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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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) ...
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Projective model structure for a directed diagram
In short :
If $M$ is a model category and $D$ is a small directed category, a theorem states that the projective model structure on $M^D$ exists. However, I am stuck on some difficulty at the end of ...
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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)...
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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's "more concise algebraic topology".
We have some model ...
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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 (...
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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{...
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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 ...
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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}...
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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 ...
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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 ...
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Is it possible to define the Serre mod $\mathscr{C}$ model category structure on $r$-reduced simplicial sets as Cisinski model structure?
I think it is quite straightforward to show that a $\bmod\mathscr{C}$ model structure on $r$-reduced simplicial sets is a Bousfield localization of the transferred Kan-Quillen model structure.
However,...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 $...
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Proving some properties of the localization functor in the stable homotopy category.
I am trying to understand the paper named " Localization with respect to Certain Periodic Homology Theories"
Here is the part of it I am trying to understand the proof of proposition 1.5 in ...
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Cofibration and homotopy equivalence imply deformation retract?
Theorem 8 of Strom's Notes on Cofibrations II we have these conditions on a map for a lift to exist.
He references his Notes on Cofibrations Theorem 3 for the proof however the assumptions have ...
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Counterexamples in model categories
Can it happen that a model category has factorisations which can not be made functorial, but still it admits a (co)fibrant replacement functor?
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Existence of certain cofibrant replacement functors
Suppose we have a model category whose factorisations are not required to be functorial (as defined by Dwyer and Spalinski, for instance). In this case, when we choose (still non-functorial) cofibrant ...
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Filling an outer horn in a quasicategory when the first edge or the last edge is an equivalence
I convinced myself some time ago that the following holds: given an outer horn in a quasicategory, if the first or last edge (the map $0\to 1$ or $n-1\to n$) is an equivalence then the horn has a ...
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Why use $\infty$-categories over model categories?
Let's say that I know (roughly) how derived categories help us solve problems. After all,
we want to consider chain complexes up to homotopy equivalence, and the derived category literally lets us do ...
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Is quotient of fibration a fibration
Assume $f:X\to Y$ is a fibration, and given quotient maps $X\to X/R_X,Y\to Y/R_Y$ is compatible with $f$, is there some condition to ensure that $\bar{f}:X/R_X\to Y/R_Y$ being a fibration?
My aim is ...
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Example of a non-closed model category in the sense of Quillen
When Quillen introduced model categories in [1], he differentiated between model categories and closed model categories, where the latter has stronger assumptions.
Today, the term 'model category' ...
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homotopies between cofibrant resolutions
Let $X$ be a finite-dimensional noetherian separated scheme, let $\mathcal{U}=\{U_i\}$ be an affine cover and $\mathcal{N}$ the nerve of this cover. For any coherent sheaf $\mathcal{F}$ on $X$, we ...
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Proof of Proposition A.2.6.15 in Higher Topos Theory
I am reading Lurie's Higher Topos Theory and I need some help to understand a part of the proof of Proposition A.2.6.15 (A.2.6.13 in the published version).
In the proposition, we are working with a ...
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Iterated homotopy pullbacks
I would like a reference for the following fact, which I believe to be true.
Consider the simplicial model category of Kan complexes with the Quillen model structure, and suppose given a commutative ...
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comparing homotopy colimits of "equivalent" diagrams
Motivation
Let us work in a fixed model category $\mathcal{C}$. I'm interested in homotopy colimits with indexing categories $\mathcal{I}=\bullet \leftarrow\bullet\to \bullet $ and $\mathcal{I}=\...
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Why does tensor product satisfy pushout-product axiom?
Example 11.4 in this paper claims that the tensor product of chain complexes of bimodules (over not-necessarily-commutative rings) satisfies the pushout-product axiom (the first condition of a Quillen ...
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Reference for loop space objects
Okay so, hi!
I would need to have some discussion related to loop space objects in a project of mine, but the only reference I can find for loop space objects is https://ncatlab.org/nlab/show/loop+...
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When does a left Quillen functor preserve weak equivalences?
I am interested in an answer to the following question: suppose we have a left Quillen functor $L: \mathcal{C} \rightarrow \mathcal{D}$ between symmetric monoidal model categories $\mathcal{C}$ and $\...
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CW complexes with cofibration morphisms
Let $CW$ be the category of CW complexes and $CW_{cof}$ be the wide subcategory whose objects are CW complexes and morphisms given by inclusions of subcomplexes (i.e. cofibrations in the model ...
