Articles of model categories

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, Ch. II, §4] asserts that the following data define a model structure on $\mathcal{M}$: The weak equivalences are the quasi-isomorphisms (= homology isomorphisms). The cofibrations are the monomorphisms […]

The empty set in homotopy theoretic terms (as a simplicial set/top. space)

I am currently confused about the empty set in terms of its path components and how this fits into the Quillen adjunction between topological spaces and simplicial sets. Probably, one of my definitions are not precisely correct: The category of simplicial sets is a cofibrantly generated model category with generating acyclic cofibrations the inclusions of […]

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 map $f:X\to Y$ factors as the composite of an acyclic cofibration and a fibration The authors proceed as follows. Let $Z_0=X$ […]

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 $M^\bullet$? Thinking about algebraic topology (a circle and an annulus) I was thinking of tensoring $M^\bullet$ by $R$ (over $R$!) but this is […]

Closed model categories in the sense of Quillen 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 finite limits and finite colimits. The two factorisation systems are now required to be functorial. These changes do make the two definitions genuinely different, since […]