Kerodon

Proof. Let $\lambda $ be a cardinal of exponential cofinality $\geq \kappa $ such that $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ are essentially $\lambda $-small. It follows from Proposition 9.1.5.8 (together with Proposition 9.1.4.1) that $\operatorname{\mathcal{C}}$ is $\kappa $-filtered if and only if, for every $\kappa $-small simplicial set $L$, the limit functor $\varprojlim : \operatorname{Fun}(L, \operatorname{\mathcal{S}}^{< \lambda } ) \rightarrow \operatorname{\mathcal{S}}^{< \lambda }$ preserves $\operatorname{\mathcal{C}}$-indexed colimits. Since $F$ is right cofinal, it will suffice to show that $\varprojlim $ preserves $\operatorname{\mathcal{D}}$-indexed colimits, which follows from our assumption that $\operatorname{\mathcal{D}}$ is $\kappa $-filtered. $\square$

Proof. Let $J$ be a set of cardinality $\kappa $ and let $\operatorname{\mathcal{J}}$ be the corresponding indiscrete category (that is, the category having object set $\operatorname{Ob}(\operatorname{\mathcal{J}}) = J$ and $\operatorname{Hom}_{\operatorname{\mathcal{J}}}(j,j') = \ast $ for every pair of elements $j,j' \in J$). Then the nerve $\operatorname{N}_{\bullet }(\operatorname{\mathcal{J}})$ is a contractible Kan complex. Setting $\widetilde{\operatorname{\mathcal{C}}} = \operatorname{N}_{\bullet }(\operatorname{\mathcal{J}}) \times \operatorname{\mathcal{C}}$, it follows that the projection map $\pi : \widetilde{\operatorname{\mathcal{C}}} \rightarrow \operatorname{\mathcal{C}}$ is a trivial Kan fibration. We will complete the proof by showing that $\widetilde{\operatorname{\mathcal{C}}}$ satisfies condition $(\ast )$. Let $K$ be a $\kappa $-small simplicial subset of $\widetilde{\operatorname{\mathcal{C}}}$, so that the inclusion map $K \hookrightarrow \widetilde{\operatorname{\mathcal{C}}}$ can be identified with a pair of morphisms

Since $J$ has cardinality $\kappa $, we can choose an element $j \in J$ which is not of the form $f(x)$ for any vertex $x \in K$. It follows that $f$ admits a unique extension $\overline{f}: K^{\triangleright } \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{J}})$ which carries the cone point of $K^{\triangleright }$ to the element $j \in J$. Our assumption that $\operatorname{\mathcal{C}}$ is $\kappa $-filtered guarantees that $g$ admits an extension $\overline{g}: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$. We complete the proof by observing that the pair $(\overline{f}, \overline{g} )$ determines a monomorphism of simplicial sets $K^{\triangleright } \rightarrow \widetilde{\operatorname{\mathcal{C}}}$. $\square$

Proof of Theorem 9.1.6.2. Let $\operatorname{\mathcal{C}}$ be a $\kappa $-filtered $\infty $-category; we wish to show that there exists a $\kappa $-directed partially ordered set $(A, \leq )$ and a right cofinal functor $\operatorname{N}_{\bullet }(A) \rightarrow \operatorname{\mathcal{C}}$ (the reverse implication follows from Proposition 9.1.6.1 and Example 9.1.1.7). Without loss of generality, we may assume that $\kappa $ is regular (otherwise, we can replace $\kappa $ by its successor $\kappa ^{+}$: see Corollary 9.1.5.9). Choose a trivial Kan fibration $\pi : \widetilde{\operatorname{\mathcal{C}}} \rightarrow \operatorname{\mathcal{C}}$ which satisfies condition $(\ast )$ of Lemma 9.1.6.4. Then $\pi $ is right cofinal (Corollary 7.2.1.13). Since the collection of right cofinal morphisms is closed under composition (Proposition 7.2.1.6), we can replace $\operatorname{\mathcal{C}}$ by $\widetilde{\operatorname{\mathcal{C}}}$ and thereby reduce to proving Theorem 7.2.1.6 in the special case where the $\infty $-category $\operatorname{\mathcal{C}}$ satisfies condition $(\ast )$ of Lemma 9.1.6.4.

