9.6.5 Weak Factorization Systems
We now formulate the $\infty $-categorical counterpart of Definition 9.6.0.3.
Definition 9.6.5.1. 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:
- $(0)$
The collections $S_{L}$ and $S_{R}$ are closed under retracts (in the $\infty $-category $\operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}})$).
- $(1)$
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 $g \in S_{R}$.
Example 9.6.5.3. Let $\operatorname{\mathcal{C}}$ be a category and let $S_ L$ and $S_ R$ be collections of morphism of $\operatorname{\mathcal{C}}$. Then $(S_ L, S_ R)$ is a weak factorization system on $\operatorname{\mathcal{C}}$ (in the sense of Definition 9.6.0.3) if and only if it is a weak factorization system on the $\infty $-category $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ (in the sense of Definition 9.6.5.1).
Proposition 9.6.5.5. 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 9.6.5.4, 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 $S_{L}$.
$\square$
Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. It follows from Proposition 9.6.5.5 that a weak factorization system $(S_ L, S_ R)$ on $\operatorname{\mathcal{C}}$ is determined by the collection of morphisms $S_ L$. Conversely, given a collection of morphisms $W$ of $\operatorname{\mathcal{C}}$, one can ask if there exists a weak factorization system $(S_ L, S_ R)$ with $S_ L = W$. For this condition to be satisfied, $W$ must have various closure properties.
Definition 9.6.5.6. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $W$ be a collection of morphisms of $\operatorname{\mathcal{C}}$. We say that $W$ is weakly saturated if it satisfies the following conditions:
- $(1)$
The collection $W$ is closed under pushouts: that is, for every pushout diagram
\[ \xymatrix@C =50pt@R=50pt{ C \ar [d]^{w} \ar [r] & C' \ar [d]^{w'} \\ D \ar [r] & D' } \]
in the $\infty $-category $\operatorname{\mathcal{C}}$, if $w$ belongs to $W$, then $w'$ also belongs to $W$.
- $(2)$
The collection $W$ is closed under the formation of retracts (in the $\infty $-category $\operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{C}})$).
- $(3)$
The collection $W$ is closed under small transfinite composition (Definition 9.6.1.1).
Example 9.6.5.8. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $X \in \operatorname{\mathcal{C}}$ be an object, and let $W$ be the collection of all morphisms $w: C \rightarrow D$ of $\operatorname{\mathcal{C}}$ such that $X$ is weakly $w$-local. Then $W$ is weakly saturated. This follows from Proposition 9.6.2.13, Variant 9.6.2.12, and Proposition 9.6.2.16.
Example 9.6.5.9. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $T$ be a collection of morphisms of $\operatorname{\mathcal{C}}$, and let $S$ be the collection of all morphisms of $\operatorname{\mathcal{C}}$ which are weakly left orthogonal to $T$. Then $S$ is weakly saturated. See Propositions 9.6.4.14, 9.6.4.15, and 9.6.4.16 with Remark 9.6.5.11.
Example 9.6.5.10. 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 $\operatorname{\mathcal{C}}$. This is a special case of Example 9.6.5.9, since $S_{L}$ is the collection of morphisms which are weakly left orthogonal to $S_{R}$ (Proposition 9.6.5.5).
Using the small object argument of ยง9.6.3, we have the following converse of Example 9.6.5.10.
Theorem 9.6.5.12 (Existence of Weak Factorization Systems). Let $\operatorname{\mathcal{C}}$ be a presentable $\infty $-category and let $W$ be a small collection of morphisms of $\operatorname{\mathcal{C}}$. 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 9.6.5.11) and $S_{R}$ is the collection of morphisms which are weakly right orthogonal to $W$.
Proof of Theorem 9.6.5.12.
The collection $S_{L}$ is closed under retracts by construction, and $S_{R}$ is closed under retracts by virtue of Proposition 9.6.4.14. Example 9.6.5.9 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 presentable guarantees that $\widetilde{\operatorname{\mathcal{C}}}$ is also presentable (Proposition 9.5.4.1). 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. Applying Theorem 9.6.3.3, 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 weakly $\widetilde{W}$-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 7.1.4.27), the morphism $f$ is a transfinite pushout of morphisms belonging to $W$, and therefore belongs to $S_{L}$. Our assumption that $\widetilde{Y}$ is weakly $\widetilde{W}$-local guarantees that $g$ belongs to $S_{R}$ (Remark 9.6.4.12).
$\square$