Holmstrom Model category

nLab Quillen equivalence

Toen Essen talk: Any model category is naturally enriched over the homotopy cat of simplicial sets. This exposition is by the way a very concise intro to some key concepts in model cats, including localization and infinity-cat thinking, and homotopy limits.

http://ncatlab.org/nlab/show/global+model+structure+on+functors

http://mathoverflow.net/questions/78400/do-we-still-need-model-categories

References: Dwyer and Spalinski, Hovey’s book, Goerss and Schemmerhorn.

Hirschhorn: Model cats and their localizations

Hirschhorn et al: Homotopy Limit Functors on Model Categories and Homotopical Categories (AMS)

Dwyer-Spalinski in the homotopy theory folder: Model categories, Homotopy limits brief intro, localization wrt a homology theory: very brief intro on p. 54.

Definition: A model category is a category $C$ with all small limits and colimits, together with a model structure on $C$ . A model structure on a category consists of three subcategories, called cofibrations (cofibs), fibrations (fibs), and weak equivalences (WEs), and two functorial factorizations $(\alpha, \beta)$ and $(\gamma, \delta)$ , satisfying:

(2-out-of-3) If two of $f$ , $g$ , $g \circ f$ are WEs, then so is the third.
(Retracts) The three classes of morphisms are closed under retracts.
(Lifting) Trivial cofibs have the LLP wrt fibs, and cofibs have the LLP wrt trivial fibs.
(Factorization) For any morphism $f$ , $\alpha(f)$ is a cofib, $\beta(f)$ is a trivial fib, $\gamma(f)$ is a trivial cofib, and $\delta(f)$ is a fib.

This is the definition given in Hovey, which differs slightly from earlier definitions, for example Quillen’s original definition. Some people (ref?) have suggested that the first axiom should be replaced by a (4-out-of-6) axiom, to obtain a more general setting for homotopy theory in some cases.

Hovey suggests (p. 21) that the 2-category of model categories might behave like a model category.

Examples of model cats:

Sset: Simplicial sets
Top: Topological spaces
Chain complexes of R-modules
Modules over a Frobenius ring
Cochain complexes of comodules over a Hopf algebra