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 $i\in I$.
By the small object argument, any set $I$ of morphisms in a locally presentable category generates a combinatorial weak factorization system.
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).
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 |
Last revised on February 13, 2019 at 02:21:04. See the history of this page for a list of all contributions to it.