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 ...

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6
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2answers
187 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 - ) ...
5
votes
1answer
165 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 ...
3
votes
1answer
86 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 ...
0
votes
1answer
63 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 ...
17
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0answers
347 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 ...
7
votes
0answers
140 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
0answers
52 views

Tensoring a connective chain complex with a simplicial set

Let $\mathrm{Ch}_{\geq 0}(R)$ be the category of chain complexes of $R$-modules concentrated in nonnegative degrees, equipped with the projective model structure. By a general theorem about model ...
4
votes
0answers
42 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
120 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
141 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
0answers
79 views

Why is the Quillen model structure so painful to find?

Proving that the category of simplicial sets carries the Quillen model structure is undoubtedly difficult; the book by May and Ponto "A more concise course in algebraic topology" makes a considerable ...
3
votes
0answers
36 views

If weak-equivalences in a model category are generated by an equivalence relation $\sim$ on hom-sets…

Suppose that $C$ is a bicomplete category, and $\sim$ is an equivalence relation defined on every hom-set of $C$, compatible with composition. Then we can define the quotient category $C/\sim$ whose ...
3
votes
0answers
177 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
173 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
59 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
101 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
0answers
88 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
213 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
0answers
73 views

Correct definition of model category

When answering this question, In a model category, is the full subcategory of fibrant objects a reflective subcategory? I realized that I wasn't even sure what the correct definition of a model ...
2
votes
0answers
84 views

Is the Quillen Injective Model Structure on the category of positive cochain complexes of R-modules (co)fibrantly generated?

Denoted with $Ch^+_R$ the category of positive cochain complexes of R-modules (for a commutative ring $R$), it admits a model structure where: weak-equivalences are quasi-isomorphisms; cofibrations ...
2
votes
0answers
93 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
38 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
225 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 ...
1
vote
0answers
33 views

Gluing along infinitely many trivial cofibrations

I am in a situation where I have a space Y obtained from a space X by gluing on infinitely many trivial cells. That is, I have a collection of maps $f_\alpha: A_\alpha \to X$ each of which is a ...
1
vote
0answers
32 views

projective model structure on presheaves , hom-functors are always cofibrant

Why hom-functors are always cofibrant in the projective model structure in $[\cal T,\cal V]$? The claim is here on page 5.
1
vote
0answers
26 views

Quillen equivalence vs $\infty$-categorical equivalence

It's well known that a simplicial model category presents an $\infty$-category by the homotopy coherent nerve construction. (I am drawing my knowledge and terminology from what little of Lurie's ...
1
vote
0answers
39 views

Model category that doesn't admit functorial factorizations?

I guess it's a modern convention that model categories are typically required to have functorial factorizations. In the cofibrantly generated case, the factorizations constructed by the small object ...
1
vote
0answers
37 views

Composition of Dual Maps in (rigid) Monoidal Categories

If $X, Y$ are objects in monoidal category $\mathcal{C}$ which have left duals $X^∗, Y^∗$ and $f : X → Y $ is a morphism in $\mathcal{C}$, then the left dual map $f^∗ : Y^∗ → X^∗$ of $f$ is given by: ...
1
vote
0answers
79 views

“Stable model categories are categories of modules” - Clarification about a few things

I was reading Schwede and Shipley's "Stable model categories are categories of modules", I needed clarification about a few things: 1 - When they say that stable model categories are categories of ...
1
vote
0answers
37 views

Proof of Quillen's lemma about minimal fibrations

In Hovey's book $\textit{Model categories}$ (available online here http://www.math.rochester.edu/people/faculty/doug/otherpapers/hovey--model-cats.pdf) one finds the classical result that any ...
1
vote
0answers
36 views

Criterion for projectively cofibrant diagrams?

Let $\mathbf{D}$ be a small category, $\mathbf{Set}$ the category of sets and $\mathbf{sSet}$ the category of simplicial sets with its usual model structure. The category of functors $\mathbf{D}\to ...
1
vote
0answers
42 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 $ ...
1
vote
0answers
98 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 ...
1
vote
0answers
29 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
1
vote
0answers
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 ...
1
vote
0answers
176 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 & ...
0
votes
0answers
45 views

Weak equivalence iff isomorphism in homotopy category?

I know that a weak equivalence becomes an isomorphism in the homotopy category but is the opposite direction true? Suppose we have a map $f: C\rightarrow D$ in a model category. If $f$ becomes an ...
0
votes
0answers
54 views

Limits in a Model Category

I've become interested in how the axioms of a model category have changed, since originally posed by Quillen; in particular, that Quillen only originally required finite limits and colimits, however ...
0
votes
0answers
23 views

Is the category of H-bicomodules within the monoidal category of H-bimodules equivalent to the category of left H-comodules

Fix $\mathbb{k}$ a field. Let $H$ be a $\mathbb{k}$-quasi-bialgebra. Is there an equivalence $ {}_H^H \mathcal{M}_H^H \cong {}^H \mathcal{M}$ where ${}_H^H \mathcal{M}_H^H$ is the category of ...
-1
votes
0answers
16 views

Assigning groups to a general model category

The two most fundamental model categories are the category of simplicial sets and the category of chain complexes. Most other model categories are constructed from these in some way. In both these ...