Kerodon

Proof. Since $\operatorname{\mathcal{C}}$ is essentially finite, we can choose a categorical equivalence $U_0: \operatorname{\mathcal{D}}_0 \rightarrow \operatorname{\mathcal{C}}$, where $\operatorname{\mathcal{D}}_0$ is a finite simplicial set. Arguing as in the proof of Proposition 4.1.3.2, we can factor $U_0$ as a composition $\operatorname{\mathcal{D}}_0 \xrightarrow {j} \operatorname{\mathcal{D}}\xrightarrow {U} \operatorname{\mathcal{C}}$, where $U$ is an inner fibration and $j$ is a transfinite composition of pushouts of inner horn inclusions. More precisely, we may assume that $\operatorname{\mathcal{D}}= \operatorname{\mathcal{D}}_{\beta }$ for some ordinal $\beta $ and some well-ordered diagram

having the following properties:

• For every ordinal $\alpha < \beta $, there is a pair of integers $0 < i < n$ and a pushout diagram of simplicial sets

  \[ \xymatrix@R =50pt@C=50pt{ \Lambda ^{n}_{i} \ar [r] \ar [d] & \operatorname{\mathcal{D}}_{\alpha } \ar [d] \\ \Delta ^{n} \ar [r]^-{ \sigma _{\alpha } } & \operatorname{\mathcal{D}}_{\alpha +1}. } \]

  In what follows, we will identify $\sigma _{\alpha }$ with an $n$-simplex of $\operatorname{\mathcal{D}}$ and write $\tau _{\alpha }$ for its $i$th face $d^{n}_{i}( \sigma _{\alpha } )$.

• For every nonzero limit ordinal $\lambda \leq \beta $, the comparison map $\varinjlim _{\alpha < \lambda } \operatorname{\mathcal{D}}_{\alpha } \rightarrow \operatorname{\mathcal{D}}_{\lambda }$ is an isomorphism.

Let us say that a simplicial subset $S \subseteq \operatorname{\mathcal{D}}$ is saturated if, whenever $S$ contains the simplex $\tau _{\alpha }$, then it also contains the simplex $\sigma _{\alpha }$. If $S$ is any simplicial subset of $\operatorname{\mathcal{D}}$, then there is a smallest saturated simplicial subset $S^{+}$ which contains $S$. Moreover, if $S$ is finite, then $S^{+}$ is also finite.

Since $U$ is an inner fibration, $\operatorname{\mathcal{D}}$ is an $\infty $-category. Since $U_0$ and $j$ are categorical equivalences of simplicial sets, the functor $U$ is an equivalence of $\infty $-categories, and therefore admits a homotopy inverse $U^{-1}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$. Let $L \subseteq \operatorname{\mathcal{D}}$ be the smallest saturated simplicial subset which contains both $\operatorname{\mathcal{D}}_0$ and the image of the diagram $(U^{-1} \circ f_0): K_0 \rightarrow \operatorname{\mathcal{D}}$. For $\alpha \leq \beta $, set $L_{\alpha } = L \cap \operatorname{\mathcal{D}}_{\alpha }$. Then the inclusion map $\operatorname{\mathcal{D}}_0 = L_0 \hookrightarrow L_{\beta } = L$ can be realized as a transfinite composition of morphisms $L_{\alpha } \hookrightarrow L_{\alpha +1}$, each of which is either a pushout of an inner horn inclusion (if the simplex $\sigma _{\alpha }$ is contained in $L$) or an isomorphism (if $\sigma _{\alpha }$ is not contained in $L$). In particular, the inclusion $\operatorname{\mathcal{D}}_0 \hookrightarrow L$ is inner anodyne, so that $U|_{L}: L \rightarrow \operatorname{\mathcal{C}}$ is a categorical equivalence of simplicial sets.

