Questions tagged [model-categories]

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 topological spaces.

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Cofibrant replacement in coarse model structure onG-spaces

This question is about the Borel model category structure on $G$-spaces for a topological group $G$. This is also sometimes called the coarse projective model structure. In this article https://...
Fabio Neugebauer's user avatar
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Morphism inducing isomorphism in homotopy category is weak equivalence

In Goerss-Jardine's book "Simplicial Homotopy Theory" they prove in Lemma 4.1 of chapter II that for a simplicial model category $\mathcal{C}$ the statement $$\text{ If }f:X\to Y \text{ in } ...
Fabio Neugebauer's user avatar
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Weak equivalence of filtered Colimit

Given a model category $C$, I have two functors $F,G:\mathbb{N}\rightarrow C$, where see $\mathbb{N}$ as sequence category. Question: Given a natural transformation $J:F\rightarrow G$ and suppose the $...
Mukilraj K's user avatar
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Bar construction for cocartesian monoidal structure is calculated by pushout

$\DeclareMathOperator\colim{colim}$ This is a statement in Lurie's Higher Algebra 5.2.2.4. Proposition 3.2.4.7 in HA said that the monoidal structure on $\text{CAlg}(\mathcal{C})$ is cocartesian. I ...
Xiong Jiangnan's user avatar
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1 answer
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Equivalence between the category $\text{Ho}(\mathcal{M})$ and $\text{Ho}(\mathcal{M}_{\mathcal{F}})$

Let $(\mathcal{M}, \mathcal{W}, \mathcal{C},\mathcal{F})$ be a model category and $\text{Ho}(\mathcal{M})$ its homotopy category. Now we consider $ \mathcal{M}_{\mathcal{F}}$ the full subcategory of ...
bml64's user avatar
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Different definitions of sequential homotopy colimits

There are two notions of homotopy limits, one for triangulated categories and one for model categories and I wonder whether these two coincide. More concretely, let $\mathcal{T}$ be a triangulated ...
Alexey Do's user avatar
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Impact of homotopical algebra on homological agebra

I was recently reading through Dwyer and Spalinski's notes introducing model categories. In it is shown a different way to describe $Ext(A,B)$ in homotopical terms (see proposition 7.3, I am not ...
DevVorb's user avatar
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An example of non-small object in Top, illustrated in Hovey's modelcategory textbook 49p

The below is the description in Hovey's textbook at 49p that the Sierpinski space is not small in the category Top. "The Sierpinski space, consisting of two points where exactly one of them is ...
Learning Student's user avatar
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1 answer
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Small object argument for arbirary sets of morphisms

I am working through the expository paper by Dwyer and Spalinski on model categories (https://ncatlab.org/nlab/files/DwyerSpalinski_HomotopyTheories.pdf) and trying to prove the small object argument (...
YordanToshev's user avatar
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Is exact cylinder a cylinder object

This is a question about Cisinski's model structure ex nihilo, chapter 2.4 in his book Higher categories and homotopical algebra. It defines what is an exact cylinder $I$ in the presheaf category. ...
fyx1123581347's user avatar
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A weak homotopy category for CW complexes

I am working on Weibel's K-book and have, while meditating phantom maps and weak homotopies, asked myself if in the same way as their is a category $Ho(Top)$ identifying homotopic maps, is their a ...
DevVorb's user avatar
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Path space and cofibration in topological spaces

This question came to me while reading "Model Categories and their Localization", specifically the definition of path object in a general model category. Given a topological space $X$, we ...
Alessandro's user avatar
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Diagrams in a model category

Let $\mathcal{M}$ be a model category with class of weak equivalences $W$ and $\mathcal{J}$ a small category. The category of diagrams $\operatorname{Fun}(\mathcal{J}, \mathcal{M})$ inherits a class ...
Brendan Murphy's user avatar
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Singular Chain complex of Smash Product

\begin{align} \Delta(X\wedge Y,pt)=\quad? \end{align} Question: Is there a nice way (similar to the Eilenberg-Zilber map) to compute the singular chain complex of a smash product of pointed spaces up ...
Nico's user avatar
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Functorial Homotopy category

The context is in model category, we define the homotopy category of a model category $\operatorname{Ho}\mathcal{C}$ by a functor $$\begin{align} \pi: \mathcal{C} &\to \operatorname{Ho} \mathcal{C}...
Donky Dang's user avatar
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1 answer
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Proof criticize: $Sing(X)$ is fibrant

I am trying to prove the question in the title where definitions are given below. Definition (fibration). A map $f:X\to Y$ in the category of simplicial sets is a fibration if it satisfies right ...
Mizutsuki's user avatar
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Classifying cofibrations in Mor(M) where M is a model category

If $M$ is a model category the category of morphisms in $M$ has a model category structure where the fibrations and the weak equivalences are the levelwise fibrations and levelwise weak equivalences. ...
frogorian-chant's user avatar
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1 answer
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When are Endpoint Maps for Path Space Objects Fibrations?

