Kerodon

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$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Corollary 4.4.3.18. 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.

Proof. It follows from Remark 4.1.1.14 that $\operatorname{\mathcal{C}}$ is an $\infty $-category. Moreover, each of the projection maps $\operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}(n)$ carries the core $\operatorname{\mathcal{C}}^{\simeq }$ into $\operatorname{\mathcal{C}}(n)^{\simeq }$ (Remark 4.4.3.5), so $\operatorname{\mathcal{C}}^{\simeq }$ is contained in the limit $\varprojlim _{n} \operatorname{\mathcal{C}}(n)^{\simeq }$. To prove the reverse inclusion, it will suffice to show that the limit $\varprojlim _{n} \operatorname{\mathcal{C}}(n)^{\simeq }$ is a Kan complex (Corollary 4.4.3.17). This follows from Proposition 3.3.9.1, since each of the transition maps $\operatorname{\mathcal{C}}(n)^{\simeq } \rightarrow \operatorname{\mathcal{C}}(n-1)^{\simeq }$ is a Kan fibration (Proposition 4.4.3.7). $\square$