By construction, the composition $U^{-1} \circ f_0$ can be regarded as a morphism of simplicial sets $g: K_0 \rightarrow L$. Set $f_{1} = U|_{L} \circ g$. Since $U^{-1}$ is a homotopy inverse to $U$, $f_0$ and $f_{1}$ are isomoprhic when regarded as objects of the $\infty $-category $\operatorname{Fun}(K, \operatorname{\mathcal{C}})$. Let $Q$ be a finite simplicial set equipped with a monomorphism $e: \Delta ^1 \hookrightarrow Q$ which exhibits $Q$ as a localization of $\Delta ^1$ with respect to its nondegenerate edge (for example, we can take $Q$ to be the quotient of $\Delta ^3$ appearing in Corollary 6.3.2.8). Let $q_0$ and $q_1$ be the vertices of $Q$ given by the source and target of the edge $e$. Our assumption that $f_0$ and $f_1$ are isomorphic is then equivalent to the assertion that there exists a diagram $h: Q \rightarrow \operatorname{Fun}(K_0, \operatorname{\mathcal{C}})$ satisfying $h(q_0) = f_0$ and $h(q_1) = f_1$. Let us identify $h$ with a morphism of simplicial sets $H: Q \times K_0 \rightarrow \operatorname{\mathcal{C}}$.

Note that the projection map $Q \rightarrow \Delta ^0$ is a categorical equivalence of simplicial sets (see Corollary 6.3.2.7), so the inclusion $\{ q_1 \} \hookrightarrow Q$ has the same property. Form a pushout diagram

so that the left vertical map determines a categorical equivalence $L \hookrightarrow K$ (Remark 4.5.5.13). By construction, there is a unique morphism of simplicial sets $f: K \rightarrow \operatorname{\mathcal{C}}$ satisfying $f|_{L} = U|_{L}$ and $f|_{ (Q \times K_0) } = H$. Since $U|_{L}$ is a categorical equivalence, it follows that $f$ is a categorical equivalence. We conclude by observing that $f_0$ can be recovered as the composition $f \circ u$, where $u$ is the monomorphism given by the composition

Proof. Choose a categorical equivalence $F: \operatorname{\mathcal{C}}_0 \rightarrow \operatorname{\mathcal{C}}$, where $\operatorname{\mathcal{C}}_0$ is a finite simplicial set. By virtue of Lemma 9.2.7.2, we can assume that $W = F(W_0)$, where $W_0$ is a collection of morphisms of $\operatorname{\mathcal{C}}_0$. In this case, we can identify $\operatorname{\mathcal{C}}[W^{-1}]$ with a localization of $\operatorname{\mathcal{C}}_0$ with respect to $W_0$, which is essentially finite by virtue of Remark 6.3.2.10. $\square$

Proof. The implication $(2) \Rightarrow (1)$ is trivial, since every categorical equivalence of simplicial sets is a weak homotopy equivalence (Remark 4.5.4.4). We now prove the converse. Assume that there exists a weak homotopy equivalence $f: K \rightarrow X$, where $K$ is a finite simplicial set. Then $f$ factors as a composition $K \xrightarrow {i} \operatorname{\mathcal{C}}\xrightarrow {F} X$, where $\operatorname{\mathcal{C}}$ is an $\infty $-category and $i$ is a categorical equivalence of simplicial sets (Proposition 4.1.3.2). Let $W$ be the collection of all edges of $K$. Since $f$ is a weak homotopy equivalence, it exhibits the Kan complex $X$ as a localization of $K$ with respect to $W$ (Proposition 6.3.1.21). It follows that the functor $F$ exhibits $X$ as a localization of $\operatorname{\mathcal{C}}$ with respect to $i(W)$. Since $\operatorname{\mathcal{C}}$ is an essentially finite $\infty $-category, Proposition 9.2.7.3 implies that $X$ is also an essentially finite $\infty $-category. $\square$

Proof. Using Lemma 9.2.7.2, we can construct a commutative diagram of simplicial sets

where the vertical maps are categorical equivalences, the upper horizontal maps are monomorphisms, and the simplicial sets $K_0$, $K$, and $K_1$ are finite. Our assumption that (9.10) is a categorical pushout square then guarantees that the induced map $K_0 \amalg _{K} K_1 \rightarrow \operatorname{\mathcal{C}}_{01}$ is a categorical equivalence (Proposition 4.5.5.9), so that the $\infty $-category $\operatorname{\mathcal{C}}_{01}$ is also essentially finite. $\square$

Proof. Since the initial object $\emptyset \in \operatorname{\mathcal{QC}}$ is essentially finite, it will suffice to show that essential finiteness is closed under the formation of pushouts (see Corollary 7.6.2.42). This is a reformulation of Proposition 9.2.7.6 (see Example 7.6.3.4). $\square$

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 9.2.7.7. 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 4.5.4.10 and Example 7.6.1.20).

