factorization system over a subcategory
k-ary factorization system, ternary factorization system
factorization system in a 2-category
factorization system in an (∞,1)-category
Roughly speaking, a factorization system on a category consists of two classes of maps, $L$ and $R$, such that every map factors into an $L$-map followed by an $R$-map, and the $L$-maps and $R$-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.
algebraic weak factorization systems (AWFS) 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 (i.e. a possibly non-strict 2-category) one wants instead the notion of factorization system in a 2-category, which is like a Cat-enriched OFS “up to isomorphism.” The situation in an n-category is analogous; see for instance orthogonal factorization system in an (∞,1)-category and orthogonal factorization system in a derivator.
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.
A strict factorization system is one in which the factorizations are specified uniquely on the nose, rather than merely up to isomorphism.
A factorization system over a subcategory is a common generalization of orthogonal and strict factorization systems, which requires uniqueness of factorizations only up to specified zigzags.
Particular examples of factorization systems of various sorts can be found on the individual pages referred to above.
The above notion of “binary” factorization system can be generalized to factor a morphism into more than two factors.
The orthogonal 3-ary version is a ternary factorization system.
This has a generalization to a k-ary factorization system.
The corresponding 3-ary version for weak factorization systems is closely related to the notion of model category (one of the main applications of weak factorization systems).
The factorization systems were probably first introduced in
S. MacLane, Duality for groups, Bull. Amer. Math. Soc. 56, (1950). 485–516, MR0049192, doi
J. R. Isbell, Some remarks concerning categories and subspaces, Canad. J. Math. 9 (1957), 563–577; MR0094405
Ross Street, Notes on factorization systems, (pdf)