9.3.7 Uniqueness of Factorizations
Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $S_{L}$ and $S_{R}$ be collections of morphisms of $\operatorname{\mathcal{C}}$, and let $f: X \rightarrow Z$ be a morphism which factors as a composition
\[ X \xrightarrow { f_{L} } Y \xrightarrow { f_{R} } Z \]
where $f_{L}$ belongs to $S_{L}$ and $f_{R}$ belongs to $S_{R}$. Our goal in this section is to show that, if $S_{L}$ is left orthogonal to $S_{R}$, then this factorization is essentially unique (Theorem 9.3.7.2).
Notation 9.3.7.1. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $S_{L}$ and $S_{R}$ be collections of morphisms of $\operatorname{\mathcal{C}}$. 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 diagrams
\[ \xymatrix@C =50pt@R=50pt{ & Y \ar [dr]^{f_{R}} & \\ X \ar [ur]^{ f_{L} } \ar [rr]^-{ f } & & Z } \]
where $f$ belongs to $S_{L}$, and $\operatorname{Fun}_{R}( \Delta ^2, \operatorname{\mathcal{C}})$ the full subcategory of $\operatorname{Fun}(\Delta ^2, \operatorname{\mathcal{C}})$ spanned by those diagrams where $g$ belongs to $S_{R}$. We let $\operatorname{Fun}_{LR}( \Delta ^2, \operatorname{\mathcal{C}})$ denote the intersection $\operatorname{Fun}_{L}( \Delta ^2, \operatorname{\mathcal{C}}) \cap \operatorname{Fun}_{R}( \Delta ^2, \operatorname{\mathcal{C}})$.
We can now formulate our main result.
Theorem 9.3.7.2. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $S_{L}$ and $S_{R}$ be collections of morphisms of $\operatorname{\mathcal{C}}$. If $S_{L}$ is left orthogonal to $S_{R}$, then the restriction functor
\[ D: \operatorname{Fun}_{LR}( \Delta ^2, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}}) \quad \quad \sigma \mapsto d^{2}_{1}(\sigma ) \]
is fully faithful. The converse holds if $S_{L}$ and $S_{R}$ contain all identity morphisms of $\operatorname{\mathcal{C}}$.
The proof of Theorem 9.3.7.2 will require some preliminaries. We begin by giving another description of the space of solutions to a lifting problem.
Notation 9.3.7.3. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category containing morphisms $f: A \rightarrow B$ and $g: X \rightarrow Y$. We let ${}_{f}\operatorname{Fun}(\Delta ^3, \operatorname{\mathcal{C}})_{g}$ denote the the iterated fiber product
\[ \{ f\} \times _{ \operatorname{Fun}( \operatorname{N}_{\bullet }( \{ 0 < 1 \} ) } \operatorname{Fun}( \Delta ^3, \operatorname{\mathcal{C}}) \times _{\operatorname{Fun}_{ \operatorname{N}_{\bullet }( \{ 2 < 3 \} )}} \{ g\} , \]
whose objects can be identified with diagrams
\[ \xymatrix@R =50pt@C=50pt{ A \ar [d]^{f} \ar [r] & X \ar [d]^{g} \\ B \ar [ur] \ar [r] & Y } \]
in the $\infty $-category $\operatorname{\mathcal{C}}$.
Lemma 9.3.7.4. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category containing morphisms $f: A \rightarrow B$ and $g: X \rightarrow Y$. Then precomposition with the inclusion map $\operatorname{N}_{\bullet }( \{ 1 < 2 \} ) \hookrightarrow \Delta ^3$ induces a trivial Kan fibration of simplicial sets ${}_{f}\operatorname{Fun}(\Delta ^3, \operatorname{\mathcal{C}})_{g} \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}( B, X )$. In particular, the simplicial set ${}_{f}\operatorname{Fun}(\Delta ^3, \operatorname{\mathcal{C}})_{g}$ is a Kan complex.
Proof.
By construction, we have a pullback diagram of simplicial sets
\[ \xymatrix@R =50pt@C=50pt{ \empty {}_{f}\operatorname{Fun}(\Delta ^3, \operatorname{\mathcal{C}})_{g} \ar [d] \ar [r] & \operatorname{Fun}( \Delta ^3, \operatorname{\mathcal{C}}) \ar [d] \\ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(B,X) \ar [r] & \operatorname{Fun}( \operatorname{Spine}[3], \operatorname{\mathcal{C}}), } \]
where the right vertical map is a trivial Kan fibration (see Example 1.5.7.7).
