Variant 7.6.5.9 (Sequential Colimits in $\operatorname{\mathcal{S}}$). Suppose we are given a collection of Kan complexes $\{ X(n) \} _{n \geq 0}$ and morphisms $f_ n: X(n) \rightarrow X(n+1)$, which we view as a diagram
\[ X(0) \xrightarrow { f_0 } X(1) \xrightarrow { f_1} X(2) \xrightarrow { f_2} X(3) \rightarrow \cdots \]
Let $\varinjlim _{n} X(n)$ denote the colimit of this diagram (formed in the ordinary category of simplicial sets). Then $\varinjlim _{n} X(n)$ is also a Kan complex, which is also a colimit of the associated diagram $\operatorname{N}_{\bullet }( \operatorname{\mathbf{Z}}_{\geq 0} ) \rightarrow \operatorname{\mathcal{S}}$. See Variant 9.1.7.4.