# Tagged Questions

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### 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 ...
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This question is about the definition of the duality 2-functor in Hovey's book on Model categories, Section 1.4. There he defines the 2-category of categories with adjunctions as follows: objects ...
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### Inequivalent Model Categories

A Model Category is, informally, a category where a "reasonable" notion of homotopy can be developed. I'm curious to know when two model categories are considered equivalent to each other. Thanks for ...
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### 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 ...
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### 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 ...
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### Generalized Reedy model structure

If $\mathcal{R}$ is a generalized Reedy category in the sense of Berger-Moerdijk, one can endow the fonctor category $[\mathcal{R},\mathcal{C}]$ with the generalized Reedy model structure. When is it ...
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Let me first give an abstract definition of the homotopy colimit. Let $C$ be a cofibrantly generated model category and let $D$ be a small category. There is an adjunction $$... 0answers 38 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 ... 1answer 28 views ### The definition of a (pseudo-)Dwyer map I've looked up the definition in Raptis Homotopy Theory of Posets, on the nlab and even Cinskies a classe des morphismes de Dwyer n'est pas stable par rétractes, but none of these make sense to me. ... 0answers 21 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 0answers 61 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 ... 1answer 84 views ### Model structure on sSet Which is the model structure on  \text{sSet}  (category of simplicial sets) that makes \text{sSet} Quillen equivalent to the category  \text{Cat}  (of small categories) by the adjunction ... 1answer 29 views ### Does There Exist an Induced Model Strucutre via Ordinary Equivalence? If F:\mathcal{M} \to \mathcal{C} is an equivalence of categories, and \mathcal{M} is a model category, does \mathcal{C} inherit a model structure from \mathcal{M} via F? If not, is there a ... 1answer 139 views ### The projective model structure on chain complexes Let \mathcal{A} be an abelian category with enough projective objects and let \mathcal{M} be the category of chain complexes in \mathcal{A} concentrated in non-negative degrees. Quillen [1967, ... 0answers 75 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 ... 1answer 73 views ### How to define the category of model structures of a category? It is possible to come up with different model structures for a fixed category. Let \mbox{Models}\left(\mathcal{C}\right) be the category of all model structures of \mathcal{C}, which has as ... 1answer 151 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 ... 2answers 104 views ### Problem understanding a proof in Model Categories by Hovey I have serious problems understanding this proof from the book Model Categories, by Mark Hovey: Here's a list of things I don't understand: He's trying to prove the assertion by contradiction, ... 3answers 213 views ### Cylinder object in the model category of chain complexes Let \text{Ch}⁺(R) be the category of non-negative chain complexes of R-modules where R is a commutative ring. What is a cylinder object, in the sense of model categories, for a given complex ... 0answers 214 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 ... 2answers 100 views ### Functor between categories with weak equivalance. A homotopical category is category with a distinguished class of morphism called weak equivalence. A class W of morphisms in \mathcal{C} is a weak equivalence if: All identities are ... 0answers 124 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 & ... 1answer 219 views ### Why does the definition of homotopy cartesian involve factorisations Setup: A diagram$$\require{AMScd} \begin{CD} X @>>> Y\\ @VVV @VV{f}V\\ U @>>> V \end{CD} in a (proper) model category is called homotopy cartesian if there exists a ...
I just came across a strange property of morphisms that are preserved under pullbacks, and it made me wonder. Consider a model category $\mathcal{M}$. Because the fibrations are exactly the maps that ...