Warning 4.7.4.17. Corollary 4.7.4.16 is false in the case $\kappa = \aleph _0$. If $S$ is a finite simplicial set, there usually does not exist a weak homotopy equivalence $S \rightarrow X$, where $X$ is a Kan complex which is also a finite simplicial set. For example, take $S = \Delta ^2 / \operatorname{\partial \Delta }^{2}$, so that the geometric realization $| S |$ is homeomorphic to a sphere of dimension $2$. If a Kan complex $X$ is equipped with a weak homotopy equivalence $f: S \rightarrow X$, then the homotopy group $\pi _{2}(X)$ is an infinite cyclic group (generated by the homotopy class $[f]$), so that the Kan complex $X$ must contain infinitely many $2$-simplices.
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