

Given an abelian category A, is there a model structure on the category of complexes C(A) (or K(A) ("classical" homotopy category)) such that its homotopy category "is" the derived category D(A)?
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35 views

Homotopy product

Sorry if this is a trivial question. Let $ \mathfrak{X} $ be a model category such that all objects of $ \mathfrak{X} $ are fibrant. Then we have a total derived functor of the product $ ...
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138 views

Homotopy limits

Let $\mathfrak C$ be a Grothendieck category and let ${\bf D}=\mathrm{D}(\frak C)$ be its derived category, that is, consider the injective model structure on the category $\mathrm{Ch}(\frak C)$ of ...
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1answer
35 views

Contractible objects in model categories [closed]

Please, what is the definition of contractible object in a (closed) model category (if it exists)?
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1answer
108 views

Homological algebra (homotopical approach)

I have gone through a couple of courses in homological algebra, in the context of derived functors, abelian categories,... Now I would like to watch it from another perspective: my main interest is ...
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88 views

Origins of the name “Q” and “R” for cofibrant and fibrant replacement functors.

In a model category $\mathscr M$ (in the modern sense, i.e. closed and with functorial factorizations), there is a notion of fibrant and cofibrant replacement functors. Specifically, for any object ...
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2answers
77 views

Duality 2-functor on adjunctions

This question is about the definition of the duality 2-functor in Hovey's book on Model categories, Section 1.4. There he defines the 2-category of categories with adjunctions as follows: objects ...
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1answer
66 views

Inequivalent Model Categories

A Model Category is, informally, a category where a "reasonable" notion of homotopy can be developed. I'm curious to know when two model categories are considered equivalent to each other. Thanks for ...
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58 views

Cofibrantly generated categories, cardinals, and the Thomason model structure on $\mathbf{Cat}$.

When reading up the general theory of cofibrantly generated model categories, anything related to the small object argument relies on the choice of some cardinal $\lambda$. I.e. we need ...
3
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0answers
56 views

Can injective and projective model structures be the same ?

Is there a non trivial example where injective and projective model structures coincide ? By non trivial I mean an example of a functor category on a non discrete base category where I guess ...
4
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1answer
68 views

Question on the uniqueness of a homotopy colimit up to unique isomorphism

Let me first give an abstract definition of the homotopy colimit. Let $C$ be a cofibrantly generated model category and let $D$ be a small category. There is an adjunction $$ ...
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Coherence between lifts in a model category

I'm afraid that my question won't be specific but I can edit it depending on the reactions I will get. In a functor category $[\mathcal{C},\mathcal{D}]$ (typically a presheaves category) with a model ...
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1answer
33 views

The definition of a (pseudo-)Dwyer map

I've looked up the definition in Raptis Homotopy Theory of Posets, on the nlab and even Cinskies a classe des morphismes de Dwyer n'est pas stable par rétractes, but none of these make sense to me. ...
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0answers
23 views

Explicit fibrant replacement

Do you know an explicit fibrant replacement in the injective model structure on a functor category (I'm essentially interested in the case of presheaves of groupoids) ? Best
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75 views

Injective model structure

I equip the category of presheaves $[\mathcal{D}^{op},\text{Gpd}]$ with the injective model structure ($\mathcal{D}$ is just any small category). In this structure, weak equivalences and cofibrations ...
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2answers
116 views

Model structure on sSet

Which is the model structure on $ \text{sSet} $ (category of simplicial sets) that makes $\text{sSet}$ Quillen equivalent to the category $ \text{Cat} $ (of small categories) by the adjunction ...
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2answers
74 views

Right homotopic maps iff chain homotopic

Assume the model structure on $Ch(R)$ (chain complexes of left modules over the ring $R$) in which fibrations are dimensionwise epimorphisms (i.e. surjections) and weak equivalences are homology ...
2
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57 views

The stable category of modules over quasi-Frobenius ring as a homotopy category

I'm studying homotopical algebra and I'm trying to prove the following fact: If $R$ is a quasi-Frobenius ring (for $R$-modules one has "projective module$\iff$ injective module") then the stable ...
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86 views

coproduct of base and fiber is weakly equivalent to total space of a fibration in stable model category

Let $C$ be a proper pointed model category such that for any $X, Y \in C$ the natural morphisms $$QX \coprod QY \to X \coprod Y \to RX \times RY$$ are all weak equivalences (here $Q $ and $R$ are ...
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1answer
33 views

Does There Exist an Induced Model Strucutre via Ordinary Equivalence?

