Roughly speaking, a factorization system on a category consists of two classes of maps, and , such that every map factors into an -map followed by an -map, and the -maps and -maps satisfy some lifting or diagonal fill-in property. The various ways of filling in the details give rise to many kinds of factorization systems:
In most of the literature, “factorization system” unqualified refers specifically to orthogonal factorization systems (OFS).
weak factorization systems (WFS) are “more general” than orthogonal ones, in the sense that every OFS is also a WFS. But since the most important examples of WFS (those that occur in model categories are not OFS, intuitively they are more or less independent concepts.
natural weak factorization systems (NWFS) are a strengthened “algebraic” version of WFS in which the factorizations are functorial and the two classes of maps are algebraic.
In an enriched category it is natural to consider enriched (orthogonal) factorization systems. The enriched version of WFS falls under enriched model categories.
In a bicategory one wants instead “bicategorial factorization systems,” which are like Cat-enriched OFS “up to isomorphism,” and analogously in an n-category.
In a strict 2-category there is also the notion of an enhanced (orthogonal) factorization system?, of which the main example is (bo,ff) in Cat.
Particular examples of factorization systems of various sorts can be found on the individual pages referred to above.