# Kerodon

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Remark 4.1.1.14. Suppose we are given an inverse system of simplicial sets

$\cdots \rightarrow X(4) \rightarrow X(3) \rightarrow X(2) \rightarrow X(1) \rightarrow X(0),$

where each of the transition maps $X(n) \rightarrow X(n-1)$ is an inner fibration. Then each of the projection maps $\varprojlim _{n} X(n) \rightarrow X(m)$ is an inner fibration. In particular, if any of the simplicial sets $X(m)$ is an $\infty$-category, then the inverse limit $\varprojlim _{n} X(n)$ is also an $\infty$-category.