# nLab combinatorial weak factorization system

Combinatorial weak factorization systems

# Combinatorial weak factorization systems

## 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.

## References

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.

structuresmall-set-generatedsmall-category-generatedalgebraicized
weak factorization systemcombinatorial wfsaccessible wfsalgebraic wfs
model categorycombinatorial model categoryaccessible model categoryalgebraic model category
construction methodsmall object argumentsame as $\to$algebraic small object argument

Last revised on February 13, 2019 at 02:21:04. See the history of this page for a list of all contributions to it.