Kerodon

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 9.1.9.2, it will suffice to show that the restriction map

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

Proof. By construction, we have a pullback diagram of simplicial sets

where the right vertical map is a trivial Kan fibration (see Example 1.4.7.7). $\square$

Proof. By construction, the diagram $\widetilde{f}: \Delta ^2 \rightarrow \operatorname{\mathcal{C}}$ is left Kan extended from the simplicial subset $\Delta ^1 \subset \Delta ^2$. Applying Corollary 7.3.6.9, we deduce that the restriction functor $\operatorname{Fun}( \Delta ^2, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}})$ determines a trivial Kan fibration $R: \operatorname{Hom}_{\operatorname{Fun}(\Delta ^2, \operatorname{\mathcal{C}}) }( \widetilde{f}, \widetilde{g} ) \rightarrow \operatorname{Hom}_{ \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}}) }(f, \operatorname{id}_{X} )$. Similarly, the $1$-simplex $\operatorname{id}_{X}$ can be viewed as a diagram $\Delta ^1 \rightarrow \operatorname{\mathcal{C}}$ which is right Kan extended from the vertex $\{ 1\} \subset \Delta ^2$, so the evaluation functor $\operatorname{ev}_{1}: \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}$ induces a trivial Kan fibration $Q: \operatorname{Hom}_{\operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}}) }(f, \operatorname{id}_{X} ) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(B, X )$. We are therefore reduced to showing that the composite map

is a homotopy equivalence, which follows from Lemma 9.1.9.6. $\square$

Proof. Let $\alpha : \Delta ^1 \times \Delta ^1 \hookrightarrow \Delta ^3$ be the morphism of simplicial sets given on vertices by the formula $\alpha (i,j) = 2i+j$ (see Definition 9.1.5.1). Then precomposition with $\alpha $ induces an isofibration of $\infty $-categories $U: \operatorname{Fun}( \Delta ^3, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}( \Delta ^1 \times \Delta ^1, \operatorname{\mathcal{C}})$, which restricts to an isofibration $U_0: {}_{f}\operatorname{Fun}(\Delta ^3, \operatorname{\mathcal{C}})_{g} \rightarrow \operatorname{Hom}_{ \operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{C}}) }( f, g )$. Since the source and target of $U_0$ are Kan complexes, it is a Kan fibration (Corollary 4.4.3.10). The desired result now follows by applying Corollary 3.3.7.3 to the diagram of Kan complexes

since $T$ is a homotopy equivalence (Lemma 9.1.9.8). $\square$

Proof. By virtue of Proposition 3.3.7.4, it will suffice to show that every fiber of $\theta $ is a contractible Kan complex. Let $D: \operatorname{Fun}( \Delta ^2, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}})$ denote the functor given by precomposition with the inclusion map $\Delta ^1 \simeq \operatorname{N}_{\bullet }( \{ 0 < 2 \} ) \hookrightarrow \Delta ^2$, and let $E: \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}$ be given by evaluation at the final vertex $1 \in \Delta ^1$. Let $\gamma : \Delta ^2 \times \Delta ^1 \rightarrow \Delta ^2$ be the morphism of simplicial sets given on vertices by the formula $\gamma (i,j) = \begin{cases} 1 & \text{ if } (i,j) = (0,2) \\ i & \text{ otherwise. } \end{cases}$. Then the composition