If $F:\mathcal{M} \to \mathcal{C}$ is an equivalence of categories, and $\mathcal{M}$ is a model category, does $\mathcal{C}$ inherit a model structure from $\mathcal{M}$ via $F$? If not, is there a ...
7
votes
1answer
223 views

The projective model structure on chain complexes

Let $\mathcal{A}$ be an abelian category with enough projective objects and let $\mathcal{M}$ be the category of chain complexes in $\mathcal{A}$ concentrated in non-negative degrees. Quillen [1967, ...
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34 views

2 out of 3 axiom and simplicial sets

Let $i\colon\mathcal W\to\mathcal C$ be the inclusion of a subcategory. Unless I'm mistaken, the 2 out of 3 axiom for $\mathcal W$ to be a category of weak equivalences can be expressed as the ...
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1answer
103 views

Are cofibrant/fibrant replacements homotopically unique?

Let $\mathcal{M}$ be a model category in the old sense, i.e. with factorisations that are not necessarily functorial, and let $X$ be an object in $\mathcal{M}$. The category of cofibrant replacements ...
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0answers
79 views

Are pulation squares “weak equivalences”?

Let $\cal C$ a category where a square is a pullback if and only if it is a pushout. Is there a model structure on $\cal C^\to$ (the category of arrows of $\cal C$) where the class of weak ...
4
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1answer
137 views

The two-sided simplicial bar construction is Reedy-cofibrant

Let $\mathcal{M}$ be a simplicial model category and let $\mathbb{C}$ be a small category. For a diagram $F : \mathbb{C} \to \mathcal{M}$ and a weight $G : \mathbb{C}^\mathrm{op} \to \mathbf{sSet}$, ...
3
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1answer
74 views

How to define the category of model structures of a category?

It is possible to come up with different model structures for a fixed category. Let $\mbox{Models}\left(\mathcal{C}\right)$ be the category of all model structures of $\mathcal{C}$, which has as ...
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16 views

Homotopy equivalence of C-modules

A topological leftmodule $_CX$ is a topological functor $C \to Top$ for a Category $C$. A morphism $_CX \to _CY$ of leftmodules is a natural transformation of functors. Now such a morphism is called ...
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1answer
76 views

Question about Lemma 7.7.1 from Hirschhorn's Model Categories and Their Localizations

The Lemma states the following. Let M be a model category. If $g:X\rightarrow Y$ is a weak equivalence between cofibrant objects in M, then there is a functorial factorization of $g$ as $g=ji$ where ...
5
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1answer
158 views

Homotopic maps in a directed system induce homotopic maps on colimit?

Let $(A_i,f_i)$ be a directed system of CW-complexes with colimit $A$. Further, let $g_i:A_i\to A_{i+1}$ be maps such that $g_{i+1}f_i=f_{i+1}g_i$ and $f_i\simeq g_i$. This might or might not be the ...
4
votes
2answers
108 views

Problem understanding a proof in Model Categories by Hovey

I have serious problems understanding this proof from the book Model Categories, by Mark Hovey: Here's a list of things I don't understand: He's trying to prove the assertion by contradiction, ...
5
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3answers
305 views

Cylinder object in the model category of chain complexes

Let $\text{Ch}⁺(R)$ be the category of non-negative chain complexes of $R$-modules where $R$ is a commutative ring. What is a cylinder object, in the sense of model categories, for a given complex ...
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255 views

Closed model categories in the sense of Quillen [1969] vs the modern sense

The modern definition of (closed) model category differs in two ways from Quillen's 1969 definition: Model categories are now required to be complete and cocomplete, whereas Quillen only asked for ...
2
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2answers
121 views

Functor between categories with weak equivalance.

A homotopical category is category with a distinguished class of morphism called weak equivalence. A class $W$ of morphisms in $\mathcal{C}$ is a weak equivalence if: All identities are ...
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146 views

Homotopy pullback square implies weak equivalence of homotopy fibres

I am quite confused about the following situation: suppose that we have a map $f \colon X \to Y$. Its homotopy fibre is defined as the pullback of the following of diagram: \begin{matrix} Ff & ...
4
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2answers
161 views

The empty set in homotopy theoretic terms (as a simplicial set/top. space)

I am currently confused about the empty set in terms of its path components and how this fits into the Quillen adjunction between topological spaces and simplicial sets. Probably, one of my ...
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A fibrant-objects structure on $\bf Top$

One can define (Paragraph 1.5, page 10) a fibrant-object structure on a suitable cartesian closed category of topological spaces $\bf Top$, called the $\pi_0$-fibrant structure: A ...
4
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134 views

A model structure on $\bf Cat$

Define a model structure on $\bf Cat$ by the following rules: A weak equivalence is an equivalence of categories; A cofibration is a functor which is injective on objects; A fibration is a functor ...
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In a model category, is the full subcategory of fibrant objects a reflective subcategory?

I apologize in advance if my question is utterly stupid, but I can't resist asking it. So... Is it true that in a model category ( - for example $\mbox{Set}_\Delta$ with the Joyal model structure - ) ...
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1answer
269 views

Why does the definition of homotopy cartesian involve factorisations

Setup: A diagram $$\require{AMScd} \begin{CD} X @>>> Y\\ @VVV @VV{f}V\\ U @>>> V \end{CD}$$ in a (proper) model category is called homotopy cartesian if there exists a ...