Stratified model categories
The fourth axiom of a model category states that given a commutative square of maps, say $i:A\to B,\ g:B\to Y,\ f:A\to X$, and $p:X\to Y$ 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 $h:B\to X$ 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.