category theory

# Contents

## Idea

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:

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.

## References

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)

Revised on March 23, 2012 07:15:40 by Urs Schreiber (82.172.178.200)