Model categories are categories with three distinguished classes of morphisms: the weak equivalences, the fibrations and the cofibrations. They provide a natural setting for [tag:homotopy-theory] in an arbitrary category, by mimicking the usual properties of (co)fibrations and weak equivalences in ...

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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 ...
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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, ...
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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 ...
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Not every over-under-category is cocomplete

Something is wrong between me and Hirschhorn: point 3 of this result (in the book Model categories and their localizations): 7.6.4. Homotopy in undercategories and overcategories. Theorem ...
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Doubt about proof of factorization $f=pi$, where $i$ is acyclic cofibration and $p$ is fibration

I try to understand a proof in More Concise Algebraic Topology: Localization, completions and model categories by May & Ponto (pdf). The proof is on page 262, and it is for the statement Any ...
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Kan fibrations and surjectivity

I have a basic question on the usual model structure on simplicial sets. What is the relation between being a Kan (trivial maybe ?) fibration and surjectivity ? Surjectivity here means either ...
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Is the homotopy category of modules over a quasi-Frobenius ring (pre)additive?

Let $R$ be a quasi-Frobenius ring and $Mod_R$ the category of $R$-modules. One can prove that it admits a model structure whose weak-equivalences are stable equivalences, whose cofibrations are monos ...