nLab factorization system




Roughly speaking, a factorization system on a category consists of two classes of maps, LL and RR, such that every map factors into an LL-map followed by an RR-map, and the LL-maps and RR-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:

Particular examples of factorization systems of various sorts can be found on the individual pages referred to above.

Higher-ary factorization systems

The above notion of “binary” factorization system can be generalized to factor a morphism into more than two factors.

Cylinder 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:

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.