$\square$
Notation 9.3.7.5. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category containing morphisms $f: A \rightarrow B$ and $g: X \rightarrow Y$. We let $\widetilde{f} = s^{1}_{1}(f)$ and $\widetilde{g} = s^{1}_{0}(g)$ denote the degenerate $2$-simplices of $\operatorname{\mathcal{C}}$ depicted in the diagram
\[ A \xrightarrow {f} B \xrightarrow {\operatorname{id}} B \quad \quad X \xrightarrow {\operatorname{id}} X \xrightarrow {g} Y. \]
Let $\beta : \Delta ^2 \times \Delta ^1 \rightarrow \Delta ^3$ denote the morphism of simplicial sets given on vertices by the formulae
\[ \beta (0,0) = 0 \quad \quad \beta (1,0) = 1 = \beta (2,0) \quad \quad \beta (0,1) = 2 = \beta (1,1) \quad \quad \beta (2,1) = 3. \]
Then precomposition with $\beta $ determines a functor $\operatorname{Fun}( \Delta ^3, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}( \Delta ^2 \times \Delta ^1, \operatorname{\mathcal{C}})$, which restricts to a map of Kan complexes $T: {}_{f}\operatorname{Fun}(\Delta ^3, \operatorname{\mathcal{C}})_{g} \rightarrow \operatorname{Hom}_{ \operatorname{Fun}(\Delta ^2,\operatorname{\mathcal{C}})}( \widetilde{f}, \widetilde{g} )$. Concretely, $T$ carries a $3$-simplex
\[ \xymatrix@R =50pt@C=50pt{ A \ar [d]^{f} \ar [r]^-{u} & X \ar [d]^{g} \\ B \ar [ur]^{v} \ar [r]^-{w} & Y } \]
to the morphism in $\operatorname{Fun}(\Delta ^2, \operatorname{\mathcal{C}})$ depicted in the diagram
\[ \xymatrix@R =50pt@C=50pt{ A \ar [r]^-{f} \ar [d]^{u} & B \ar [d]^{v} \ar [r]^-{\operatorname{id}_{B}} & B \ar [d]^{w} \\ X \ar [r]^-{\operatorname{id}_{X}} & X \ar [r]^-{g} & Y. } \]
Lemma 9.3.7.6. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category containing morphisms $f: A \rightarrow B$ and $g: X \rightarrow Y$. Then the comparison map
\[ T: {}_{f}\operatorname{Fun}(\Delta ^3, \operatorname{\mathcal{C}})_{g} \rightarrow \operatorname{Hom}_{ \operatorname{Fun}(\Delta ^2,\operatorname{\mathcal{C}})}( \widetilde{f}, \widetilde{g} ) \]
of Notation 9.3.7.5 is a homotopy equivalence.
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
\[ {}_{f}\operatorname{Fun}(\Delta ^3, \operatorname{\mathcal{C}})_{g} \xrightarrow {T} \operatorname{Hom}_{ \operatorname{Fun}(\Delta ^2,\operatorname{\mathcal{C}})}( \widetilde{f}, \widetilde{g} ) \xrightarrow {R} \operatorname{Hom}_{ \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}}) }(f, \operatorname{id}_{X} ) \xrightarrow {Q} \operatorname{Hom}_{\operatorname{\mathcal{C}}}( B, X ) \]
is a homotopy equivalence, which follows from Lemma 9.3.7.4.
$\square$
In the situation of Lemma 9.3.7.6, restriction to the “long edge” of $\Delta ^2$ determines an inner fibration of $\infty $-categories $\operatorname{Fun}( \Delta ^2, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}})$ (Corollary 4.1.4.2), and therefore induces a Kan fibration of mapping spaces
\[ \operatorname{Hom}_{ \operatorname{Fun}( \Delta ^2, \operatorname{\mathcal{C}}) }( \widetilde{f}, \widetilde{g} ) \rightarrow \operatorname{Hom}_{ \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}}) }( f, g ). \]
Lemma 9.3.7.7. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category containing a lifting problem $\sigma :$
9.24
\begin{equation} \begin{gathered}\label{equation:other-comparison-map-for-Sol} \xymatrix@R =50pt@C=50pt{ A \ar [d]^{f} \ar [r] & X \ar [d]^{g} \\ B \ar [r] \ar@ {-->}[ur]^{s} & Y, } \end{gathered} \end{equation}
which we identify with a morphism from $f$ to $g$ in the $\infty $-category $\operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}})$. Then the comparison map $T: {}_{f}\operatorname{Fun}(\Delta ^3, \operatorname{\mathcal{C}})_{g} \rightarrow \operatorname{Hom}_{ \operatorname{Fun}(\Delta ^2,\operatorname{\mathcal{C}})}( \widetilde{f}, \widetilde{g} )$ of Notation 9.3.7.5 restricts to a homotopy equivalence of Kan complexes
\[ T_0: \operatorname{Sol}( \sigma ) \rightarrow \operatorname{Hom}_{ \operatorname{Fun}(\Delta ^2,\operatorname{\mathcal{C}})}( \widetilde{f}, \widetilde{g} ) \times _{ \operatorname{Hom}_{\operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}}) }(f,g) } \{ \sigma \} . \]
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 9.3.4.1). 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 3.3.7.5 to the diagram of Kan complexes
\[ \xymatrix@R =50pt@C=50pt{ \empty {}_{f}\operatorname{Fun}(\Delta ^3, \operatorname{\mathcal{C}})_{g} \ar [dr]^{U_0} \ar [rr]^{T} & & \operatorname{Hom}_{ \operatorname{Fun}(\Delta ^2,\operatorname{\mathcal{C}})}( \widetilde{f}, \widetilde{g} ) \ar [dl] \\ & \operatorname{Hom}_{ \operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{C}}) }( f, g), & } \]
since $T$ is a homotopy equivalence (Lemma 9.3.7.6).
