Proposition 4.6.7.22 (02J2)

Proposition 4.6.7.22. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $f: X \rightarrow Y$ be a morphism in $\operatorname{\mathcal{C}}$. The following conditions are equivalent:

$(1)$: The morphism $f$ is an isomorphism from $X$ to $Y$ in the $\infty $-category $\operatorname{\mathcal{C}}$ (Definition 1.4.6.1).
$(2)$: The morphism $f$ is final when regarded as an object of the slice $\infty $-category $\operatorname{\mathcal{C}}_{/Y}$.
$(2')$: The morphism $f$ is final when regarded as an object of the oriented fiber product $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \{ Y\} $.
$(3)$: The morphism $f$ is initial when regarded as an object of the coslice $\infty $-category $\operatorname{\mathcal{C}}_{X/}$.
$(3')$: The morphism $f$ is initial when regarded as an object of the oriented fiber product $\{ X\} \operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}$.

Proof. The equivalences $(2) \Leftrightarrow (2')$ and $(3) \Leftrightarrow (3')$ follow from Corollaries 4.6.4.18 and 4.6.7.20. We will complete the proof by showing that $(1) \Leftrightarrow (3)$; the equivalence $(1) \Leftrightarrow (2)$ follows by applying the same argument in the $\infty $-category $\operatorname{\mathcal{C}}^{\operatorname{op}}$. By virtue of Corollary 4.6.7.13, condition $(3)$ is equivalent to the requirement that the restriction map $\operatorname{\mathcal{C}}_{f/} \rightarrow \operatorname{\mathcal{C}}_{X/}$ is a trivial Kan fibration: that is, every lifting problem

4.62

\begin{equation} \begin{gathered}\label{equation:limit-of-point} \xymatrix@R =50pt@C=50pt{ \operatorname{\partial \Delta }^{n} \ar [r] \ar [d] & \operatorname{\mathcal{C}}_{f/} \ar [d] \\ \Delta ^ n \ar@ {-->}[ur] \ar [r] & \operatorname{\mathcal{C}}_{X/} } \end{gathered} \end{equation}

admits a solution. Using the isomorphism of simplicial sets

\[ (\Delta ^1 \star \operatorname{\partial \Delta }^ n) \coprod _{ \{ 0\} \star \operatorname{\partial \Delta }^ n } ( \{ 0\} \star \Delta ^ n ) \simeq \Lambda ^{n+2}_{0} \]

supplied by Lemma 4.3.6.15, we can identify (4.62) with a lifting problem

\[ \xymatrix@R =50pt@C=50pt{ \Lambda ^{n+2}_{0} \ar [r]^-{\sigma _0} \ar [d] & \operatorname{\mathcal{C}}\ar [d] \\ \Delta ^{n+2} \ar@ {-->}[ur]^{\sigma } \ar [r] & \Delta ^0, } \]

where $\sigma _0$ carries the initial edge $\Delta ^1 \simeq \operatorname{N}_{\bullet }( \{ 0 < 1 \} ) \subseteq \Lambda ^{n+2}_{0}$ to the morphism $f$. The equivalence $(1) \Leftrightarrow (3)$ now follows from the criterion of Theorem 4.4.2.6. $\square$