Kerodon

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

Corollary 4.5.6.21. Suppose we are given an inverse system of $\infty $-categories

\[ \cdots \rightarrow \operatorname{\mathcal{C}}(3) \rightarrow \operatorname{\mathcal{C}}(2) \rightarrow \operatorname{\mathcal{C}}(1) \rightarrow \operatorname{\mathcal{C}}(0) \]

where each of the transition functors $\operatorname{\mathcal{C}}(n) \rightarrow \operatorname{\mathcal{C}}(n-1)$ is an isofibration. Then the limit $\operatorname{\mathcal{C}}= \varprojlim _{n} \operatorname{\mathcal{C}}(n)$ is an $\infty $-category, whose core $\operatorname{\mathcal{C}}^{\simeq }$ is the inverse limit $\varprojlim _{n} \operatorname{\mathcal{C}}(n)^{\simeq }$. In other words, a morphism of $\operatorname{\mathcal{C}}$ is an isomorphism if and only if its image in each $\operatorname{\mathcal{C}}(n)$ is an isomorphism.