Stratified model categories

Jan Spaliński¹

¹ Faculty of Mathematics and Information Science Warsaw University of Technology Pl. Politechniki 1 00-661 Warszawa, Poland

Fundamenta Mathematicae, Tome 178 (2003) no. 3, pp. 217-236.

Résumé

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.

Détail

DOI : 10.4064/fm178-3-3

Mots-clés : fourth axiom model category states given commutative square maps say cofibration fibration either weak equivalence lifting map exists many model categories conditions either above weak equivalence embedded infinite number conditions which imply existence lifting roughly weak equivalence condition split between there similar modification fifth axiom call model categories stratified simplest model categories have property moreover under assumptions category associated category simplicial sets family adjoint functors has structure postnikov decompositions n types exist category