Kerodon

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

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. $\square$

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

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. $\square$

Proof. Since the collection of compact objects of $\operatorname{\mathcal{S}}$ is closed under finite colimits (Corollary 9.2.2.23), this follows from the observation that $\Delta ^0 \in \operatorname{\mathcal{S}}$ is compact (Example 9.2.2.4). $\square$

Proof. The $\infty $-category $\operatorname{\mathcal{S}}$ admits small filtered colimits (Corollary 7.4.3.10), and every object of $\operatorname{\mathcal{S}}_{\mathrm{fin}}$ is compact when viewed as an object of $\operatorname{\mathcal{S}}$ (Corollary 9.2.7.5). Recall that every (small) simplicial set $X$ can be realized as the colimit of a (small) filtered diagram $ \{ X_{\alpha } \} $, where each $X_{\alpha }$ is a finite simplicial set (Remark 3.6.1.8). Applying Proposition 3.3.6.4, we conclude that $\operatorname{Ex}^{\infty }(X)$ can be realized as the colimit of a (small) filtered diagram $\{ \operatorname{Ex}^{\infty }(X_{\alpha } ) \} $ in the category of simplicial sets, and therefore also in the $\infty $-category $\operatorname{\mathcal{S}}$ (Variant 9.1.6.4). If $X$ is a Kan complex, then it is homotopy equivalent to $\operatorname{Ex}^{\infty }(X)$ (Proposition 3.3.6.7), and can therefore be realized as the colimit of a small filtered diagram of essentially finite Kan complexes. The desired result now follows from the criterion of Corollary 9.2.3.6. $\square$

Proof. Combine Corollary 9.2.7.6 with Proposition 9.2.6.13. $\square$

Proof. If $X$ is essentially $\kappa $-small, then there exists a weak homotopy equivalence $K \rightarrow X$, where $K$ is a $\kappa $-small simplicial set. In this case, $X$ can be viewed as a colimit of the constant diagram $K \rightarrow \{ \Delta ^0 \} \hookrightarrow \operatorname{\mathcal{S}}_{< \lambda }$ (Example 7.1.2.9). This proves the implication $(1) \Rightarrow (2)$. Note that the object $\Delta ^0 \in \operatorname{\mathcal{S}}_{< \lambda }$ corepresents the identity functor $\operatorname{id}: \operatorname{\mathcal{S}}_{< \lambda } \rightarrow \operatorname{\mathcal{S}}_{< \lambda }$, and is therefore $(\kappa ,\lambda )$-compact. Consequently, the implication $(2) \Rightarrow (3)$ follows from Proposition 9.2.2.21.

We will complete the proof by showing that $(3)$ implies $(1)$. Suppose that $X$ is $(\kappa ,\lambda )$-compact when viewed as an object of the $\infty $-category $\operatorname{\mathcal{S}}_{< \lambda }$; we wish to show that $X$ is essentially $\kappa $-small. Without loss of generality we may assume that $X$ is $\lambda $-small. Using our assumption that $\kappa \trianglelefteq \lambda $, we can choose a $\lambda $-small collection of $\kappa $-small simplicial subsets $\{ X_{\alpha } \} _{\alpha \in A}$ such that every $\kappa $-small simplicial subset of $X$ is contained in some $X_{\alpha }$ (Lemma 9.1.7.18). We regard $A$ as a partially ordered set, with $\alpha \leq \beta $ if $X_{\alpha }$ is contained in $X_{\beta }$. The partially ordered set $(A, \leq )$ is then $\kappa $-directed (Remark 9.1.7.19). Let $\operatorname{Ex}^{\infty }: \operatorname{Set_{\Delta }}\rightarrow \operatorname{Set_{\Delta }}$ be the functor defined in Construction 3.3.6.1. Since $\operatorname{Ex}^{\infty }$ preserves filtered colimits (Proposition 3.3.6.4), we can identify $\operatorname{Ex}^{\infty }(X)$ with the colimit of the diagram $\{ \operatorname{Ex}^{\infty }( X_{\alpha } ) \} _{\alpha \in A}$ in the category of Kan complexes, and therefore also in the $\infty $-category $\operatorname{\mathcal{S}}_{< \lambda }$ (Variant 9.1.6.4). Note that each $\operatorname{Ex}^{\infty }(X_{\alpha } )$ is a $\kappa $-small Kan complex, and therefore $(\kappa ,\lambda )$-compact as an object of the $\infty $-category $\operatorname{\mathcal{S}}_{< \lambda }$. Applying Lemma 9.2.6.12, we deduce that there exist some index $\alpha \in A$ such that $X$ is a retract of $\operatorname{Ex}^{\infty }(X_{\alpha } )$ in the $\infty $-category $\operatorname{\mathcal{S}}_{< \lambda }$. In particular, $X$ can be realized as the colimit of a diagram

in the $\infty $-category $\operatorname{\mathcal{S}}_{< \lambda }$, for some (homotopy idempotent) endomorphism $e$ of the Kan complex $\operatorname{Ex}^{\infty }( X_{\alpha } )$. Forming the colimit of this diagram in the ordinary category of simplicial sets, we obtain a $\kappa $-small Kan complex which is homotopy equivalent to $X$ (Variant 7.6.5.9). $\square$

Proof of Proposition 9.2.7.11. Apply Proposition 9.2.7.12 in the special case where $\lambda = \operatorname{\textnormal{\cjRL {t}}}$ is a strongly inaccessible cardinal (see Example 9.1.7.11). $\square$

Proof. We proceed as in the proof of Corollary 9.2.7.6. We assume that $\lambda $ is uncountable (otherwise, $\kappa = \lambda $ and there is nothing to prove). Note that $\operatorname{\mathcal{S}}_{< \lambda }$ admits $\lambda $-small colimits (Corollary 7.4.3.10), and that objects of $\operatorname{\mathcal{S}}_{< \kappa }$ are $(\kappa ,\lambda )$-compact when viewed as objects of $\operatorname{\mathcal{S}}_{< \lambda }$ (for $\kappa $ uncountable, this is Proposition 9.2.7.12). Our assumption that $\kappa \trianglelefteq \lambda $ guarantees that every $\lambda $-small simplicial set $X$ can be realized as the colimit of a $\lambda $-small $\kappa $-filtered diagram $\{ X_{\alpha } \} $, where each $X_{\alpha }$ is $\kappa $-small (see Lemma 9.1.7.18). Applying Proposition 3.3.6.4, we conclude that $\operatorname{Ex}^{\infty }(X)$ can be realized as the colimit of a (small) filtered diagram $\{ \operatorname{Ex}^{\infty }(X_{\alpha } ) \} $ in the category of simplicial sets, and therefore also in the $\infty $-category $\operatorname{\mathcal{S}}_{< \lambda }$ (Variant 9.1.6.4). If $X$ is a Kan complex, then it is homotopy equivalent to $\operatorname{Ex}^{\infty }(X)$ (Proposition 3.3.6.7), and can therefore be realized the colimit of a $\lambda $-small, $\kappa $-filtered diagram of Kan complexes which are essentially $\kappa $-small (or essentially finite, in the case $\kappa = \aleph _0$). The desired result now follows from the criterion of Proposition 9.2.3.3. $\square$

Proof. Apply Corollary 9.2.7.13 in the special case where $\lambda = \operatorname{\textnormal{\cjRL {t}}}$ is a strongly inaccessible cardinal (here the inequality $\kappa < \lambda $ guarantees that $\kappa \triangleleft \lambda $; see Example 9.1.7.11). $\square$