9.1.6 Weak Factorization Systems
Throughout this text, we have frequently made use of the fact that every morphism of simplicial sets $h: X \rightarrow Z$ admits a factorization $X \xrightarrow {f} Y \xrightarrow {g} Z$, where $g$ is some sort of fibration and the morphism $f$ has innocuous properties. In this section, we develop a general framework for results of this type.
Definition 9.1.6.1. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. A weak 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 lifting problem
\[ \xymatrix@R =50pt@C=50pt{ A \ar [d]^{f} \ar [r] & X \ar [d]^{g} \\ B \ar [r] \ar@ {-->}[ur] & Y } \]
in $\operatorname{\mathcal{C}}$ admits a solution, provided that $f \in S_{L}$ and $g \in S_{R}$.
- $(3)$
The collections $S_{L}$ and $S_{R}$ are closed under retracts (in the $\infty $-category $\operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}})$).
Example 9.1.6.2. Let $\operatorname{\mathcal{C}}= \operatorname{Set_{\Delta }}$ be the category of simplicial sets. We have already encountered several examples of weak factorization systems $(S_{L}, S_{R} )$ on $\operatorname{\mathcal{C}}$:
We can take $S_{R}$ to be the collection of Kan fibrations and $S_{L}$ the collection of anodyne morphisms (Proposition 3.1.7.1).
We can take $S_{R}$ to be the collection of inner anodyne morphisms and $S_{L}$ the collection of inner fibrations (Proposition 4.1.3.2).
We can take $S_{R}$ to be the collection of left fibrations and $S_{L}$ the collection of left anodyne morphisms (Proposition 4.2.4.8).
We can take $S_{R}$ to be the collection of right fibrations and $S_{L}$ the collection of left anodyne morphisms (Variant 4.2.4.9).
We can take $S_{R}$ to be the collection of trivial Kan fibrations and $S_{L}$ the collection of monomorphisms (Exercise 3.1.7.11).
In the situation of Definition 9.1.6.1, the collections $S_{L}$ and $S_{R}$ are determined by one another.
Proposition 9.1.6.5. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $(S_{L}, S_{R})$ be a weak factorization system on $\operatorname{\mathcal{C}}$, and let $h: X \rightarrow Z$ be a morphism of $\operatorname{\mathcal{C}}$. Then $h$ belongs to $S_{L}$ if and only if it is weakly left orthogonal to $S_{R}$, and $h$ belongs to $S_{R}$ if and only if it is weakly right orthogonal to $S_{L}$.
Proof.
We will prove the first assertion; the second follows by a similar argument. Assume that $h$ is weakly left orthogonal to $S_{R}$; we wish to show that $h$ belongs to $S_{L}$ (the reverse implication is immediate from the definition). By virtue of Remark 9.1.6.4, we may assume that $\operatorname{\mathcal{C}}$ is (the nerve of) an ordinary category. The morphism $h$ admits a factorization $X \xrightarrow {f} Y \xrightarrow {g} Z$, where $f \in S_{L}$ and $g \in S_{R}$. Since $h$ is weakly left orthogonal to $g$, the lifting problem
\[ \xymatrix@R =50pt@C=50pt{ X \ar [d]^{h} \ar [r]^-{f} & Y \ar [d]^{g} \\ Z \ar@ {-->}[ur]^-{i} \ar [r]^-{\operatorname{id}} & Z } \]
admits a solution. We then have a commutative diagram
\[ \xymatrix@R =50pt@C=50pt{ X \ar [r]^-{ \operatorname{id}} \ar [d]^{h} & X \ar [d]^{f} \ar [r]^-{ \operatorname{id}} & X \ar [d]^{h} \\ Z \ar [r]^-{i} & Y \ar [r]^-{ g} & Z } \]
which exhibits $h$ as a retract of $f$, so that $h$ also belongs to $S_{L}$.
$\square$
Corollary 9.1.6.6. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $(S_{L}, S_{R})$ be a weak factorization system on $\operatorname{\mathcal{C}}$. Then $S_{L}$ is a weakly saturated collection of morphisms of $\operatorname{\mathcal{C}}$ (see Definition 9.1.3.19).
