Stratified model categories

Jan Spaliński

Fundamenta Mathematicae, Tome 178 (2003) no. 3, p. 217.

Résumé

The fourth axiom of a model category states that given a commutative square of maps, say i: A → B, g: B → Y, f: A → X, and p: X → Y such that gi = pf, if i is a cofibration, p a fibration and either i or p is a weak equivalence, then a lifting (i.e. a map h: B → X such that ph = g and hi = f) exists. We show that for many model categories the two conditions that either i or p above is a weak equivalence can be embedded in an infinite number of conditions which imply the existence of a lifting (roughly, the weak equivalence condition can be split between i and p). There is a similar modification of the fifth axiom. We call such model categories "stratified" and show that the simplest model categories have this property. Moreover, under some assumptions a category associated to the category of simplicial sets by a family of adjoint functors has this structure. Postnikov decompositions and n-types exist in any such category.

Détail

Classification : 55R05, 55U35, 55P05, 55P10
Mots-clés : stratified model category, weak equivalence, fibration, cofibration, Postnikov decomposition, -type