Proposition 9.2.7.3. Let $\operatorname{\mathcal{S}}_{\mathrm{fin}}$ denote the smallest full subcategory of $\operatorname{\mathcal{S}}$ which contains $\Delta ^0$ and is closed under finite colimits. Then a Kan complex $X$ belongs to $\operatorname{\mathcal{S}}_{\mathrm{fin}}$ if and only if it is essentially finite.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. Assume first that $X$ is essentially finite: that is, there exists a weak homotopy equivalence $K \rightarrow X$, where $K$ is a finite simplicial set. Then $X$ can be realized as the colimit of the constant diagram $K \rightarrow \{ \Delta ^0 \} \hookrightarrow \operatorname{\mathcal{S}}$ (Example 7.1.2.9), and therefore belongs to $\operatorname{\mathcal{S}}_{\mathrm{fin}}$.
To prove the converse, it will suffice to show that the collection of essentially finite Kan complexes is closed under finite colimits (in the $\infty $-category $\operatorname{\mathcal{S}}$). Since the initial object $\emptyset \in \operatorname{\mathcal{S}}$ is finite (and therefore essentially finite), it will suffice to show that the collection of essentially finite Kan complexes is closed under pushouts (Corollary 7.6.2.30). By virtue of Example 7.6.3.3 (and Corollary 5.6.5.18), this is equivalent to the following more concrete assertion:
- $(\ast )$
Suppose we are given a homotopy pushout diagram of Kan complexes
9.4\begin{equation} \begin{gathered}\label{equation:snoops} \xymatrix@R =50pt@C=50pt{ X \ar [r]^{f_0} \ar [d]^{f_1} & X_0 \ar [d] \\ X_1 \ar [r] & X_{01}. } \end{gathered} \end{equation}If $X$, $X_0$, and $X_1$ are essentially finite, then $X_{01}$ is also essentially finite.
To prove $(\ast )$, we may assume without loss of generality that $X_0 = \operatorname{Ex}^{\infty }(K_0)$ and $X_1 = \operatorname{Ex}^{\infty }(K_1)$, for some finite simplicial sets $K_0$ and $K_1$ (Remark 9.2.7.2). Choose a weak homotopy equivalence $u: K \rightarrow X$, where $K$ is a finite simplicial set. Since $K$ is finite, the morphisms $f_0 \circ u$ and $f_1 \circ u$ factor through $\operatorname{Ex}^{n}(K_0)$ and $\operatorname{Ex}^{n}(K_1)$ for some integer $n \gg 0$. Using Proposition 3.3.6.7, we see that the diagram (9.4) induces a weak homotopy equivalence from the homotopy pushout $\operatorname{Ex}^{n}(K_0) { \coprod }_{K}^{\mathrm{h}} \operatorname{Ex}^{n}(K_1)$ to the Kan complex $X_{01}$. Combining Propositions 3.3.4.8 and 3.4.2.5 with Theorem 3.3.5.1, we obtain weak homotopy equivalences
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Ex}^{n}(K_0) { \coprod }_{K}^{\mathrm{h}} \operatorname{Ex}^{n}(K_1) \\ \operatorname{Sd}^{n}( \operatorname{Ex}^{n}( K_0 ) ) \coprod _{\operatorname{Sd}^{n}(K) }^{\mathrm{h}} \operatorname{Sd}^{n}( \operatorname{Ex}^{n}(K_1) ) \ar [u] \ar [d] \\ K_0 \coprod _{ \operatorname{Sd}^{n}(K)}^{\mathrm{h} } K_1. } \]
It follows that $X_{01}$ is weakly homotopy equivalent to the finite simplicial set $K_0 \coprod _{ \operatorname{Sd}^{n}(K)}^{\mathrm{h} } K_1$, and is therefore essentially finite. $\square$