Let $A$ be the collection of all simplicial subsets $L \subseteq \operatorname{\mathcal{C}}$ which are isomorphic to $K^{\triangleright }$, for some $\kappa $-small simplicial set $K$. To avoid confusion, we use the symbol $\alpha $ to represent an element of $A$, and we will write $L_{\alpha }$ for the corresponding simplicial subset of $\operatorname{\mathcal{C}}$. By assumption, we can write $L_{\alpha }$ as a join $K_{\alpha } \star \{ C_{\alpha } \} $, where $K_{\alpha }$ is a finite simplicial subset of $\widetilde{\operatorname{\mathcal{C}}}$ and $C_{\alpha }$ is an object of $\operatorname{\mathcal{C}}$.

Note that condition $(\ast )$ of Lemma 9.1.6.4 can be restated as follows:

• Every $\kappa $-small simplicial subset $K \subseteq \operatorname{\mathcal{C}}$ is equal to $K_{\alpha }$, for some element $\alpha \in A$.

Let us regard $A$ as a partially ordered set, where elements $\alpha ,\beta \in A$ satisfy $\alpha \leq \beta $ if and only if $L_{\alpha }$ is contained in $L_{\beta }$ (as simplicial subsets of $\operatorname{\mathcal{C}}$). If $A_0$ is any $\kappa $-small subset of $A$, our assumption that $\kappa $ is regular guarantees that $\bigcup _{\alpha \in A_0} L_{\alpha }$ is also $\kappa $-small, and therefore coincides with $K_{\beta }$ for some element $\beta \in A$. In particular, we have $L_{\alpha } \subseteq K_{\beta } \subset L_{\beta }$, so that $\beta $ is an upper bound for $A_0$. Allowing $A_0$ to vary, we conclude that the partially ordered set $A$ is $\kappa $-directed.

To every $n$-simplex $\sigma = ( \alpha _0 \leq \cdots \leq \alpha _ n)$ of $\operatorname{N}_{\bullet }(A)$, we associate an $n$-simplex $F(\sigma )$ of $L_{ \alpha _{n} } \subseteq \operatorname{\mathcal{C}}$ by the following recursive procedure:

• If $n=0$, then $\sigma $ can be identified with an element $\alpha \in A$. In this case, we define $F(\sigma )$ to be the object $C_{\alpha } \in \operatorname{\mathcal{C}}$.

• Suppose that $n > 0$, and let $\sigma ' = d^{n}_ n(\sigma )$ denote the $(n-1)$-simplex $( \alpha _0 \leq \cdots \leq \alpha _{n-1} )$ of $\operatorname{N}_{\bullet }(A)$. Then $F(\sigma )$ is the unique $n$-simplex $\Delta ^{n} \rightarrow L_{ \alpha _{n} }$ whose restriction to $\Delta ^{n-1}$ coincides with $F(\sigma ')$ and which carries vertex $n \in \Delta ^ n$ to the cone point $C_{\alpha _ n} \in L_{\alpha _ n}$.

Regarding each $F(\sigma )$ as a simplex of the $\infty $-category $\operatorname{\mathcal{C}}$, we observe that the construction $\sigma \mapsto F(\sigma )$ is compatible with face and degeneracy operators and therefore determines a functor of $\infty $-categories $F: \operatorname{N}_{\bullet }(A) \rightarrow \operatorname{\mathcal{C}}$.

We will complete the proof by showing that the functor $F$ is right cofinal. To verify this, we will use the criterion of Theorem 7.2.3.1. Let $C$ be an object of $\operatorname{\mathcal{C}}$; we wish to show that the $\infty $-category $\operatorname{N}_{\bullet }(A) \times _{ \operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{C/}$ is weakly contractible. We will prove something a bit stronger: the $\infty $-category $\operatorname{N}_{\bullet }(A) \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{C/}$ is $\kappa $-filtered (this is sufficient, by virtue of Proposition 9.1.1.13). To prove this, let $S$ be any $\kappa $-small simplicial set and suppose that we are given a diagram $g: S \rightarrow \operatorname{N}_{\bullet }(A) \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{C/}$; we wish to show that $g$ can be extended to a morphism $\overline{g}: S^{\triangleright } \rightarrow \operatorname{N}_{\bullet }(A) \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{C/}$. Unwinding the definitions, we can identify $g$ with a pair of diagrams

satisfying $g_1|_{S} = F \circ g_0$, where $g_1$ carries the cone point of $S^{\triangleleft }$ to the object $C \in \operatorname{\mathcal{C}}$. Note that the union $K = \operatorname{im}( g_1 ) \cup \bigcup _{s \in S} L_{ g_0(s) }$ is a $\kappa $-small simplicial subset of $\operatorname{\mathcal{C}}$. Using $(\ast ')$, we can write $K = K_{\alpha }$ for some element $\alpha \in A$. Since the image of $g_{1}$ is contained in $K_{\alpha }$, it admits a canonical extension

