Corollary 4.6.7.24 (02J5)

Corollary 4.6.7.24. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $Y$ be an object of $\operatorname{\mathcal{C}}$. The following conditions are equivalent:

$(1)$

The object $Y \in \operatorname{\mathcal{C}}$ is final.

$(2)$

There exists a functor $F: \operatorname{\mathcal{C}}^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$ satisfying $F|_{\operatorname{\mathcal{C}}} = \operatorname{id}_{\operatorname{\mathcal{C}}}$ and for which the composition

\[ \Delta ^1 \simeq \{ Y\} ^{\triangleright } \hookrightarrow \operatorname{\mathcal{C}}^{\triangleright } \xrightarrow {F} \operatorname{\mathcal{C}} \]

is the identity morphism $\operatorname{id}_{Y}$ (in particular, $F$ carries the cone point of $\operatorname{\mathcal{C}}^{\triangleright }$ to the object $Y$).

$(3)$

The inclusion map $\{ Y\} \hookrightarrow \operatorname{\mathcal{C}}$ is right anodyne.

Proof. The equivalence $(1) \Leftrightarrow (2)$ is a reformulation of Corollary 4.6.7.23. We next show that $(2)$ implies $(3)$. If condition $(2)$ is satisfied, then we have a commutative diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ \{ Y\} \ar [r] \ar [d] & \{ Y\} ^{\triangleright } \ar [r] \ar [d] & \{ Y\} \ar [d] \\ \operatorname{\mathcal{C}}\ar [r] & \operatorname{\mathcal{C}}^{\triangleright } \ar [r]^-{F} & \operatorname{\mathcal{C}}} \]

where the horizontal compositions are the identity. Since the inclusion $\{ Y\} ^{\triangleright } \hookrightarrow \operatorname{\mathcal{C}}^{\triangleright }$ is right anodyne (Lemma 4.3.7.8), it follows that the inclusion $\{ Y\} \hookrightarrow \operatorname{\mathcal{C}}$ is also right anodyne.

We now complete the proof by showing that $(3)$ implies $(2)$. Suppose that the inclusion $\{ Y\} \hookrightarrow \operatorname{\mathcal{C}}$ is right anodyne; we wish to show that there exists a functor $F: \operatorname{\mathcal{C}}^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$ satisfying $F|_{ \operatorname{\mathcal{C}}} = \operatorname{id}_{\operatorname{\mathcal{C}}}$ and $F|_{ \{ Y\} ^{\triangleright } } = \operatorname{id}_ Y$. For this, it will suffice to show that the inclusion map

\[ \operatorname{\mathcal{C}}\coprod _{ \{ Y\} } \{ Y\} ^{\triangleright } \hookrightarrow \operatorname{\mathcal{C}}^{\triangleright } \]

is inner anodyne, which is a special case of Proposition 4.3.6.4. $\square$