Kerodon

Proof. We first observe that $S_{R}$ is closed under the formation of relative diagonals: that is, if a functor $G: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is essentially $(n-1)$-categorical, then the relative diagonal of $G$ (formed in the $\infty $-category $\operatorname{\mathcal{QC}}$) has the same property. Using Exercise 7.6.3.13, we can identify the relative diagonal of $G$ with the inclusion map $\iota : \operatorname{\mathcal{C}}\hookrightarrow \operatorname{\mathcal{C}}\times ^{\mathrm{h}}_{\operatorname{\mathcal{D}}} \operatorname{\mathcal{C}}$. For $n \geq 1$, it follows from Variant 4.8.6.15 that $\iota $ is essentially $(n-2)$-categorical, and therefore also essentially $(n-1)$-categorical (Remark 4.8.6.6). If $n \leq 0$, then the functor $G$ is fully faithful, so $\iota $ is an equivalence of $\infty $-categories.

It follows from Remarks 4.8.5.16, 4.8.5.17, and 4.8.5.18 that $S_{L}$ and $S_{R}$ are invariant under isomorphism. Theorem 4.8.8.3 asserts that every functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{E}}$ admits a factorization $\operatorname{\mathcal{C}}\xrightarrow {F_{L}} \operatorname{\mathcal{D}}\xrightarrow {F_{R}} \operatorname{\mathcal{E}}$, where $F_{L}$ belongs to $S_{L}$ and $F_{R}$ belongs to $S_{R}$. We will complete the proof by showing that $S_{L}$ is left orthogonal to $S_{R}$. By virtue of (the dual of) Corollary 9.2.7.21, it will suffice to show that $S_{L}$ is weakly left orthogonal to $S_{R}$: that is, every lifting problem

in the $\infty $-category $\operatorname{\mathcal{QC}}$ admits a solution, provided that $F$ is categorically $n$-connective and $G$ is essentially $(n-1)$-categorical. By virtue of Corollary 5.6.5.18, we may assume that (9.22) arises from a commutative diagram in the category of simplicial sets. Using Corollary 4.5.2.23, we can further assume that $F$ is a monomorphism of simplicial sets and that $G$ is an isofibration. In this case, the lifting problem (9.22) already admits a solution in the category of simplicial sets: see Corollary 4.8.7.18 and Remark 4.8.7.19. $\square$

Proof. Recall that a morphism of Kan complexes is $n$-connective if and only if it is categorically $n$-connective (Example 4.8.7.3), and $(n-1)$-truncated if and only if it is essentially $(n-1)$-categorical (Example 4.8.6.3). It follows immediately from Proposition 9.2.9.4 that $S_{L}$ and $S_{R}$ are closed under isomorphism, and that $S_{L}$ is left orthogonal to $S_{R}$. To complete the proof, it suffices to show that every morphism of Kan complexes $f: X \rightarrow Z$ admits a factorization $X \xrightarrow {f_{L}} Y \xrightarrow {f_{R}} Z$, where $f_{L}$ is $n$-connective and $f_{R}$ is $(n-1)$-truncated. This is the content of Corollary 4.8.8.9. $\square$

Proof. Condition $(1)$ of Definition 9.2.9.1 guarantees that $D$ is surjective on objects, and condition $(2)$ guarantees that $D$ is fully faithful (Theorem 9.2.8.2). Applying the criterion Theorem 4.6.2.21, we deduce that $D$ is an equivalence of $\infty $-categories. Condition $(3)$ of Definition 9.2.9.1 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 of $\infty $-categories (see Corollary 4.4.5.3). Applying Proposition 4.5.5.20, we conclude that $D$ is a trivial Kan fibration. $\square$

Proof. Let $W$ be the collection of all isomorphisms in $\operatorname{\mathcal{C}}$, and set $S_{L}^{+} = S_{L} \cup W$ and $S_{R}^{+} = S_{R} \cup W$. Using Corollary 9.2.7.14, we deduce that $S_{L}^{+}$ is left orthogonal to $S_{R}^{+}$, so that $(S_{L}^{+}, S_{R}^{+} )$ is also a factorization system on $\operatorname{\mathcal{C}}$. Let $\operatorname{Fun}_{LR}(\Delta ^2, \operatorname{\mathcal{C}})$ be as in Notation 9.2.8.1 and define $\operatorname{Fun}_{LR}^{+}( \Delta ^2, \operatorname{\mathcal{C}})$ similarly. We then have a commutative diagram

where both of the vertical maps are trivial Kan fibrations (Proposition 9.2.9.6). It follows that the inclusion map $\operatorname{Fun}_{LR}( \Delta ^2, \operatorname{\mathcal{C}}) \hookrightarrow \operatorname{Fun}_{LR}^{+}( \Delta ^2, \operatorname{\mathcal{C}})$ is an equivalence of $\infty $-categories. Since $\operatorname{Fun}_{LR}( \Delta ^2, \operatorname{\mathcal{C}})$ is a replete full subcategory of $\operatorname{Fun}_{LR}^{+}( \Delta ^2, \operatorname{\mathcal{C}})$, we must have $\operatorname{Fun}_{LR}( \Delta ^2, \operatorname{\mathcal{C}}) = \operatorname{Fun}_{LR}^{+}( \Delta ^2, \operatorname{\mathcal{C}})$. In particular, if $f: X \rightarrow Y$ is an isomorphism in $\operatorname{\mathcal{C}}$, then the degenerate $2$-simplices $s^{1}_{0}(f)$ and $s^{1}_{1}(f)$ are both contained in $\operatorname{Fun}_{LR}( \Delta ^2, \operatorname{\mathcal{C}})$, so that $f$ is contained in both $S_{L}$ and $S_{R}$. $\square$