$\square$
Corollary 9.3.7.8. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category containing $2$-simplices $\sigma $ and $\tau $. Suppose that the initial edge $f = d^{2}_{2}(\sigma )$ is left orthogonal to the final edge $g = d^{2}_{0}(\tau )$. Then the restriction map $\theta : \operatorname{Hom}_{ \operatorname{Fun}( \Delta ^2, \operatorname{\mathcal{C}}) }(\sigma , \tau ) \rightarrow \operatorname{Hom}_{ \operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{C}}) }(f,g)$ is a trivial Kan fibration.
Proof.
By virtue of Proposition 3.3.7.6, 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 $\gamma (i,j) = \begin{cases} 1 & \end{cases}$
$\square$
if $$(i,j) = (0,2)$ $
i otherwise.
$$. Then the composition
\[ \Delta ^2 \times \Delta ^1 \xrightarrow {\gamma } \Delta ^2 \xrightarrow {\sigma } \operatorname{\mathcal{C}} \]
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 5.3.7.4 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 9.3.7.8 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 9.3.7.7 guarantees that every fiber of $\theta $ is homotopy equivalent to the space of solutions to some lifting problem
\[ \xymatrix@R =50pt@C=50pt{ A \ar [d]^{f} \ar [r] & X \ar [d]^{g} \\ B \ar [r] \ar@ {-->}[ur]^{s} & Y, } \]
which is contractible by virtue of our assumption that $f$ is left orthogonal to $g$.
Proof of Theorem 9.3.7.2.
Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $S_{L}$ and $S_{R}$ be collections of morphisms of $\operatorname{\mathcal{C}}$. Suppose first that $S_{L}$ is left orthogonal to $S_{R}$. In this case Corollary 9.3.7.8 guarantees that the restriction map
\[ \theta : \operatorname{Hom}_{ \operatorname{Fun}( \Delta ^2, \operatorname{\mathcal{C}}) }(\sigma , \sigma ') \rightarrow \operatorname{Hom}_{ \operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{C}}) }( d^{2}_{1}(\sigma ), d^{2}_{1}(\sigma ') ) \]
is a homotopy equivalence whenever $\sigma $ belongs to $\operatorname{Fun}_{L}( \Delta ^2, \operatorname{\mathcal{C}})$ and $\sigma '$ belongs to $\operatorname{Fun}_{R}( \Delta ^2, \operatorname{\mathcal{C}})$. It follows that the functor
\[ D: \operatorname{Fun}_{LR}( \Delta ^2, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}}) \quad \quad \sigma \mapsto d^{2}_{1}(\sigma ) \]
is fully faithful.
We now prove the converse. Assume that $D$ is fully faithful and that $S_{L}$ and $S_{R}$ contain all identity morphisms of $\operatorname{\mathcal{C}}$; we wish to show that $S_{L}$ is left orthogonal to $S_{R}$. Suppose we are given a lifting problem $\tau :$
\[ \xymatrix@R =50pt@C=50pt{ A \ar [d]^{f} \ar [r] & X \ar [d]^{g} \\ B \ar [r] \ar@ {-->}[ur]^{s} & Y } \]
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 $\widetilde{f} = s^{1}_{1}(f)$ and $\widetilde{g} = s^{1}_{0}(g)$ denote the degenerate $2$-simplices of $\operatorname{\mathcal{C}}$ defined in Notation 9.3.7.3. 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 9.3.7.7 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 3.3.7.6.
$\square$