9.1.8 Factorization Systems
We now consider a variant of Definition 9.1.6.1.
Definition 9.1.8.1. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. A factorization system on $\operatorname{\mathcal{C}}$ is a pair $(S_{L}, S_{R})$, where $S_{L}$ and $S_{R}$ are collections of morphisms of $\operatorname{\mathcal{C}}$ which satisfy the following conditions:
- $(1)$
For every morphism $h: X \rightarrow Z$ of $\operatorname{\mathcal{C}}$, there exists a $2$-simplex
\[ \xymatrix@R =50pt@C=50pt{ & Y \ar [dr]^{g} & \\ X \ar [ur]^{f} \ar [rr]^{h} & & Z } \]
where $f$ belongs to $S_{L}$ and $g$ belongs to $S_{R}$.
- $(2)$
Every morphism $f \in S_{L}$ is left orthogonal to every morphism $g \in S_{R}$. (Definition 9.1.7.4).
- $(3)$
The collections $S_{L}$ and $S_{R}$ are closed under retracts (in the $\infty $-category $\operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}})$).
Example 9.1.8.3 (Trivial Factorization Systems). Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $W$ be the collection of all isomorphisms in $\operatorname{\mathcal{C}}$, and let $A$ denote the collection of all morphisms in $\operatorname{\mathcal{C}}$. Then the pairs $(W,A)$ and $(A,W)$ are factorization systems on $\operatorname{\mathcal{C}}$ (see Corollary 9.1.7.14).
Exercise 9.1.8.5. Let $\operatorname{Set}$ denote the category of sets and let $\operatorname{\mathcal{C}}= \operatorname{N}_{\bullet }(\operatorname{Set})$ be the associated $\infty $-category. Let $S$ be the collection of surjective functions, and let $I$ be the collection of injective functions. Show that:
The pair $(S, I )$ is a factorization system on $\operatorname{\mathcal{C}}$.
The pair $(I,S)$ is a weak factorization system on $\operatorname{\mathcal{C}}$.
The pair $(I,S)$ is not a factorization system on $\operatorname{\mathcal{C}}$.
In the situation of Definition 9.1.8.1, either of the collections $S_{L}$ and $S_{R}$ can be recovered from the other.
Proposition 9.1.8.6. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $(S_{L}, S_{R})$ be a factorization system on $\operatorname{\mathcal{C}}$, and let $f$ be a morphism of $\operatorname{\mathcal{C}}$. The following conditions are equivalent:
- $(1)$
The morphism $f$ belongs to $S_{L}$.
- $(2)$
The morphism $f$ is left orthogonal to $S_{R}$.
- $(3)$
The morphism $f$ is weakly left orthogonal to $S_{R}$.
Proof.
The implication $(1) \Rightarrow (2)$ is immediate from the definition, the implication $(2) \Rightarrow (3)$ follows from Remark 9.1.7.7, and the implication $(3) \Rightarrow (1)$ follows from Proposition 9.1.6.5 (together with Remark 9.1.8.4).
$\square$
Corollary 9.1.8.7. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category which admits pushouts and let $(S_{L}, S_{R} )$ be a weak factorization system on $\operatorname{\mathcal{C}}$. The following conditions are equivalent:
- $(1)$
The pair $(S_{L}, S_{R} )$ is a factorization system on $\operatorname{\mathcal{C}}$.
- $(2)$
For every $2$-simplex
\[ \xymatrix@R =50pt@C=50pt{ & Y \ar [dr]^{g} & \\ X \ar [ur]^{f} \ar [rr]^{h} & & Z } \]
of $\operatorname{\mathcal{C}}$, if $f$ and $h$ belong to $S_{L}$, then $g$ also belongs to $S_{L}$.
- $(3)$
For every morphism $f: X \rightarrow Y$ which belongs to $S_{L}$, the relative codiagonal $\gamma _{X/Y}: Y \coprod _{X} Y \rightarrow Y$ also belongs to $S_{L}$.
