---
tag: 36J8PH2B0T
kind: concept
name: Factorization system
msc: [18A32]
sources:
  - url: https://en.wikipedia.org/wiki/Factorization_system
same_as:
  wikidata: Q5428747
  wikipedia: Factorization system
---

## Summary

A factorization system on a category is a pair $(E, M)$ of classes of morphisms.
Every morphism factors as a member of $E$ followed by a member of $M$, in an essentially unique way.
The two classes determine each other through a unique [[lifting property|AZK4RD8XV0]].
The standard example in sets is (surjections, injections), which factors a function through its image.
A weaker variant, the weak factorization system, drops uniqueness and is a basic ingredient of model categories.

## Explanations

### L1 human

Every function factors as a surjection onto its image followed by the inclusion of the image.
A factorization system abstracts this: two kinds of maps such that every map is first one kind, then the other.
The factorization is unique up to a unique isomorphism of the middle object.
Groups have the same system with surjective and injective homomorphisms, and the middle object is the image subgroup.