TLDR: When an object is fibrant in a model category, the endpoint maps for a very good path space object are fibrations. When is the converse true? That is, when can we conclude that an object is ...
Joe's user avatar
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5 votes
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What does it mean for a model category to present a higher category

A model category serves as an abstract/ axiomatic framework for homotopy theory. A higher category, in particular an $(\infty,1)$ category, I'm using the model of quasicategories, is a category with ...
Secher Nbiw's user avatar
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1 answer
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Composition of Paths in a Path Space Object

TLDR: Is there a way to talk about composition of paths in an arbitrary model category? Intuition: In topological spaces, we can compose paths. That is, if $\gamma_1$ is a path from $x$ to $y$, and $\...
Joe's user avatar
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Example of a factorisation of functors $F = HK$ for which the Kan extension of $F$ along $K$ is not $H$.

I was reading Emily Riehl's book: Categorical Homotopy theory, and I encountered exercise 1.1.3: Exercise 1.1.3: Construct a toy example to illustrate that if $F$ factors through $K$ along some ...
julio_es_sui_glace's user avatar
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Are chain complexes with degree-wise infinite direct product still cofibrant?

A theorem I have seen a number of times is that chain complexes over a field decompose into a spheres and disk decomposition (ref here). That is, a chain complex $C$ is a direct sum of the chain ...
richokicked800goals's user avatar
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Model Structure on Constant-free Symmetric Operads

I am currently trying to find a reference for the assertion that the category of positive / constant-free (meaning $\cal{O}(0)=\emptyset$ is the initial object) symmetric operads $\operatorname{Opd}_\...
Jonas Linssen's user avatar
3 votes
1 answer
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Is this map a cofibration in a model category?

Consider the map $$i\colon A\sqcup A\to A$$ where $i=(Id_{A},Id_{A})$. Is this map a cofibration? Actually, I know that isomorphisms are cofibrations and pushouts of cofibrations are cofibrations. But ...
T. Wildwolf's user avatar
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principal bundles references

I am studying the second chapter of this paper and even though I have a small experience with principal $G$-bundles I am struggling with some theorems that I have not seen so far. Is there a modern ...
T. Wildwolf's user avatar
2 votes
0 answers
92 views

Category objects in $\infty$-groupoids vs Complete Segal Spaces

From my understanding, one way to motivate Complete Segal Spaces is to see them as $(\infty,1)$-category objects inside the $(\infty,1)$-category of spaces, or rather $\infty$-groupoids, represented ...
Elies's user avatar
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Homotopy theory of subcatogory of cofibrant objects?

Let $(C,\mathcal{W},\mathcal{C},\mathcal{F})$ be a model category. The homotopy category is defined as the localization of category $C$ with respect to $\mathcal{W}$, denoted by $C[\mathcal{W}^{-1}]$ ...
questionmark's user avatar
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2 answers
137 views

Understanding a proof of Ken Brown's lemma

I'm reading a book on homotopical algebra, and trying my best to understand the various proofs etc but I'm new to model categories and weak factorisation systems, so there are a lot of details I'm ...
t_kln's user avatar
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Meaning of "functorial" in Proposition 1.3.5 of Hovey's Model Categories

For any category $C$, with a terminal object $*$, denote by $C_*$ the coslice category $*/C$. There is an adjoint pair, denoted by $V_C\dashv U_C:C_*\to C$, where $U_C$ is the forgetful functor. Let $...
Jerry Scott's user avatar
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$\pi_*:Ho(Top_*)\to Set^{\mathbb N}$ preserves limits

Consider $\pi_*:Ho(Top_*)\to Set^{\mathbb N}$ defined by $[X,x]\mapsto \{\pi_n(X,x)\}_n$ from the homotopy category of the pointed topological spaces. I showed that this is a conservative functor and ...
raisinsec's user avatar
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2 answers
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Clarify on Lemma 1.2.2, Hovey's Model Categories

Let $\mathscr C$ be a model category, and denote as usual by $\operatorname{Ho}\mathscr C$ the localization of $\scr C$ with respect to the weak equivalences. For any category $\scr D$, let $\mathrm{...
Jerry Scott's user avatar
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Special short exact sequences of chain complexes

For simplicity let's say I am working in the category of chain complexes of $R$ modules. I have a chain map $i:C\rightarrow D$, which I know to be a weak equivalence (i.e. $i_*$ is an isomorphism of ...
Chris's user avatar
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1 answer
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Slice model category and initial/terminal objects

Consider $(M,Fib,Cof,WE)$ a model category, that is a complete/cocomplete category with 3 wide subcategories satisfying certain properties. Now consider $X\in\operatorname{Ob}M$ and the slice category ...
raisinsec's user avatar
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3 votes
1 answer
64 views

Embedding of model categories

Let $\sf C$ be a category with all small limits and colimits. Let $*$ be the terminal object, and denote by $\sf C_*$ the under category $*/\sf C$. Define the functor $I:\sf C\to C_*$ by setting $I(c)$...
Jerry Scott's user avatar
2 votes
1 answer
39 views

