nLab combinatorial weak factorization system

Combinatorial weak factorization systems

Combinatorial weak factorization systems


A weak factorization system (L,R)(L,R) on a locally presentable category is combinatorial if it is cofibrantly generated by a set of morphisms. That is, there is a set II of morphisms such that for any morphism ff, we have fRf\in R if and only if ff has the right lifting property with respect to all iIi\in I.


By the small object argument, any set II of morphisms in a locally presentable category generates a combinatorial weak factorization system.



Algebraic model structures: Quillen model structures, mainly on locally presentable categories, and their constituent categories with weak equivalences and weak factorization systems, that can be equipped with further algebraic structure and “freely generated” by small data.

weak factorization systemcombinatorial wfsaccessible wfsalgebraic wfs
model categorycombinatorial model categoryaccessible model categoryalgebraic model category
construction methodsmall object argumentsame as \toalgebraic small object argument

Last revised on February 13, 2019 at 07:21:04. See the history of this page for a list of all contributions to it.