nLab combinatorial weak factorization system

Definition

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

Construction

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

Properties

• Combinatorial wfs can be both right- and left-lifted along adjunctions; see transferred model structure.

Related concepts

• A combinatorial model category is a model category whose two constituent weak factorization systems are combinatorial.

• A more general notion is an accessible weak factorization system.

• Any combinatorial weak factorization system can be equipped with the structure of an algebraic weak factorization system (generally in more than one way).

References

• M. Makkai, J. Rosický, Cellular categories, 2013, arXiv:1304.7572

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.

structure	small-set-generated	small-category-generated	algebraicized
weak factorization system	combinatorial wfs	accessible wfs	algebraic wfs
model category	combinatorial model category	accessible model category	algebraic model category
construction method	small object argument	same as $\to$	algebraic small object argument