Reedy fibrant replacement for Segal categories

Suppose $C$ is a Segal category, that is a functor $X: \Delta^{op} \rightarrow Spaces$ so that $X_0$ is discrete and satisfies the Segal condition $X_k \cong X_1 \times_{X_0} \cdots \times_{X_0} X_1$. ...
user39598's user avatar
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3 votes
0 answers
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Reference request: Tensoring with Barratt-Eccles is $\Sigma_* $-cofibrant replacement

In some different places (e.g. in Berger & Fresse, Combinatorial operad actions on cochains), it is stated as a classical result that the Barratt-Eccles operad $\mathcal{E}$ (in chain complexes) ...
user1029251's user avatar
3 votes
1 answer
46 views

Confusion regarding $Der_R(X,M)$ and $A\ltimes M$

I am trying to prove the following claim from the paper "Model Categories and Simplicial Methods" by Goerss and Schemmerhorn. Given a commutative $R$-algebra's $A, X$ (not necessarily unital)...
joe's user avatar
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3 votes
2 answers
163 views

Relative cell complexes in the undercategory $B/\mathscr{M}$ are relative cell complexes in $\mathscr{M}$ - why must we also assume $B$ is a complex?

$\newcommand{\M}{\mathscr{M}}\newcommand{\I}{\mathcal{I}}\newcommand{\J}{\mathcal{J}}$The context: the following is claimed in J. May and Ponto's "more concise algebraic topology". We have ...
FShrike's user avatar
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3 votes
1 answer
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Existence of abelianization functor

When does the abelianzation functor exists? I am reading Quillen's paper "On the (co-)homology of commutative rings" and in it he states that when $\mathscr{C}$ is an algebraic category (...
joe's user avatar
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1 answer
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Show naturality of $\infty$-natural transformation

Working with the model category of complete Segal spaces $\text{CSS}$, which has as its underlying category the category of simplicial presheaves on $\Delta$, one has a suitable internal hom in $\text{...
h3fr43nd's user avatar
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3 votes
0 answers
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Kahler differentials give a left Quillen functor

Is there a reference for the fact that the functor of Kahler differentials is a left Quillen functor on the category of CDGA (over a field of characteristic $0$)? Remark: I mean the model structure on ...
Grisha Taroyan's user avatar
2 votes
1 answer
55 views

Are the two definitions of local objects in the context of left Bousfield localization equivalent?

Given a simplicial model category $\mathcal{M}$ and a set $S$ of morphisms in it, we can define the concept of $S$-local objects. In nLab page (below Definition 3.2), a fibrant object $X\in \mathrm{Ob}...
Yining Chen's user avatar
2 votes
1 answer
81 views

Homotopy colimit formula deduced from Coend Quillen bifunctor

Reading the article on Quillen bifunctors on the Nlab (https://ncatlab.org/nlab/show/Quillen+bifunctor) I stumbled upon the following claim: Let $\mathcal{C}$ be a combinatorial simplicial model ...
h3fr43nd's user avatar
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1 answer
59 views

Definition of homotopy equivalence in an arbitrary model category

I'm reading notes on Whitehead's Theorem, and I'm slightly confused on the definition of homotopy equivalence. Let $f: X\to Y$ be a morphism in a model category $(C, Cof, Fib, W)$. We say that $f$ is ...
FazeZizek's user avatar
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4 votes
1 answer
132 views

Cech model structure and the homotopy descent condition

Let $\text{Cart}$ be the category of cartesian spaces which has as its objects the collection of sets $U$ for which there exists $n \in \mathbb{N}$ so that $U \subset \mathbb{R}^n$ and $U$ is ...
h3fr43nd's user avatar
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0 answers
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"left homotopic" and "left homotopy" as in Quillen's book Homotopical algebra

I am reading Quillen's book Homotopical algebra. This is mainly about the terminology. Fix a model category $\mathcal{C}$. In definition $3$ (page $1.4$) Quillen defines what does it mean to say that ...
Praphulla Koushik's user avatar
1 vote
0 answers
111 views

What is the takeaway of cofibrant generation?

I have recently begun reading about model categories. In particular, I have been using Balchin's A Handbook of Model Categories as a reference, and the following quote has been quite perplexing. A ...
JJ Hoo's user avatar
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2 votes
1 answer
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Is there a proof that uses (co)ends solely to establish the derived adjoint correspondence of e.g. deformable functors?

In Riehl's book "Categorical homotopy theory" (the pdf may be downloaded on https://emilyriehl.github.io/books/) Exercise 2.2.15 on page 21 is given as follows: Suppose $F \dashv G$ is an ...
h3fr43nd's user avatar
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1 vote
1 answer
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Cofibrant approximation of maps. [Hirschhorn 8.1.23]

In [Hirschhorn 8.1.23] (page 142) the author claims the following: Let $g: X \to Y$ be a any map in a general model category $\mathcal{M}$ (with functorial factorization), then there exists cofibrant ...
lola's user avatar
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1 answer
77 views

Pullback of a section of a trivial fibration along a fibration

Assume we have a category of fibrant objects and a pullback square in it, where both vertical maps are fibrations and the lower horizontal map is a weak equivalence, call it $i$. Assume further that $...
Roland's user avatar
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