Proof. We will show that $S_{L}$ is closed under retracts; the analogous statement for $S_{R}$ follows by a similar argument. By virtue of Proposition 9.2.9.6, the restriction map

is a trivial Kan fibration. It therefore admits a section $\operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}_{LR}( \Delta ^2, \operatorname{\mathcal{C}})$, which carries each morphism $f: X \rightarrow Z$ of $\operatorname{\mathcal{C}}$ to a $2$-simplex $\sigma _{f}:$

where $f_{L} \in S_{L}$ and $f_{R} \in S_{R}$. We will complete the proof by showing that $f$ belongs to $S_{L}$ if and only if $f_{R}$ is an isomorphism in $\operatorname{\mathcal{C}}$. One direction is clear: if $f_{R}$ is an isomorphism, then $f$ is isomorphic to $f_{L}$ in the $\infty $-category $\operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}})$, and therefore belongs to $S_{L}$ by virtue of our assumption that $S_ L$ is closed under isomorphism. For the converse, assume that $f$ belongs to $S_{L}$. Since $\operatorname{id}_{Z}$ belongs to $S_{R}$ (Corollary 9.2.9.7), the degenerate $2$-simplex $\widetilde{f} = s^{1}_{1}(f)$ can be regarded as an object of $\operatorname{Fun}_{LR}( \Delta ^2, \operatorname{\mathcal{C}})$ satisfying $D( \widetilde{f} ) = f = D( \sigma _{f} )$. Since $D$ is an equivalence of $\infty $-categories, the $2$-simplex $\sigma _{f}$ is isomorphic to $\widetilde{f}$ as an object of the $\infty $-category $\operatorname{Fun}_{LR}( \Delta ^2, \operatorname{\mathcal{C}})$. It follows that $f_{R} = d^{2}_{0}( \sigma _{f} )$ is isomorphic to $\operatorname{id}_{Z} = d^{2}_{0}( \widetilde{f} )$ as an object of the $\infty $-category $\operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}})$, so that $f_{R}$ is an isomorphism (Example 4.4.1.14). $\square$

Proof. The only nontrivial point is to verify that $S_{L}$ and $S_{R}$ are closed under retracts, which follows from Corollary 9.2.9.8. $\square$

Proof. The implication $(1) \Rightarrow (2)$ is immediate from the definition, the implication $(2) \Rightarrow (3)$ follows from Remark 9.2.7.7, and the implication $(3) \Rightarrow (1)$ follows from Proposition 9.2.6.5 (together with Corollary 9.2.9.9). $\square$

Proof. We first show that $(1) \Rightarrow (2)$. Assume that $(S_{L}, S_{R} )$ is a factorization system and consider a $2$-simplex

of $\operatorname{\mathcal{C}}$. If $f$ and $h$ belong to $S_{L}$, then they are left orthogonal to $S_{R}$. Applying Corollary 9.2.7.15, we deduce that $g$ is also left orthogonal to $S_{R}$, so that $g \in S_{L}$ by virtue of Proposition 9.2.9.11.

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

where $f'$ is a pushout of $f$. Since $S_{L}$ is weakly saturated (Corollary 9.2.6.6), 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 right orthogonal to $S_{L}$, and we wish to show that $g$ is right orthogonal to $S_{L}$. This follows by combining assumption $(3)$ with Corollary 9.2.7.21. $\square$

Proof. By assumption, the collection $S_{L}$ is weakly left orthogonal to $S_{R}$. Applying Corollary 9.2.7.23, 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 isomorphism, the collections $\widetilde{S}_{L}$ and $\widetilde{S}_{R}$ have the same property (see Corollary 8.5.1.13). We will complete the proof by showing that the pair $( \widetilde{S}_{L}, \widetilde{S}_{R} )$ satisfies condition $(1)$ of Definition 9.2.6.1. 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 $\sigma :$

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 $\widetilde{\sigma }:$

in the $\infty $-category $\operatorname{\mathcal{E}}$. By construction, we have $\widetilde{f} \in \widetilde{S}_{L}$ and $\widetilde{g} \in \widetilde{S}_{R}$. $\square$

Proof. Combine Proposition 9.2.9.13 with Example 9.2.9.3. $\square$

Proof. By virtue of Corollary 9.2.7.21 (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 9.2.6.7, 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 9.2.7.21, 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 9.2.7.20 guarantees that $g$ is right orthogonal to $W$. Allowing $g$ to vary, we conclude that $(S_{L}, S_{R} )$ is a factorization system. $\square$

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 $(1)$. $\square$

Proof. Since $S_{L}$ and $S_{R}$ contain identity morphisms and are closed under isomorphism, the collections $S_{L}^{K}$ and $S_{R}^{K}$ have the same properties. By virtue of Theorem 9.2.9.17, it will suffice to show that the restriction map

is an equivalence of $\infty $-categories. This follows from Remark 4.5.1.16, since $D_ K$ is obtained by applying the functor $\operatorname{Fun}(K, \bullet )$ to the restriction map $D: \operatorname{Fun}_{LR}(\Delta ^2, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}})$. $\square$