$(d)$

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 $(K,x)$ and $(K',x')$ be pointed simplicial sets 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 4.5.5.12). The left half of the diagram is a categorical pushout square by virtue of Proposition 6.3.2.4. Applying Proposition 4.5.5.8, 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 4.5.5.12). Applying Proposition 4.5.5.8, 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 (9.11) is a categorical pushout square (Example 4.5.5.12), it follows from $(d)$ that $K$ is also good. $\square$

Proof. Let $h^{\operatorname{\mathcal{C}}}: \operatorname{\mathcal{QC}}\rightarrow \operatorname{\mathcal{S}}$ be the functor corepresented by $\operatorname{\mathcal{C}}$; we wish to show that the functor $h^{\operatorname{\mathcal{C}}}$ is finitary. By virtue of Corollary 9.2.1.13, it will suffice to show that the functor $h^{\operatorname{\mathcal{C}}}$ preserves the colimit of every diagram $\mathscr {F}: \operatorname{N}_{\bullet }(A) \rightarrow \operatorname{\mathcal{QC}}$ indexed by a small directed partially ordered set $(A,\leq )$. Using Corollary 5.6.5.18, we can reduce to the case where $\mathscr {F}$ factors through the simplicial subset $\operatorname{N}_{\bullet }(\operatorname{QCat}) \subset \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{QCat}) = \operatorname{\mathcal{QC}}$: that is, it is given by a diagram in the ordinary category of simplicial sets.

Choose a categorical equivalence $f: K \rightarrow \operatorname{\mathcal{C}}$, where $K$ is a finite simplicial set. Using Proposition 5.6.6.17, we can assume that $h^{\operatorname{\mathcal{C}}}$ is obtained as the homotopy coherent nerve of the simplicially enriched functor $\operatorname{Fun}(K, \bullet )^{\simeq }: \operatorname{QCat}\rightarrow \operatorname{Kan}$. In particular, we have a commutative diagram of $\infty $-categories

Consequently, to show that the functor $h^{\operatorname{\mathcal{C}}}$ preserves the colimit of the diagram $\mathscr {F}$, it will suffice to show that the vertical maps and upper horizontal map in this diagram preserve $\operatorname{N}_{\bullet }(A)$-indexed colimits. For the vertical maps, this follows from Corollary 9.1.6.3 and Variant 9.1.6.4. For the upper horizontal map, it follows from Variant 9.2.0.5. $\square$

Proof. If $\operatorname{\mathcal{C}}$ is finitely dominated, then it is a retract (in the $\infty $-category $\operatorname{\mathcal{QC}}$) of an essentially finite $\infty $-category $\operatorname{\mathcal{D}}$. Since $\operatorname{\mathcal{D}}$ is a compact object of $\operatorname{\mathcal{QC}}$ (Proposition 9.2.7.9), it follows that $\operatorname{\mathcal{C}}$ is also a compact object of $\operatorname{\mathcal{QC}}$ (Remark 9.2.5.18).

For the converse, we proceed as in the proof of Proposition 9.2.6.7. Using Proposition 4.1.3.2, we can choose a functor $Q: \operatorname{Set_{\Delta }}\rightarrow \operatorname{QCat}$ which preserves small filtered colimits, which carries each simplicial set $X$ to an $\infty $-category $Q(X)$ equipped with an inner anodyne map $u_ X: X \hookrightarrow Q(X)$. Using Corollary 9.1.6.3, we see that the functor

also preserves small filtered colimits. If $\operatorname{\mathcal{C}}$ is a compact object of $\operatorname{\mathcal{QC}}$, it follows that the functor

preserves small filtered colimits. In particular, the equivalence $u_{\operatorname{\mathcal{C}}}: \operatorname{\mathcal{C}}\rightarrow Q( \operatorname{\mathcal{C}})$ factors up to homotopy through $Q( \operatorname{\mathcal{C}}_0 )$ for some finite simplicial subset $\operatorname{\mathcal{C}}_0 \subseteq \operatorname{\mathcal{C}}$. It follows that $\operatorname{\mathcal{C}}$ is a retract of the essentially finite $\infty $-category $Q( \operatorname{\mathcal{C}}_0 )$, and is therefore finitely dominated. $\square$

