---
tag: TA39C6P91W
kind: definition
name: Factorization system
msc: [18A32]
defines: 36J8PH2B0T
uses: [A2DE66937D, AZK4RD8XV0, 01RVCMSN5N, 09NWETVTNY]
sources:
  - url: https://en.wikipedia.org/wiki/Factorization_system
---

## Definition

Let $\mathcal{C}$ be a [[category|A2DE66937D]], and let $E$ and $M$ be classes of morphisms of $\mathcal{C}$.
The pair $(E, M)$ is an *orthogonal factorization system* if these three conditions hold.

1. Both $E$ and $M$ contain all isomorphisms and are closed under composition.
2. Every morphism $f$ of $\mathcal{C}$ factors as $f = m \circ e$ with $e \in E$ and $m \in M$.
3. Every $e \in E$ is orthogonal to every $m \in M$: it has the [[left lifting property|AZK4RD8XV0]] with respect to $m$, with a unique filler.

## Remarks

Conditions 1 to 3 imply that the factorization in clause 2 is unique up to a unique isomorphism.
They also imply that $E$ is exactly the class of morphisms orthogonal to all of $M$, and conversely.
A *weak factorization system* asks instead for factorizations and for liftings without uniqueness.
It also asks that $E$ and $M$ be exactly the classes of maps with the left and the right lifting property against each other.

Example: in the category of sets, let $E$ be the [[surjections|01RVCMSN5N]] and $M$ the [[injections|09NWETVTNY]].
A function $f \colon A \to B$ factors as $A \to f(A) \to B$.
For orthogonality, take a square with $e \colon A \to B$ surjective, $m \colon C \to D$ injective, and $v \circ e = m \circ u$.
Define $d(b) = u(a)$ for any $a$ with $e(a) = b$; this does not depend on $a$, since $m$ is injective and $m(u(a)) = v(b)$.
Then $d$ is the unique filler.

Non-example: in sets, let $E$ and $M$ both be the class of all functions.
Conditions 1 and 2 hold, but condition 3 fails.
For $e \colon \emptyset \to \set{*}$ and $m \colon \set{0, 1} \to \set{*}$, the square with $v = \id$ has two fillers $\set{*} \to \set{0, 1}$.