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Dold-Kan correspondence, Model structures and Homology
I am fairly new to the concept of model categories, simplicial sets, etc. And so there is some questions, which may be obivuous, that I need to clarify.
Consider the cateogry of simplicial abelian ...
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The class $l(F)$ of morphisms which have the left lifting property with respect to $F$ is stable under transfinite compositions.
I am reading Cisinski's Higher Categories and Homotopical Algebra and I am having trouble trying to verify some claims there. My background in category theory is not very solid. I would like some help ...
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Rotman's Algebraic Topology Lemma 9.11
This is the Lemma 9.11 of Rotman's "An Introduction to Algebraic Topology".
The topic where I found this simple lemma of homological algebra is the Theorem of Acyclic Models. So we are ...
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When does the left adjoint of the base change functor between categories of algebras over operads preserve quasi-isomorphisms?
I have been thinking about the following question: given a morphism of coloured dg-operads $$\phi: P \longrightarrow Q$$
we derive a lax morphism between their respective monads $T= T_P \...
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What is meant by the phrase "inclusion of categories induces an equivalence of categories"?
I am looking through Hovey's "Model categories" and the above mentioned phrase comes up several times. Here is a couple of examples:
Does this phrase have a strict meaning? Is there a ...
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$f$ - isomorphism in a model category $\implies$ $f$ is fibration, cofibration and weak equivalence?
Why must an isomorphism in a model category be fibration, cofibration and weak equivalence simultaneously?
Thank you.
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Is it customary to call a zig-zag of quasi-isomorphisms a weak homotopy equivalence? [closed]
Is it customary to call a zig-zag of quasi-isomorphisms a weak homotopy equivalence?
$$ A_0 \leftarrow A_1 \rightarrow \dotsb \leftarrow A_{k-1}\rightarrow A_k $$
Could someone, please, elaborate on ...
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Natural transformation between sheaves in homotopy theory
Firstly a small disclaimer. I am not an expert in the theory of higher sheaves and their presentation in the model categories, so please feel free to correct all inaccuracies in the question itself!
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Homotopy Limit is the Limit in the Homotopy Category
I am trying to understand the homotopy limit. This question naturally appears to my mind. Let $I$ be a small category and $\mathcal{X}$ is an $I$-diagram of simplicial sets. There is a functor from ...
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Nisnevich local model structure is a Bousfield localisation of global projective model structure
Suppose $\Delta^{op}Psh(Sm/k)$ is the category of simplicial presheaves on the category $Sm/k$ of finite type smooth schemes over $k$, endowed with Nisnevich topology. $\Delta^{op}Psh(Sm/k)$ has two ...
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Simpliciality of projective model structure on simplicial cofibrantly generated model categories
In Model Categories and Their Localizations, Definition 11.7.2, for $M$ a simplicial cofibrantly generated model category, $C$ a small category, Hirschhorn gives the following simplicial structure on $...
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Is the composition of hom-spaces entirely determined by the tensor in a simplicial model category?
Let $\mathcal{C}$ be a simplicial model category, as defined in https://ncatlab.org/nlab/show/simplicial+model+category. We can prove that the $\mathrm{Hom}$ functor is determined by the tensor, ...
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To check factorization in axiom in cofibration category, it suffices to check factorization of codiagonal map.
Say we have category $C$ with class of morphisms weak equivalence and cofibration. And say we have that (Acylic) cofibrations are closed under pushouts, composition. And that initial object is in $C$ ...
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Global Fibrations are Local Fibration in Local Injecive model Structure
Suppose $T$ is a small site with enough points and $\Delta^{op}Shv(T)$ is the category of simplicial sheaves on $T$ endowed with local injective model structure. Here a map $f: \mathcal{X} \to \...
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Splitting of square diagram in chain complexes
Let us consider a square in chain complexes over a field $k$ \begin{array}{ccc}A & \xrightarrow{f} & B \\ \downarrow{g} & & \downarrow{h} \\ C & \xrightarrow{k} & D\end{array}
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Are weak equivalences characterized by the homotopy fibers in an arbitrary model category?
Given a model category $\mathcal{C}$ we can define homotopy pullbacks like the loop space
$$\begin{array}{ccc}
\Omega_x(X) & \rightarrow & *\\
\downarrow & & \;\;\downarrow_x\\
* & ...
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Long exact sequence of homotopy groups groups in simplicial sets - reference request
I believe it is well-known that for a based map $f:X\to Y$ of simplicial sets (possibly with some extra hypotheses on $X$ and $Y$), there is a long exact sequence
$$
\ldots \to \pi_n(F)\to \pi_n(X) \...
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