Proof. By virtue of Proposition 8.5.5.7, it will suffice to show that the $\infty $-category $\operatorname{\mathcal{QC}}_{ \mathrm{fin} }$ is essentially $\aleph _1$-small. Since the collection of isomorphism classes of finite simplicial sets is countable (Variant 4.9.4.21), the collection of isomorphism classes of objects of $\operatorname{\mathcal{QC}}_{ \mathrm{fin} }$ is also countable. It will therefore suffice to show that the $\infty $-category $\operatorname{\mathcal{QC}}_{ \mathrm{fin} }$ is locally $\aleph _1$-small. Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be essentially finite $\infty $-categories; we wish to show that the morphism space $\operatorname{Hom}_{ \operatorname{\mathcal{QC}}} ( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \simeq \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})^{\simeq }$ is essentially $\aleph _1$-small (see Remark 5.5.3.5). Without loss of generality, we may assume that $\operatorname{\mathcal{D}}$ is $\aleph _1$-small. Since $\operatorname{\mathcal{C}}$ is essentially finite, there is a categorical equivalence $e: K \rightarrow \operatorname{\mathcal{C}}$, where $K$ is a finite simplicial set. Then composition with $e$ induces a homotopy equivalence from $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})^{\simeq }$ to the Kan complex $\operatorname{Fun}( K, \operatorname{\mathcal{D}})^{\simeq }$, which is $\aleph _1$-small by virtue of Corollary 4.9.4.19. $\square$

Proof. Let $\operatorname{\mathcal{E}}\subseteq \operatorname{\mathcal{QC}}_{< \lambda }$ be the smallest full subcategory which contains $\Delta ^1$ and is closed under $\kappa $-small colimits. Applying Proposition 9.2.7.8, we conclude that every essentially finite $\infty $-category belongs to $\operatorname{\mathcal{E}}$. Let $Q: \operatorname{Set_{\Delta }}\rightarrow \operatorname{QCat}$ be as in the proof of Proposition 9.2.7.12. If $\operatorname{\mathcal{C}}$ is a $\kappa $-small simplicial set, then it can be realized as the colimit of a $\kappa $-small filtered diagram $\{ \operatorname{\mathcal{C}}_{\alpha } \} $ of finite simplicial sets $\operatorname{\mathcal{C}}_{\alpha }$, so that $Q( \operatorname{\mathcal{C}})$ is the colimit of a $\kappa $-small filtered diagram $\{ Q(\operatorname{\mathcal{C}}_{\alpha } ) \} $ in the $\infty $-category $\operatorname{\mathcal{QC}}_{< \lambda }$. Since each $Q(\operatorname{\mathcal{C}}_{\alpha } )$ is essentially finite, it follows that $Q( \operatorname{\mathcal{C}})$ belongs to $\operatorname{\mathcal{E}}$. If $\operatorname{\mathcal{C}}$ is an $\infty $-category, then it is equivalent to $Q(\operatorname{\mathcal{C}})$ and therefore also belongs to $\operatorname{\mathcal{E}}$. This establishes the implication $(1) \Rightarrow (2)$.

The implication $(2) \Rightarrow (3)$ follows from Proposition 9.2.5.24. We will complete the proof by showing that $(3)$ implies $(1)$. Using Variant 9.1.7.21, we can write $\operatorname{\mathcal{C}}$ as the colimit of a diagram

where $(A, \leq )$ is a $\lambda $-small $\kappa $-directed partially ordered set and each $\operatorname{\mathcal{C}}_{\alpha }$ is a $\kappa $-small $\infty $-category. Corollary 9.1.6.3 implies that $\operatorname{\mathcal{C}}$ is the also the colimit of the diagram $\{ \operatorname{\mathcal{C}}_{\alpha } \} $ in the $\infty $-category $\operatorname{\mathcal{QC}}_{< \lambda }$. If $\operatorname{\mathcal{C}}$ is a $(\kappa ,\lambda )$-compact object of $\operatorname{\mathcal{QC}}_{< \lambda }$, it follows that the identity functor $\operatorname{id}_{\operatorname{\mathcal{C}}}$ factors up to homotopy through some $\operatorname{\mathcal{C}}_{\alpha }$. In particular, $\operatorname{\mathcal{C}}$ is a retract of $\operatorname{\mathcal{C}}_{\alpha }$, and can therefore be realized as the colimit of a diagram

in the $\infty $-category $\operatorname{\mathcal{S}}_{< \lambda }$, for some (homotopy idempotent) endomorphism $e$ of the $\infty $-category $\operatorname{\mathcal{C}}_{\alpha }$. Forming the colimit of this diagram in the ordinary category of simplicial sets, we obtain a $\kappa $-small $\infty $-category which is equivalent to $\operatorname{\mathcal{C}}$ (Corollary 9.1.6.3). $\square$