can be regarded as a morphism $e: \widetilde{f} \rightarrow \sigma $ in the $\infty $-category $\operatorname{Fun}( \Delta ^2, \operatorname{\mathcal{C}})$. It follows from Corollary 5.3.7.5 that the morphism $e$ is $(E \circ D)$-cocartesian and that $D(e)$ is $E$-cocartesian. Consequently, the morphism $e$ is $D$-cocartesian (Corollary 5.1.2.6). Using Proposition 5.1.2.1, we deduce that every fiber of $\theta $ is homotopy equivalent to a fiber of the restriction map $\theta ': \operatorname{Hom}_{ \operatorname{Fun}( \Delta ^2, \operatorname{\mathcal{C}}) }(\widetilde{f}, \tau ) \rightarrow \operatorname{Hom}_{ \operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{C}}) }(f,g)$. It will therefore suffice to prove Corollary 9.1.9.10 in the special case where $\sigma = \widetilde{f}$. By a similar argument, we may also assume that $\tau $ is the degenerate $2$-simplex $\widetilde{g} = s^{1}_{0}(g)$. In this case, Lemma 9.1.9.9 guarantees that every fiber of $\theta $ is homotopy equivalent to the space of solutions to some lifting problem

which is contractible by virtue of our assumption that $f$ is left orthogonal to $g$. $\square$

Proof. Assume that condition $(1)$ is satisfied; we will prove $(2)$ (the reverse implication follows immediately from Corollary 9.1.9.10). Suppose we are given a lifting problem $\tau :$

in the $\infty $-category $\operatorname{\mathcal{C}}$, where $f$ belongs to $S_{L}$ and $g$ belongs to $S_{R}$. We wish to show that the solution space $\operatorname{Sol}(\tau )$ is contractible. Let $\widetilde{f} = s^{1}_{f}(f)$ and $\widetilde{g} = s^{1}_{0}(g)$ denote the degenerate $2$-simplices of $\operatorname{\mathcal{C}}$ defined in Notation 9.1.9.5. Since $S_{L}$ and $S_{R}$ contain all identity morphisms, we can view $\widetilde{f}$ and $\widetilde{g}$ as objects of the $\infty $-category $\operatorname{Fun}_{LR}( \Delta ^2, \operatorname{\mathcal{C}})$. Assumption $(1)$ guarantees that the Kan fibration $\operatorname{Hom}_{ \operatorname{Fun}( \Delta ^2, \operatorname{\mathcal{C}}) }( \widetilde{f}, \widetilde{g} ) \rightarrow \operatorname{Hom}_{ \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}}) }( f,g )$ is a homotopy equivalence. Lemma 9.1.9.9 supplies a homotopy equivalence of $\operatorname{Sol}( \tau )$ with the fiber $\operatorname{Hom}_{ \operatorname{Fun}( \Delta ^2, \operatorname{\mathcal{C}}) }( \widetilde{f}, \widetilde{g} )_{\tau }$, which is contractible by virtue of Proposition 3.3.7.4. $\square$

Proof of Theorem 9.1.9.2. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $S_{L}$ and $S_{R}$ be collections of morphisms of $\operatorname{\mathcal{C}}$. Assume first that the pair $(S_{L}, S_{R})$ is a factorization system on $\operatorname{\mathcal{C}}$; we wish to show that the restriction functor

is an equivalence of $\infty $-categories. Using the assumption that $(S_{L}, S_{R} )$ is a weak factorization system, we deduce that the functor $D$ is surjective on objects; in particular, it is essentially surjective. By virtue of Theorem 4.6.2.19, we are reduced to proving that $D$ is fully faithful, which is a special case of Corollary 9.1.9.11.

We now prove the converse. Assume that $D$ is an equivalence of $\infty $-categories; we wish to show that the pair $(S_ L, S_ R)$ is a factorization system on $\operatorname{\mathcal{C}}$. Using Remark 9.1.9.3, we see that $D$ 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 $f_{R} \in S_{R}$. Using Corollary 9.1.9.11, we deduce that $S_{L}$ is left orthogonal to $S_{R}$. We will complete the proof by showing that $S_{L}$ is closed under retracts (a similar argument shows that $S_{R}$ is closed under retracts). For this, it will suffice to show that a morphism $f$ of $\operatorname{\mathcal{C}}$ belongs to $S_{L}$ if and only $f_{R}$ 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 $S_{L}$. Then 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 4.4.1.13). $\square$