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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1answer
28 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
votes
1answer
84 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 ...
1
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1answer
40 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 ...
0
votes
1answer
45 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 ...
2
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1answer
63 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?
3
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1answer
56 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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33 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 $ ...
3
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0answers
136 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
32 views

Contractible objects in model categories [closed]

Please, what is the definition of contractible object in a (closed) model category (if it exists)?
4
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1answer
94 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 ...
7
votes
0answers
82 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 ...
4
votes
2answers
74 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
65 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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0answers
52 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
55 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 ...
0
votes
0answers
32 views

Generalized Reedy model structure

If $\mathcal{R}$ is a generalized Reedy category in the sense of Berger-Moerdijk, one can endow the fonctor category $[\mathcal{R},\mathcal{C}]$ with the generalized Reedy model structure. When is it ...
4
votes
1answer
65 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 $$ ...
4
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0answers
39 views

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 ...
0
votes
1answer
30 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
3
votes
0answers
72 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 ...
5
votes
2answers
111 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 ...
1
vote
2answers
68 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
votes
0answers
53 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 ...
2
votes
0answers
85 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 ...
1
vote
1answer
31 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 ...
6
votes
1answer
189 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, ...
2
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0answers
33 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 ...
3
votes
1answer
100 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 ...
3
votes
0answers
77 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
votes
1answer
122 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
votes
1answer
73 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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0answers
15 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 ...
1
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1answer
74 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
votes
1answer
157 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
107 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
votes
3answers
262 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 ...
12
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0answers
240 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
votes
2answers
108 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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0answers
140 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
votes
2answers
152 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 ...
4
votes
0answers
108 views

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
votes
0answers
133 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 ...
4
votes
0answers
89 views

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 - ) ...
1
vote
1answer
248 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 ...
3
votes
1answer
109 views

Contractible homotopy fibre for CW complexes, categorial construction of the homotopy inverse

Let $f:X\to Y$ be a map of topological spaces. Assume further that the homotopy fibre is contractible. We get a long exact sequence on the homotopy groups and if $X$ and $Y$ are connected $f$ is a ...
3
votes
2answers
265 views

Kan fibrations and surjectivity

I have a basic question on the usual model structure on simplicial sets. What is the relation between being a Kan (trivial maybe ?) fibration and surjectivity ? Surjectivity here means either ...
4
votes
1answer
164 views

Do we implicitly consider model categories to be locally small?

Do we implicitly consider model categories to be locally small? I have the impression (but am not sure) that many references on model categories assume that all the categories are locally small, but ...
2
votes
2answers
427 views

Monomorphisms and fibrations are preserved by pullback

I just came across a strange property of morphisms that are preserved under pullbacks, and it made me wonder. Consider a model category $\mathcal{M}$. Because the fibrations are exactly the maps that ...
3
votes
0answers
155 views

How are injective model structures cofibrantly generated?

I have a question about the injective model structure on functor categories. As background : If $\mathcal{M}$ is a combinatorial model category and $\mathcal{C}$ is a small category, then there are ...