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 model category, a homotopy factorization system leverages the ambient weak factorization systems to give a presentation of an orthogonal factorization system in the underlying $(\infty,1)$-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.
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).
See cylinder factorisation system.
Introductory texts:
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)
On the relationship between cones and factorisation systems:
Rudolf-E. Hoffmann, Factorization of cones, Mathematische Nachrichten 87 1 (1979) 221-238. [doi:10.1002/mana.19790870120]
Rudolf-E. Hoffmann, Factorization of cones II, with applications to weak Hausdorff spaces, in: Categorical Aspects of Topology and Analysis: Proceedings of an International Conference Held at Carleton University, Ottawa, August 11–15, 1981, Springer (2006) [doi:10.1007/BFb0092878]
A list of elementary examples of factorization systems (associated with the notions of: compact, discrete, connected, and totally disconnected spaces, dense image, induced topology, and separation axioms; finite groups being nilpotent, solvable, torsion-free, p-groups, and prime-to-p groups; injective and projective modules; injective, surjective, and split homomorphisms)
Last revised on November 16, 2023 at 12:52:53. See the history of this page for a list of all contributions to it.