Similarly, the inclusion $L_{ g_0(s) } \subseteq K_{\alpha } \subset L_{\alpha }$ guarantees that $g_0$ can be extended uniquely to a morphism $\overline{g}_{0}: S^{\triangleright } \rightarrow \operatorname{N}_{\bullet }(A)$ carrying the cone point of $S^{\triangleright }$ to the element $\alpha \in A$. We conclude by observing that the pair $( \overline{g}_0, \overline{g}_1 )$ determines a diagram $\overline{g}: S^{\triangleright } \rightarrow \operatorname{N}_{\bullet }(A) \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{C/}$ satisfying $\overline{g}|_{S} = g$. $\square$

Proof. The implication $(1) \Rightarrow (2)$ follows from Corollary 9.1.2.8 and the implication $(2) \Rightarrow (3)$ follows from Example 9.1.1.2. The implication $(3) \Rightarrow (1)$ follows by combining Theorem 9.1.6.2 (and Remark 9.1.6.5) with Corollary 7.2.2.3. $\square$

Proof. The implication $(1) \Rightarrow (2)$ follows from Example 9.1.1.6, the implication $(2) \Rightarrow (3)$ from Remark 9.1.6.11, and the implication $(3) \Rightarrow (4)$ from Corollary 9.1.1.15. We will complete the proof by showing that every $\kappa $-filtered $\infty $-category $\operatorname{\mathcal{C}}$ satisfies condition $(1)$. Without loss of generality, we may assume that $\operatorname{\mathcal{C}}$ satisfies condition $(\ast )$ of Lemma 9.1.6.4. Let $A$ be the $\kappa $-directed partially ordered set defined in the proof of Theorem 9.1.6.2. For each $\alpha \in A$, let $L_{\alpha } \subseteq \operatorname{\mathcal{C}}$ denote the corresponding subset of $\operatorname{\mathcal{C}}$. Using Corollary 4.1.3.3, we can choose an $\infty $-category $\operatorname{\mathcal{C}}_{\alpha }$ and an inner anodyne morphism $F_{\alpha }: L_{\alpha } \hookrightarrow \operatorname{\mathcal{C}}_{\alpha }$, depending functorially on $\alpha $. Applying Corollary 4.5.7.2, we see that the morphisms $F_{\alpha }$ induce an equivalence of $\infty $-categories

To complete the proof, it will suffice to show that each of the $\infty $-categories $\operatorname{\mathcal{C}}_{\alpha }$ contains a final object. By construction, there exists an isomorphism of simplicial sets $u: L_{\alpha } \simeq K^{\triangleright }$, for some $\kappa $-small simplicial set $K$. Using Corollary 4.1.3.3, we can choose a categorical equivalence $v: K \rightarrow \operatorname{\mathcal{D}}$, where $\operatorname{\mathcal{D}}$ is an $\infty $-category. Applying Corollary 4.5.8.9, we deduce that the map $v^{\triangleright }: K^{\triangleright } \rightarrow \operatorname{\mathcal{D}}^{\triangleright }$ is also a categorical equivalence of simplicial sets. Since $F_{\alpha }$ is inner anodyne, there exists a functor $G: \operatorname{\mathcal{C}}_{\alpha } \rightarrow \operatorname{\mathcal{D}}^{\triangleright }$ satisfying $G \circ F_{\alpha } = v^{\triangleright } \circ u$. Applying the two-out-of-three property (Remark 4.5.3.5), we see that $G$ is an equivalence of $\infty $-categories. Since the $\infty $-category $\operatorname{\mathcal{D}}^{\triangleright }$ has a final object (given by the cone point; see Example 4.6.7.5), it follows that $\operatorname{\mathcal{C}}_{\alpha }$ also has a final object (Corollary 4.6.7.21). $\square$