Kerodon

Proof. For every simplicial set $K$, we can choose a categorical equivalence $K \rightarrow \operatorname{\mathcal{C}}$, where $\operatorname{\mathcal{C}}$ is an $\infty $-category. Let us say that $K$ is good if the $\infty $-category $\operatorname{\mathcal{C}}$ belongs to $\operatorname{\mathcal{QC}}_{\mathrm{fin}}$ (note that this condition is invariant under equivalence, and therefore depends only on $K$). We will show that every finite simplicial set $K$ is good: that is, every essentially finite $\infty $-category belongs to $\operatorname{\mathcal{QC}}_{\mathrm{fin}}$; the converse is an immediate consequence of Corollary 9.2.8.7. Our proof proceeds in several steps:

$(a)$

The simplicial set $\Delta ^1$ is good (since it is an $\infty $-category which belongs to $\operatorname{\mathcal{QC}}_{\mathrm{fin}}$).

$(b)$

Let $f: K \rightarrow K'$ be a categorical equivalence of simplicial sets. Then $K$ is good if and only if $K'$ is good.

$(c)$

The collection of good simplicial sets is closed under finite coproducts (since $\operatorname{\mathcal{QC}}_{\mathrm{fin}}$ is closed under finite coproducts in $\operatorname{\mathcal{QC}}$; see Corollary 4.5.3.10 and Example 7.6.1.17).

$(d)$

Suppose we are given a categorical pushout square of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ K_{01} \ar [r] \ar [d] & K_0 \ar [d] \\ K_1 \ar [r] & K. } \]

If $K_0$, $K_1$, and $K_{01}$ are good, then $K$ is good. See Example 7.6.3.5.

$(e)$

Let $K$ and $K'$ be simplicial sets containing vertices $x$ and $x'$, respectively, and let $K \vee K'$ denote the pushout $K \coprod _{ \Delta ^0 } K'$. If $K$ and $K'$ are good, then $K \vee K'$ is also good. To prove this, consider the diagram

\[ \xymatrix@R =50pt@C=50pt{ \Delta ^1 \amalg \Delta ^1 \ar [r] \ar [d] & \{ x\} \amalg \{ x'\} \ar [r] \ar [d] & K \amalg K' \ar [d] \\ \Delta ^1 \ar [r] & \Delta ^0 \ar [r] & K \vee K'. } \]

The right half of the diagram is a pushout square in which the horizontal maps are monomorphisms, and therefore a categorical pushout square (Example 4.5.4.12). The left half of the diagram is a categorical pushout square by virtue of Proposition 6.3.2.4. Applying Proposition 4.5.4.8, we conclude that the outer rectangle is also a categorical pushout square. By virtue of $(d)$, it will suffice to show that the simplicial sets $\Delta ^1$, $\Delta ^1 \amalg \Delta ^1$, and $K \amalg K'$ are good. This follows from $(a)$ and $(b)$.

$(f)$

Let $n$ be a positive integer, and let $\operatorname{Spine}[n]$ denote the spine of the standard $n$-simplex $\Delta ^ n$ (Example 1.5.7.7). Then the simplicial set $\operatorname{Spine}[n]$ is good. For $n = 1$, this follows from $(a)$. The general case follows by induction on $n$, using $(e)$.

$(g)$

The standard $0$-simplex $\Delta ^0$ is good. To prove this, consider the diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{N}_{\bullet }( \{ 0 < 1 \} ) \ar [r] \ar [d] & \operatorname{Spine}[2] \ar [d] \ar [r] & \operatorname{Spine}[2] / \operatorname{N}_{\bullet }( \{ 1 < 2 \} ) \ar [d] \\ \Delta ^0 \ar [r] & \operatorname{Spine}[2] / \operatorname{N}_{\bullet }( \{ 0 < 1 \} ) \ar [r] & \Delta ^0. } \]

The left side and outer rectangle are pushout square in which the horizontal maps are monomorphisms, and therefore categorical pushout squares (Example 4.5.4.12). Applying Proposition 4.5.4.8, we see that the right side is also a categorical pushout square. Using $(d)$, we are reduced to proving that the simplicial set $\operatorname{Spine}[2]$ and its quotients $\operatorname{Spine}[2] / \operatorname{N}_{\bullet }( \{ 0 < 1 \} )$ and $\operatorname{Spine}[2] / \operatorname{N}_{\bullet }( \{ 1 < 2 \} )$ are good, which follows from $(f)$.

$(h)$

For every integer $n$, the standard $n$-simplex $\Delta ^ n$ is good. For $n=0$, this follows from $(g)$. For $n > 0$, it follows from $(b)$ and $(f)$, since the inclusion map $\operatorname{Spine}[n] \hookrightarrow \Delta ^ n$ is a categorical equivalence (Example 1.5.7.7).

We now show that every finite simplicial set $K$ is good. If $K$ is empty, this follows from $(c)$. We may therefore assume that $K$ has dimension $d \geq 0$. Proceeding by induction on $d$ and on the number of nondegenerate simplices of $K$, we may assume that there exists a pushout square

where the simplicial sets $\operatorname{\partial \Delta }^ d$ and $K_0$ are good by our inductive hypothesis, and $\Delta ^ d$ is good by virtue of $(h)$. Since (9.6) is a categorical pushout square (Example 4.5.4.12), it follows from $(d)$ that that $K$ is also good. $\square$