Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Theorem 4.4.4.4. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $f,f': X \rightarrow \operatorname{\mathcal{C}}$ be diagrams in $\operatorname{\mathcal{C}}$ indexed by a simplicial set $X$, and let $u: f \rightarrow f'$ be a natural transformation. Then $u$ is a natural isomorphism if and only if, for every vertex $x \in X$, the induced map $\operatorname{ev}_{x}(u): f(x) \rightarrow f'(x)$ is an isomorphism in the $\infty $-category $\operatorname{\mathcal{C}}$.

Proof of Theorem 4.4.4.4. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $X$ be a simplicial set, and let $u: f \rightarrow f'$ be a natural transformation between diagrams $f,f': X \rightarrow \operatorname{\mathcal{C}}$ having the property that, for every vertex $x \in X$, the induced map $u_ x: f(x) \rightarrow f'(x)$ is an isomorphism in the $\infty $-category $\operatorname{\mathcal{C}}$. We wish to show that $u$ is an isomorphism in the functor $\infty $-category $\operatorname{Fun}(X, \operatorname{\mathcal{C}})$. We will prove this by verifying the criterion of Theorem 4.4.2.5. Let $n \geq 2$ be an integer, and suppose we are given a morphism of simplicial sets $\sigma _0: \Lambda ^{n}_{0} \rightarrow \operatorname{Fun}(X, \operatorname{\mathcal{C}})$ for which the composite map

\[ \Delta ^1 \simeq \operatorname{N}_{\bullet }( \{ 0,1\} ) \hookrightarrow \Lambda ^ n_0 \xrightarrow {\sigma _0} \operatorname{Fun}(X, \operatorname{\mathcal{C}}) \]

is equal to $u$. We wish to show that $\sigma _0$ can be extended to an $n$-simplex $\sigma $ of the diagram $\infty $-category $\operatorname{Fun}(X, \operatorname{\mathcal{C}})$. Let us identify $\sigma _0$ with a morphism of simplicial sets $\rho _0: X \times \Lambda ^{n}_{0} \rightarrow \operatorname{\mathcal{C}}$; we wish to show that $\rho _0$ can be extended to a morphism $\rho : X \times \Delta ^ n \rightarrow \operatorname{\mathcal{C}}$. This follows by applying Lemma 4.4.4.8 in the special case $B = X$ and $A = \emptyset $. $\square$