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Theorem 7.2.3.1 (Joyal). Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a morphism of simplicial sets, where $\operatorname{\mathcal{D}}$ is an $\infty $-category. Then:

$(1)$

The morphism $F$ is left cofinal if and only if, for every object $X \in \operatorname{\mathcal{D}}$, the simplicial set $\operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}_{/X}$ is weakly contractible.

$(2)$

The morphism $F$ is right cofinal if and only if, for every object $X \in \operatorname{\mathcal{D}}$, the simplicial set $\operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}_{X/}$ is weakly contractible.

Proof of Theorem 7.2.3.1. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a morphism of simplicial sets, where $\operatorname{\mathcal{D}}$ is an $\infty $-category. We will show that $F$ is right cofinal if and only if, for every object $X \in \operatorname{\mathcal{D}}$, the simplicial set $\operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}_{X/}$ is weakly contractible; the analogous characterization of left cofinal morphisms follows by a similar argument.

Suppose first that $F$ is right cofinal. For every object $X \in \operatorname{\mathcal{D}}$, the projection map $\operatorname{\mathcal{D}}_{X/} \rightarrow \operatorname{\mathcal{D}}$ is a left fibration (Proposition 4.3.6.1), and therefore a cocartesian fibration (Proposition 5.1.4.14). Applying Proposition 7.2.3.12, we conclude that the projection map $\operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}_{X/} \rightarrow \operatorname{\mathcal{D}}_{X/}$ is also right cofinal. In particular, it is a weak homotopy equivalence (Proposition 7.2.1.5). Since the $\infty $-category $\operatorname{\mathcal{D}}_{X/}$ has an initial object (Proposition 4.6.6.23), it is weakly contractible, so that the fiber product $\operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}_{X/}$ is also weakly contractible.

We now prove the converse. Assume that, for every object $X \in \operatorname{\mathcal{D}}$, the simplicial set $\operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}_{X/}$ is weakly contractible. We wish to show that $F$ is right cofinal. Using Proposition 4.1.3.2, we can factor $F$ as a composition

\[ \operatorname{\mathcal{C}}\xrightarrow {F'} \operatorname{\mathcal{C}}' \xrightarrow {F''} \operatorname{\mathcal{D}}, \]

where $F'$ is inner anodyne and $F''$ is an inner fibration. Since $F'$ is left cofinal (Proposition 7.2.1.3), it will suffice to show that $F''$ is right cofinal (Proposition 7.2.1.6). For every object $X \in \operatorname{\mathcal{D}}$, Proposition 5.3.6.1 guarantees that the induced map $\operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}_{X/} \hookrightarrow \operatorname{\mathcal{C}}' \times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}_{X/}$ is a categorical equivalence. In particular, it is a weak homotopy equivalence (Remark 4.5.3.4), so that $\operatorname{\mathcal{C}}' \times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}_{X/}$ is also weakly contractible. We may therefore replace $\operatorname{\mathcal{C}}$ by $\operatorname{\mathcal{C}}'$ and thereby reduce to the case where $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is an inner fibration, so that $\operatorname{\mathcal{C}}$ is also an $\infty $-category (Remark 4.1.1.9).

Let $\operatorname{ev}_0, \operatorname{ev}_1: \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{D}}) \rightarrow \operatorname{\mathcal{D}}$ denote the functors given by evaluation at the vertices $0,1 \in \operatorname{\mathcal{D}}$, and let $\delta : \operatorname{\mathcal{D}}\hookrightarrow \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{D}})$ be the diagonal map. Note that there is a unique natural transformation from $\operatorname{id}_{ \Delta ^1 }$ to the constant map $\Delta ^1 \twoheadrightarrow \{ 1\} \hookrightarrow \Delta ^1$, which induces a natural transformation $h: \operatorname{id}_{ \operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{D}}) } \rightarrow \delta \circ \operatorname{ev}_{1}$. Let $\operatorname{\mathcal{M}}$ denote the oriented fiber product $\operatorname{\mathcal{D}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{D}}} \operatorname{\mathcal{C}}= \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{D}}) \times _{ \operatorname{Fun}( \{ 1\} , \operatorname{\mathcal{D}}) } \operatorname{Fun}(\{ 1\} , \operatorname{\mathcal{C}})$ of Construction 4.6.4.1, so that $\operatorname{ev}_{0}$ and $\operatorname{ev}_{1}$ lift to functors

\[ \operatorname{\mathcal{D}}\xleftarrow { \widetilde{\operatorname{ev}}_0 } \operatorname{\mathcal{M}}\xrightarrow { \widetilde{ \operatorname{ev}}_{1} } \operatorname{\mathcal{C}}, \]

the diagonal map $\delta $ lifts to a functor $\widetilde{\delta }: \operatorname{\mathcal{C}}\hookrightarrow \operatorname{\mathcal{M}}$, and $h$ lifts to a natural transformation $\widetilde{h}: \operatorname{id}_{\operatorname{\mathcal{M}}} \rightarrow \widetilde{\delta } \circ \widetilde{\operatorname{ev}}_1$. Note that $\widetilde{h}$ can be identified with a morphism of simplicial sets $\Delta ^1 \times \operatorname{\mathcal{M}}\rightarrow \operatorname{\mathcal{M}}$ which fits into a commutative diagram

\[ \xymatrix@R =50pt@C=50pt{ \{ 0\} \times \operatorname{\mathcal{C}}\ar [r] \ar [d]^{\widetilde{\delta }} & (\Delta ^1 \times \operatorname{\mathcal{C}}) \coprod _{ (\{ 1\} \times \operatorname{\mathcal{C}}) } (\{ 1\} \times \operatorname{\mathcal{M}}) \ar [r] \ar [d]^{\iota } & \operatorname{\mathcal{C}}\ar [d]^{ \widetilde{\delta } } \\ \{ 0\} \times \operatorname{\mathcal{M}}\ar [r] & \Delta ^1 \times \operatorname{\mathcal{M}}\ar [r]^-{ \widetilde{h} } & \operatorname{\mathcal{M}}, } \]

where the horizontal compositions are the identity. It follows that $\widetilde{\delta }$ is a retract of $\iota $. Since $\iota $ is right anodyne (Proposition 4.2.5.3), $\widetilde{\delta }$ is also right anodyne, and therefore right cofinal (Proposition 7.2.1.3).

The functor $\widetilde{\operatorname{ev}}_0: \operatorname{\mathcal{M}}\rightarrow \operatorname{\mathcal{D}}$ is a cartesian fibration (Proposition 5.3.7.1). Moreover, for each object $X \in \operatorname{\mathcal{D}}$, the fiber $\widetilde{\operatorname{ev}}_{0}^{-1} \{ X\} \simeq \{ X\} \operatorname{\vec{\times }}_{\operatorname{\mathcal{D}}} \operatorname{\mathcal{C}}$ is equivalent to the $\infty $-category $\operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}_{X/}$ (Example 5.1.6.7), and is therefore weakly contractible. Applying Corollary 6.3.5.3, we deduce that the functor $\widetilde{\operatorname{ev}}_1$ exhibits $\operatorname{\mathcal{D}}$ as a localization of $\operatorname{\mathcal{D}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{D}}} \operatorname{\mathcal{C}}$, and is therefore right cofinal (Proposition 7.2.1.9). We now observe that the functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ factors as a composition

\[ \operatorname{\mathcal{C}}\xrightarrow { \widetilde{\delta } } \operatorname{\mathcal{M}}\xrightarrow { \widetilde{\operatorname{ev}}_1 } \operatorname{\mathcal{D}}, \]

and is therefore also right cofinal (Proposition 7.2.1.6). $\square$