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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13
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276 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 ...
12
votes
2answers
832 views

Why is Top a model category?

Recall that a model category is a complete and cocomplete category with classes of morphisms called cofibrations, fibrations, and weak equivalences. These are closed under composition and satisfy ...
11
votes
1answer
427 views

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 ...
7
votes
1answer
279 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, ...
7
votes
2answers
250 views

Geometric interpretation of injective/projective resolutions?

I understand the geometric interpretation of derived functors, as well as their usefulness in giving a simple, purely algebraic description of cohomology. I also understand how resolutions are used ...
7
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0answers
97 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 ...
5
votes
1answer
133 views

What does it mean for a category to be “tensored over” another category?

What does it mean for a category to be "tensored over" another category? I was reading "Stable model categories are categories of modules" by Schwede and Shipley ...
5
votes
3answers
338 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 ...
5
votes
1answer
183 views

The Notion of “A Homotopy Theory”

Sometimes (specifically in this case I'm looking at Charles Rezk's "A Model for the Homotopy Theory of Homotopy Theory") it seems that people refer to the homotopy category of a model category as a ...
5
votes
2answers
80 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 ...
5
votes
2answers
119 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 ...
5
votes
1answer
161 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 ...
5
votes
0answers
113 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 - ) ...
4
votes
2answers
168 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
1answer
128 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 ...
4
votes
1answer
76 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
votes
1answer
165 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 $G : \mathbb{C}^\mathrm{op} \to \mathbf{sSet}$, ...
4
votes
2answers
109 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, ...
4
votes
1answer
171 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 ...
4
votes
0answers
41 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 ...
4
votes
0answers
110 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
136 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 ...
3
votes
2answers
319 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 ...
3
votes
1answer
43 views

Is the induced map $\varphi$ on the homotopy cofibers null-homotopic in this situation?

Let \begin{eqnarray} X & \xrightarrow{f} & * \\ \downarrow & & \downarrow\\ Y & \xrightarrow{g} & Z \end{eqnarray} be a (strictly) commutative diagram of pointed CW-complexes ...
3
votes
1answer
79 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)?
3
votes
1answer
78 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 ...
3
votes
1answer
114 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
1answer
43 views

Inclusions of CW-complexes are cofibrations.

Has the inclusion from the $ (n - 1) $-sphere in the $ n $-disc the left lifting property for all acyclic Serre fibrations? I am looking for a reference for this proposition, or alternatively, for an ...
3
votes
1answer
54 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] ...
3
votes
0answers
146 views

A construction with homotopy colimits and homotopy pullbacks for descent.

I have some troubles in trying to give a meaningful interpretation to the following property which is stated in this preprint by professor Rezk (see Definition 6.5) as part of the requirement for a ...
3
votes
0answers
42 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 ...
3
votes
0answers
148 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 ...
3
votes
0answers
57 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 ...
3
votes
0answers
88 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 ...
3
votes
1answer
121 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
82 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 ...
3
votes
0answers
174 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 ...
2
votes
2answers
132 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 ...
2
votes
1answer
87 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?
2
votes
1answer
85 views

How was the functorial factorization axiom “frequently misstated”?

In modern times, a model category is frequently required to have functorial factorizations of morphisms into (fibration/acyclic cofibration) and (acyclic fibration/cofibration). But the precise ...
2
votes
1answer
41 views

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 ...
2
votes
2answers
504 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 ...
2
votes
1answer
39 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
130 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 ...
2
votes
2answers
81 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
36 views

Derived categories of filtered modules

For a ${\mathbb Z}$-filtered ring ${\mathbb k}$ one can consider the category ${\mathbb k}\text{-filt}$ of ${\mathbb Z}$-filtered ${\mathbb k}$-modules, equipped with the exact structure which ...
2
votes
0answers
64 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
90 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 ...
2
votes
0answers
36 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 ...
2
votes
0answers
158 views

left inverse to trivial fibration is trivial cofibration

It is claimed that in the model category of simplicial sets (with usual model structure), a trivial fibration $X \to Y$ has a section, which is a trivial cofibration. Now, I see that there is a ...