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
155 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
163 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
85 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
60 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 ...
15
votes
0answers
317 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
128 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
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
119 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
139 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
76 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
31 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
170 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
52 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 ...
3
votes
0answers
165 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
58 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
98 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
87 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
196 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
41 views

Relation between homotopy and homology groups of realization of simplicial abelian group

Let $X$ be a simplicial abelian group, $U(X)$ the corresponding simplicial set and $|U(X)|$ its geometric realization (assumed to be path-connected). Then, for $k \geq 1$, how are $\pi_k(|U(X)|)$ and ...
2
votes
0answers
74 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
220 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
29 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
21 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
35 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
34 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
72 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
31 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
32 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
40 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
90 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
27 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
172 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
20 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 ...
0
votes
0answers
43 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
53 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
22 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 ...