Theorem 9.2.9.17 (04PT)

Theorem 9.2.9.17. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $S_{L}$ and $S_{R}$ be collections of morphisms of $\operatorname{\mathcal{C}}$, and let $\operatorname{Fun}_{LR}( \Delta ^2, \operatorname{\mathcal{C}}) \subseteq \operatorname{Fun}(\Delta ^2, \operatorname{\mathcal{C}})$ be the full subcategory of Notation 9.2.8.1. Then $(S_ L, S_ R)$ is a factorization system on $\operatorname{\mathcal{C}}$ if and only if it satisfies the following conditions:

$(1)$

The restriction map

\[ D: \operatorname{Fun}_{LR}( \Delta ^2, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}}) \quad \quad \sigma \mapsto d^{2}_{1}(\sigma ) \]

is an equivalence of $\infty $-categories.

$(2)$

Every identity morphism of $\operatorname{\mathcal{C}}$ is contained in both $S_ L$ and $S_ R$.

$(3)$

The collections $S_ L$ and $S_ R$ are closed under isomorphism (in the $\infty $-category $\operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}})$).

Proof. The necessity of $(3)$ is immediate from the definitions, and the necessity of $(1)$ and $(2)$ follow from Proposition 9.2.9.6 and Corollary 9.2.9.7, respectively. For the converse, assume that conditions $(1)$, $(2)$, and $(3)$ are satisfied. Combining $(1)$ and $(3)$ with Theorem 9.2.8.2, we deduce that $S_{L}$ is left orthogonal to $S_{R}$. We are therefore reduced to proving that the functor $D$ is surjective on objects. Assumption $(3)$ guarantees that the full subcategory $\operatorname{Fun}_{LR}( \Delta ^2, \operatorname{\mathcal{C}}) \subseteq \operatorname{Fun}( \Delta ^2, \operatorname{\mathcal{C}})$ is replete, so that $D$ is an isofibration (see Corollary 4.4.5.3). It will therefore suffice to show that $D$ is essentially surjective, which follows from assumption $(1)$. $\square$