Corollary 9.2.7.4. Let $X$ be a Kan complex. Then $X$ is essentially finite if and only if it is homotopy equivalent to a simplicial set of the form $\operatorname{Sing}_{\bullet }(Y)$, where $Y$ is a finite CW complex.
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Proof. Assume first that $X$ is essentially finite: that is, there exists a weak homotopy equivalence $f: K \rightarrow X$, where $K$ is a finite simplicial set. Then $f$ induces a homotopy equivalence of topological spaces $|K| \rightarrow |X|$ (Corollary 3.6.4.3), and therefore a homotopy equivalence of Kan compelxes $\operatorname{Sing}_{\bullet }( |K| ) \rightarrow \operatorname{Sing}_{\bullet }( |X| )$ (Example 3.1.6.3). The unit map $X \rightarrow \operatorname{Sing}_{\bullet }( |X| )$ is a weak homotopy equivalence between Kan complexes (Theorem 3.6.4.1), and therefore a homotopy equivalence (Proposition 3.1.6.13). It follows that $X$ is homotopy equivalent to $\operatorname{Sing}_{\bullet }(Y)$, where $Y = |K|$ is a finite CW complex (Remark 1.2.3.12).
To prove the converse, it will suffice to show that if $Y$ is a finite CW complex, then the singular simplicial set $\operatorname{Sing}_{\bullet }(Y)$ is essentially finite. We proceed by induction on the number of cells of $Y$ (with respect to any choice of CW decomposition). If $Y$ is empty, there is nothing to prove. Otherwise, fix a cell $U \subseteq Y$ of maximal dimension $d$, so that $Y_0 = Y \setminus U$ is a CW complex with a smaller number of cells. Choose a point $y \in Y$. Applying Theorem 3.4.6.1, we deduce that the diagram of Kan complexes
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Sing}_{\bullet }( U \setminus \{ y\} ) \ar [r] \ar [d] & \operatorname{Sing}_{\bullet }( U ) \ar [d] \\ \operatorname{Sing}_{\bullet }( Y \setminus \{ y\} ) \ar [r] & \operatorname{Sing}_{\bullet }(Y) } \]
is a homotopy pushout square of simplicial sets, and therefore a pushout square in the $\infty $-category $\operatorname{\mathcal{S}}$ (Example 7.6.3.3). Note that $U$ is contractible and $U \setminus \{ y\} $ is homotopy equivalent to a sphere $S^{n-1}$, so that $\operatorname{Sing}_{\bullet }(U)$ and $\operatorname{Sing}_{\bullet }(U \setminus \{ y\} )$ are weakly homotopy equivalent to $\Delta ^0$ and $\operatorname{\partial \Delta }^ n$, respectively. The topological space $Y \setminus \{ y\} $ is homotopy equivalent to $Y_0$, so that $\operatorname{Sing}_{\bullet }( Y \setminus \{ y\} )$ is essentially finite by virtue of our inductive hypothesis. Applying Proposition 9.2.7.3, we conclude that $\operatorname{Sing}_{\bullet }(Y)$ is also essentially finite. $\square$