Proof.
We first show that $(1) \Rightarrow (2)$. Assume that $(S_{L}, S_{R} )$ is a factorization system and consider a $2$-simplex
\[ \xymatrix@R =50pt@C=50pt{ & Y \ar [dr]^{g} & \\ X \ar [ur]^{f} \ar [rr]^{h} & & Z } \]
of $\operatorname{\mathcal{C}}$. If $f$ and $h$ belong to $S_{L}$, then they are left orthogonal to $S_{R}$. Applying Corollary 9.1.7.15, we deduce that $g$ is also left orthogonal to $S_{R}$, so that $g \in S_{L}$ by virtue of Proposition 9.1.8.6.
We now show that $(2)$ implies $(3)$. Let $f: X \rightarrow Y$ be a morphism which belongs to $S_{L}$. Then the relative codiagonal $\gamma _{X/Y}$ fits into a commutative diagram
\[ \xymatrix@R =50pt@C=50pt{ & Y \coprod _{X} Y \ar [dr]^{ \gamma _{X/Y} } & \\ Y \ar [rr]^{ \operatorname{id}_{Y} } \ar [ur]^{f'} & & Y, } \]
where $f'$ is a pushout of $f$. Since $S_{L}$ is weakly saturated (Corollary 9.1.6.6), it contains the morphisms $f'$ and $\operatorname{id}_{Y}$. If condition $(2)$ is satisfied, then $\gamma _{X/Y}$ also contains $\gamma _{X/Y}$.
We now complete the proof by showing that $(3)$ implies $(1)$. Let $g$ be a morphism of $\operatorname{\mathcal{C}}$ which belongs to $S_{R}$. Then $g$ is weakly orthogonal to $S_{L}$, and we wish to show that $g$ is orthogonal to $S_{L}$. This follows by combining assumption $(3)$ with Corollary 9.1.7.21.
$\square$
Proposition 9.1.8.8 (Lifting Factorization Systems). Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of $\infty $-categories and let $(S_{L}, S_{R} )$ be a weak factorization system on $\operatorname{\mathcal{C}}$. Let $\widetilde{S}_{L}$ denote the collection of all $U$-cocartesian morphisms $\widetilde{f}$ in $\operatorname{\mathcal{E}}$ satisfying $U( \widetilde{f} ) \in S_{L}$, and let $\widetilde{S}_{R}$ be the collection of all morphisms $\widetilde{g}$ in $\operatorname{\mathcal{E}}$ satisfying $U( \widetilde{g}) \in S_{R}$. Then the pair $( \widetilde{S}_{L}, \widetilde{S}_{R} )$ is a weak factorization system on $\operatorname{\mathcal{E}}$. If $(S_{L}, S_{R} )$ is a factorization system on $\operatorname{\mathcal{C}}$, then $( \widetilde{S}_{L}, \widetilde{S}_{R} )$ is a factorization system on $\operatorname{\mathcal{E}}$.
Proof.
