Factorization systems

Ivan Kobe

0.1 Definitions and basic properties

Definition 1

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A factorization system in a category $\mathcal{C}$ consists of two classes of morphisms $(L,R)$, such that both $L$ and $R$ contain isomorphisms and are closed under composition, and every morphism $f: C \to D$ in $\mathcal{C}$ admits a factorization into a morphism $l\in L$ followed by a morphism $r\in R$, which is unique up to unique isomorphism among such factorizations.

$\begin{tikzcd} && E \\ C &&&& D \\ && {E'} \arrow["r", from=1-3, to=2-5] \arrow["i"', from=1-3, to=3-3] \arrow["\cong", from=1-3, to=3-3] \arrow["l", from=2-1, to=1-3] \arrow["{l'}"', from=2-1, to=3-3] \arrow["{r'}"', from=3-3, to=2-5] \end{tikzcd}$

Definition 2

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If $W$ is a class of morphisms in a category $\mathcal{C}$ and $X$ is an object in $\mathcal{C}$, we define a class of morphisms $W/X$ in $\mathcal{C}/X$, given by $f \in W/X$ iff $U f \in W$, where $U : \mathcal{C}/X \to \mathcal{C}$ is the forgetful functor.

Lemma 3

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If $(L,R)$ is a factorization system in a category $\mathcal{C}$ and $X$ is an object in $\mathcal{C}$, then $(L/X,R/X)$ is a factorization system in $\mathcal{C}/X$.

Lemma 4

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If $(L,R)$ is a factorization system in a category $\mathcal{C}$, then the intersection of $L$ and $R$ is precisely the class of isomorpihsms in $\mathcal{C}$.

Lemma 5

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If $(L,R)$ is a factorization system in a category $\mathcal{C}$, then $R$ has the left cancellation property and $L$ has the right cancellation property.

Lemma 6

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$(\text{Epi},\text{Mono})$ is a factorization system in $\text{Set}$.

0.2 Orthogonality

Definition 7

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Given two morphisms $l : A \to B$ and $r : X \to Y$ in $\mathcal{C}$, we say that $l$ is left-orthogonal to $g$, or that $g$ is right-orthogonal to $l$, if for every commutative square

$\begin{tikzcd} A & X \\ B & Y, \arrow["u", from=1-1, to=1-2] \arrow["l"', from=1-1, to=2-1] \arrow["r", from=1-2, to=2-2] \arrow["d", dashed, from=2-1, to=1-2] \arrow["v"', from=2-1, to=2-2] \end{tikzcd}$

there exists a unique diagonal filler $d$ making both triangles commute. If $L$ and $R$ are two classes of maps in $\mathcal{C}$, we say that $L$ is left-orthogonal to $R$ if every morphism in $L$ is left-orthogonal to every morphism in $R$.

Lemma 8

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Given $l$ and $r$ as above, $l$ is left-orthogonal to $r$ iff the square

$\begin{tikzcd} {\emph{Hom}(B,X)} & {\emph{Hom}(A,X)} \\ {\emph{Hom}(B,Y)} & {\emph{Hom}(A,Y)} \arrow["{l^*}", from=1-1, to=1-2] \arrow["{r_*}"', from=1-1, to=2-1] \arrow["{r_*}", from=1-2, to=2-2] \arrow["{l^*}"', from=2-1, to=2-2] \end{tikzcd}$

is Cartesian in Set.

Definition 9

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Let $W$ be a class of morphisms in a category $\mathcal{C}$. The left orthogonal complement of $W$, denoted ${}^{\bot }W$, consists of those morphisms in $\mathcal{C}$ which are left orthogonal to every morphism in $W$. The right orthogonal complement of $W$, denoted $W^\bot $, consists of those morphisms in $\mathcal{C}$ which are right orthogonal to every morphism in $W$.

Lemma 10

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For every class of morphisms $W$, $W^\bot $ contains isomorphisms and is closed under limits, composition and base change, and has the left cancellation property. The left orthogonal complement enjoys dual properties.

Theorem 11

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Given two classes of maps $L,R$ in a category $\mathcal{C}$, there exists a $(L,R)-$factorization system on $\mathcal{C}$ iff every morphism in $\mathcal{C}$ has a $(L,R)-$factorization, $L$ is left-orthogonal to $R$ and both $L$ and $R$ are replete.