# 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://...
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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 ...
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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. ...
1 vote
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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 ...
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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 ...
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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 ...
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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 ...
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### 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 ...
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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 $C[\mathcal{W}^{-1}]$ ...
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### 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 ...
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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 ...
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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 ...
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### 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 $I(c)$...
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### 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 $X_k \cong X_1 \times_{X_0} \cdots \times_{X_0} X_1$. ...
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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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### 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 and Ponto's "more concise algebraic topology". We have ...
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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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### 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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### 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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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 \$...