Proposition 9.6.9.10. Let $\operatorname{\mathcal{C}}$ be an accessible $\infty $-category and let $(S_ L, S_ R)$ be a factorization system on $\operatorname{\mathcal{C}}$. Let $\mathcal{L}$ and $\mathcal{R}$ denote the full subcategories of $\operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}})$ spanned by the elements of $S_{L}$ and $S_{R}$, respectively. The following conditions are equivalent:
- $(1)$
The $\infty $-category $\mathcal{L}$ is accessibly embedded in $\operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}})$ (see Definition 9.4.7.8).
- $(2)$
The $\infty $-category $\mathcal{R}$ is accessibly embedded in $\operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}})$.
- $(3)$
Let $D: \operatorname{Fun}_{LR}( \Delta ^2, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}})$ be the trivial Kan fibration of Theorem 9.6.8.15 and let $Q: \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}_{LR}( \Delta ^2, \operatorname{\mathcal{C}})$ be a section of $D$. Then $Q$ is accessible (when regarded as a functor from $\operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}})$ to $\operatorname{Fun}( \Delta ^2, \operatorname{\mathcal{C}})$).
Proof.
We will prove that $(1) \Leftrightarrow (3)$; the equivalence $(2) \Leftrightarrow (3)$ follows by a similar argument. Assume first that $\mathcal{L}$ is an accessibly embedded subcategory of $\operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}})$. As in Notation 9.6.7.1, we let $\operatorname{Fun}_{L}( \Delta ^2, \operatorname{\mathcal{C}})$ denote the full subcategory of $\operatorname{Fun}( \Delta ^2, \operatorname{\mathcal{C}})$ spanned by those $2$-simplices
\[ \xymatrix@R =50pt@C=50pt{ & Y \ar [dr] & \\ X \ar [ur]^{f} \ar [rr] & & Z } \]
where $f$ belongs to $S_{L}$. Then $\operatorname{Fun}_{L}( \Delta ^2, \operatorname{\mathcal{C}})$ is the inverse image of $\mathcal{L}$ under the restriction functor $\operatorname{Fun}( \Delta ^2, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}})$ and is therefore an accessibly embedded subcategory of $\operatorname{Fun}( \Delta ^2, \operatorname{\mathcal{C}})$ (Corollary 9.4.8.12). In particular, $\operatorname{Fun}_{L}( \Delta ^2, \operatorname{\mathcal{C}})$ is an accessible $\infty $-category, and $Q$ is accessible when regarded a functor from $\operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}})$ to $\operatorname{Fun}( \Delta ^2, \operatorname{\mathcal{C}})$ if and only if it is accessible as a functor from $\operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}})$ to $\operatorname{Fun}_{L}( \Delta ^2, \operatorname{\mathcal{C}})$. Let $D_{+}: \operatorname{Fun}_{L}( \Delta ^2, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}})$ be the functor given on objects by the formula $D_{+}(\sigma ) = d^{2}_{1}(\sigma )$. By construction, the composition $D_{+} \circ Q$ is the identity functor on $\operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}})$. It follows from Proposition 9.6.7.4 (and the criterion of Corollary 6.2.6.5) that the identity transformation $\operatorname{id}\rightarrow D_{+} \circ Q$ is the unit of an adjunction. In particular, the functor $Q$ is a left adjoint (when regarded as a functor from $\operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}})$ to $\operatorname{Fun}_{L}( \Delta ^2, \operatorname{\mathcal{C}})$) and is therefore accessible (Corollary 9.4.7.18).
We now prove the converse. Assume that $Q$ is an accessible functor. Then the composition of $Q$ with the restriction map
\[ T: \operatorname{Fun}( \Delta ^2, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}( \operatorname{N}_{\bullet }( \{ 1 < 2\} ), \operatorname{\mathcal{C}}) \simeq \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}}) \]
is also an accessible functor. Let $\operatorname{Isom}(\operatorname{\mathcal{C}})$ denote the full subcategory of $\operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}})$ spanned by the isomorphisms (Example 4.4.1.14). It follows from the proof of Corollary 9.6.8.10 that $\mathcal{L}$ is the inverse image of $\operatorname{Isom}( \operatorname{\mathcal{C}})$ under $T \circ Q$. Since $\operatorname{Isom}(\operatorname{\mathcal{C}})$ is an accessibly embedded subcategory of $\operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}})$ (Remark 9.4.8.3), it follows that $\mathcal{L}$ is also an accessibly embedded subcategory of $\operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}})$ (Corollary 9.4.8.12).
$\square$