By assumption, the collection $S_{L}$ is weakly left orthogonal to $S_{R}$. Applying Corollary 9.1.7.23, we see that $\widetilde{S}_{L}$ is weakly left orthogonal to $\widetilde{S}_{R}$ (and left orthogonal if the pair $(S_{L}, S_{R} )$ is a factorization system on $\operatorname{\mathcal{C}}$). Since $S_{L}$ and $S_{R}$ are closed under retracts, the collections $\widetilde{S}_{L}$ and $\widetilde{S}_{R}$ have the same property (see Corollary 8.5.1.12). We will complete the proof by showing that the pair $( \widetilde{S}_{L}, \widetilde{S}_{R} )$ satisfies condition $(1)$ of Definition 9.1.6.1. Let $\widetilde{h}: \widetilde{X} \rightarrow \widetilde{Z}$ be a morphism in the $\infty $-category $\operatorname{\mathcal{E}}$, and let $h: X \rightarrow Z$ denote its image in the $\infty $-category $\operatorname{\mathcal{C}}$. Since $(S_{L}, S_{R} )$ is a weak factorization system, we can choose a $2$-simplex $\sigma :$
\[ \xymatrix@R =50pt@C=50pt{ & Y \ar [dr]^{g} & \\ X \ar [ur]^{f} \ar [rr]^{h} & & Z } \]
of $\operatorname{\mathcal{C}}$, where $f$ belongs to $S_{L}$ and $g$ belongs to $S_{R}$. Our assumption that $U$ is a cocartesian fibration guarantees that we can lift $f$ to a $U$-cocartesian morphism $\widetilde{f}: \widetilde{X} \rightarrow \widetilde{Y}$ in the $\infty $-category $\operatorname{\mathcal{E}}$. Since $\widetilde{f}$ is $U$-cocartesian, we can lift $\sigma $ to a $2$-simplex $\widetilde{\sigma }:$
\[ \xymatrix@R =50pt@C=50pt{ & \widetilde{Y} \ar [dr]^{\widetilde{g}} & \\ \widetilde{X} \ar [ur]^{\widetilde{f}} \ar [rr]^{\widetilde{h}} & & \widetilde{Z} } \]
in the $\infty $-category $\operatorname{\mathcal{E}}$. By construction, we have $\widetilde{f} \in \widetilde{S}_{L}$ and $\widetilde{g} \in \widetilde{S}_{R}$.
$\square$
Corollary 9.1.8.9. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of $\infty $-categories, let $S$ be the collection of all $U$-cocartesian morphisms of $\operatorname{\mathcal{E}}$, and let $T$ be the collection of all morphisms $f$ of $\operatorname{\mathcal{E}}$ such that $U(f)$ is an isomorphism in $\operatorname{\mathcal{C}}$. Then the pair $(S,T)$ is a factorization system on $\operatorname{\mathcal{E}}$.
Proof.
Combine Proposition 9.1.8.8 with Example 9.1.8.3.
$\square$
One can produce many examples of factorization systems using the small object argument of ยง9.1.4.
Theorem 9.1.8.10 (Existence of Factorization Systems). Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $W$ be a collection of morphisms of $\operatorname{\mathcal{C}}$. Assume that:
- $(1)$
The $\infty $-category $\operatorname{\mathcal{C}}$ is locally small and admits small colimits.
- $(2)$
The collection $W$ is small.
- $(3)$
For every morphism $w: X \rightarrow Y$ in $W$, the objects $X$ and $Y$ are $\kappa $-compact for some small cardinal $\kappa $.
Then $\operatorname{\mathcal{C}}$ admits a factorization system $( S_{L}, S_{R} )$, where $S_{R}$ is the collection of morphisms of $\operatorname{\mathcal{C}}$ which are right orthogonal to $W$.
Proof.
By virtue of Corollary 9.1.7.21 (and Proposition 
), we can enlarge $W$ to arrange that every morphism $w: A \rightarrow B$ which belongs to $W$ admits a relative codiagonal $\gamma _{A/B}: B \coprod _{A} B \rightarrow B$ which also belongs to $W$. Applying Theorem 9.1.6.7, we conclude that $\operatorname{\mathcal{C}}$ admits a weak factorization system $( S_{L}, S_{R} )$, where $S_{L}$ is the weakly saturated class of morphisms generated by $W$ and $S_{R}$ is the collection of morphisms which are weakly right orthogonal to $W$. Using Corollary 9.1.7.21, we see that a morphism $g: X \rightarrow Y$ of $\operatorname{\mathcal{C}}$ belongs to $S_{R}$ if and only if it is right orthogonal to $W$. If this condition is satisfied, then Corollary 9.1.7.20 guarantees that $g$ is right orthogonal to $W$. Allowing $g$ to vary, we conclude that $(S_{L}, S_{R} )$ is a factorization system.
$\square$