Proof.
Combine Proposition 9.1.6.5 with Corollary 9.1.5.17.
$\square$
Using the small object argument of ยง9.1.4, we can produce many examples of weak factorization systems.
Theorem 9.1.6.7 (Existence of Weak 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 object $X$ is $\kappa $-compact for some small cardinal $\kappa $.
Then $\operatorname{\mathcal{C}}$ admits a weak factorization system $( S_{L}, S_{R} )$, where $S_{L}$ is the weakly saturated collection of morphisms generated by $W$ (Remark 9.1.3.23) and $S_{R}$ is the collection of morphisms which are weakly right orthogonal to $W$.
Proof of Theorem 9.1.6.7.
The collection $S_{L}$ is closed under retracts by construction, and $S_{R}$ is closed under retracts by virtue of Proposition 9.1.5.14. Corollary 9.1.5.17 guarantees that $S_{L}$ is weakly left orthogonal to $S_{R}$. It will therefore suffice to show that every morphism $h: X \rightarrow Z$ of $\operatorname{\mathcal{C}}$ factors as a composition $X \xrightarrow {f} Y \xrightarrow {g} Z$, where $f$ belongs to $S_{L}$ and $g$ belongs to $S_{R}$.
Let $\widetilde{\operatorname{\mathcal{C}}}$ denote the slice $\infty $-category $\operatorname{\mathcal{C}}_{/Z}$, and let $\pi : \widetilde{\operatorname{\mathcal{C}}} \rightarrow \operatorname{\mathcal{C}}$ denote the projection map. Our assumption that $\operatorname{\mathcal{C}}$ is locally small guarantees that $\widetilde{\operatorname{\mathcal{C}}}$ is also locally small (Example 4.7.8.11), and our assumption that $\operatorname{\mathcal{C}}$ admits small colimits guarantees that $\widetilde{\operatorname{\mathcal{C}}}$ admits small colimits (Corollary 7.1.3.20). Let $\widetilde{W}$ denote a set of representatives for the collection of isomorphism classes of morphisms $\widetilde{w}$ of $\operatorname{\mathcal{C}}_{/Z}$ satisfying $\pi ( \widetilde{w} ) \in W$. Since $W$ is small and $\operatorname{\mathcal{C}}$ is locally small, the set $\widetilde{W}$ is also small. For every morphism $\widetilde{w}: \widetilde{A} \rightarrow \widetilde{B}$ which belongs to $\widetilde{W}$, the image $A = \pi ( \widetilde{A} )$ is a $\kappa $-compact object of $\widetilde{\operatorname{\mathcal{C}}}$ for some small cardinal $\kappa $, so that $\widetilde{A}$ is a $\kappa $-compact object of $\widetilde{\operatorname{\mathcal{C}}}$ (Remark 
). Let us identify the morphism $h$ with an object $\widetilde{X} \in \widetilde{\operatorname{\mathcal{C}}}$ satisfying $\pi ( \widetilde{X} ) = X$. Applying Theorem 9.1.4.3, we deduce that there is a morphism $\widetilde{f}: \widetilde{X} \rightarrow \widetilde{Y}$ in the $\infty $-category $\widetilde{\operatorname{\mathcal{C}}}$ which is a transfinite pushout of morphisms of $\widetilde{W}$, where $\widetilde{Y}$ is $\widetilde{W}$-local. Set $f = \pi (\widetilde{f})$, so that $\widetilde{f}$ can be identified with a diagram
\[ \xymatrix@R =50pt@C=50pt{ & Y \ar [dr]^{g} & \\ X \ar [ur]^{f} \ar [rr]^{h} & & Z } \]
in the $\infty $-category $\operatorname{\mathcal{C}}$. Since the functor $\pi $ preserves small colimits (Corollary 7.1.3.20), the morphism $f$ is a transfinite pushout of morphisms belonging to $W$, and therefore belongs to $S_{L}$. Our assumption that $\widetilde{Y}$ is $\widetilde{W}$-local guarantees that $g$ belongs to $S_{R}$ (Remark 9.1.5.12).
$\square$