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Theorem 4.6.4.8. Let $q: X \rightarrow S$ be an isofibration of simplicial sets, let $g: B \rightarrow S$ be a morphism of simplicial sets, and let $f_0, f_1: B \rightarrow X$ be morphisms satisfying $q \circ f_0 = g = q \circ f_1$. Let $A \subseteq B$ be a simplicial subset satisfying $f_0|_{A} = f_1|_{A}$. The following conditions are equivalent:

$(1)$

The diagrams $f_0$ and $f_1$ are isomorphic when regarded as objects of the $\infty $-category $\operatorname{Fun}_{A/ \, /S}(B, X)$ (see Proposition 4.1.4.6).

$(2)$

There exists a factorization of the fold map $(\operatorname{id}_ B, \operatorname{id}_ B): B \coprod _{A} B \rightarrow B$ as a composition

\[ B \coprod _{A} B \xrightarrow { (s_0, s_1)} \overline{B} \xrightarrow {\pi } B, \]

where $\pi $ is a categorical equivalence and the lifting problem

\[ \xymatrix@R =50pt@C=50pt{ B \coprod _{A} B \ar [d]_{ (s_0, s_1)} \ar [r]^-{ (f_0, f_1)} & X \ar [d]^{q} \\ \overline{B} \ar [r]^-{ g \circ \pi } \ar@ {-->}[ur]^{ \overline{f} } & S } \]

admits a solution.

$(3)$

For every factorization of the fold map $(\operatorname{id}_ B, \operatorname{id}_ B): B \coprod _{A} B \rightarrow B$ as a composition

\[ B \coprod _{A} B \xrightarrow { (s_0, s_1)} \overline{B} \xrightarrow {\pi } B, \]

where the map $(s_0, s_1): B \coprod _{A} B \rightarrow \overline{B}$ is a monomorphism, the lifting problem

\[ \xymatrix@R =50pt@C=50pt{ B \coprod _{A} B \ar [d]_{ (s_0, s_1)} \ar [r]^-{ (f_0, f_1)} & X \ar [d]^{q} \\ \overline{B} \ar [r]^-{ g \circ \pi } \ar@ {-->}[ur]^{ \overline{f} } & S } \]

admits a solution.

Proof. By virtue of Corollary 4.4.3.12, condition $(1)$ is satisfied if and only if there exists a morphism of simplicial sets $u: Q \rightarrow \operatorname{Fun}_{A/\, /S}( B, X)$, where $Q$ is a contractible Kan complex, and a pair of vertices $x_0, x_1 \in Q$ satisfying $u(x_0) = f_0$ and $u(x_1) =f_1$. Moreover, we may assume (modifying $Q$ if necessary) that the vertices $x_0$ and $x_1$ are distinct. In this case, let $\overline{B} = A \coprod _{ (Q \times A) } (Q \times B)$ and let

\[ B \coprod _{A} B \xrightarrow { (s_0, s_1)} \overline{B} \xrightarrow {\pi } B \]

be the categorical mapping cylinder described in Example 4.6.4.6. Unwinding the definitions, we see that morphisms $u: Q \rightarrow \operatorname{Fun}_{A/\, /S}( B, X)$ satisfying $u(x_0) = f_0$ and $u(x_1) = f_1$ can be identified with solutions to the lifting problem

\[ \xymatrix@R =50pt@C=50pt{ B \coprod _{A} B \ar [d]_{ (s_0, s_1)} \ar [r]^-{ (f_0, f_1)} & X \ar [d]^{q} \\ \overline{B} \ar [r]^-{ g \circ \pi } \ar@ {-->}[ur] & S. } \]

This proves that $(3) \Rightarrow (1) \Rightarrow (2)$.

We will complete the proof by showing that $(2) \Rightarrow (3)$. Assume that $(2)$ is satisfied, so that the fold map $(\operatorname{id}_ B, \operatorname{id}_ B): B \coprod _{A} B \rightarrow B$ factors as a composition

\[ B \coprod _{A} B \xrightarrow { (s_0, s_1)} \overline{B} \xrightarrow {\pi } B \]

where $\pi $ is a categorical equivalence and there exists a morphism $\overline{f}: \overline{B} \rightarrow X$ for which the diagram

\[ \xymatrix@R =50pt@C=50pt{ B \coprod _{A} B \ar [d]_{ (s_0, s_1)} \ar [r]^-{ (f_0, f_1)} & X \ar [d]^{q} \\ \overline{B} \ar [r]^-{ g \circ \pi } \ar [ur]^{ \overline{f} } & S } \]

commutes. Using Exercise 3.1.6.11, we can factor $\pi $ as a composition

\[ \overline{B} \xrightarrow {j} \overline{B}' \xrightarrow {\pi '} B, \]

where $j$ is a monomorphism and $\pi '$ is a trivial Kan fibration. Then $\pi '$ is also a categorical equivalence (Proposition 4.5.2.9), so the morphism $j$ is a categorical equivalence (Remark 4.5.2.6). Our assumption that $q$ is an isofibration guarantees that the lifting problem

\[ \xymatrix@R =50pt@C=50pt{ \overline{B} \ar [d]_{j} \ar [r]^-{ \overline{f} } & X \ar [d]^{q} \\ \overline{B}' \ar [r]^-{ g \circ \pi ' } \ar@ {-->}[ur]^{ \overline{f}' } & S } \]

admits a solution $\overline{f}': \overline{B}' \rightarrow X$.

We now show that condition $(3)$ is satisfied. Suppose that we are given another factorization of the fold map $(\operatorname{id}_ B, \operatorname{id}_ B): (\operatorname{id}_ B, \operatorname{id}_ B): B \coprod _{A} B \rightarrow B$ as a composition

\[ B \coprod _{A} B \xrightarrow {\iota } \overline{B}'' \xrightarrow {\pi ''} B, \]

where $\iota $ is a monomorphism. We wish to show that the lifting problem

\[ \xymatrix@R =50pt@C=50pt{ B \coprod _{A} B \ar [d]_{ \iota } \ar [r]^-{ (f_0, f_1)} & X \ar [d]^{q} \\ \overline{B}'' \ar [r]^-{ g \circ \pi ''} \ar@ {-->}[ur]^{\overline{f}''} & S } \]

admits a solution $\overline{f}'': \overline{B}'' \rightarrow X$. We first observe that the lifting problem

\[ \xymatrix@R =50pt@C=50pt{ B \coprod _{A} B \ar [r]^-{j \circ (s_0, s_1)} \ar [d]_{\iota } & \overline{B}' \ar [d]^{ \pi ' } \\ \overline{B}'' \ar [r]^-{\pi ''} \ar@ {-->}[ur]^{v} & B } \]

admits a solution $v: \overline{B}'' \rightarrow \overline{B}'$, since $\iota $ is a monomorphism and $\pi '$ is a trivial Kan fibration. We now conclude the proof by setting $\overline{f}'' = \overline{f}' \circ v$. $\square$