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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm178-3-3,
author = {Jan Spali\'nski},
title = {Stratified model categories},
journal = {Fundamenta Mathematicae},
volume = {177},
year = {2003},
pages = {217-236},
zbl = {1055.55018},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm178-3-3}
}
Jan Spaliński. Stratified model categories. Fundamenta Mathematicae, Tome 177 (2003) pp. 217-236. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm178-3-3/