Factorization system

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 and $M$ the injections. 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 $\set{*} \to \set{0, 1}$.