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Theorem 7.2.7.2. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. The following conditions are equivalent:

  • The $\infty $-category $\operatorname{\mathcal{C}}$ is filtered.

  • There exists a directed partially ordered set $(A,\leq )$ and a right cofinal functor $F: \operatorname{N}_{\bullet }(A) \rightarrow \operatorname{\mathcal{C}}$.

Proof of Theorem 7.2.7.2. Let $\operatorname{\mathcal{C}}$ be a filtered $\infty $-category; we wish to show that there exists a 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 7.2.7.1 and Example 7.2.5.9). Choose a trivial Kan fibration $\pi : \widetilde{\operatorname{\mathcal{C}}} \rightarrow \operatorname{\mathcal{C}}$ which satisfies condition $(\ast )$ of Lemma 7.2.7.3. Then $\pi $ is right cofinal (Corollary 7.2.1.12). 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 7.2.7.3.

Let $A$ be the collection of all simplicial subsets $L \subseteq \operatorname{\mathcal{C}}$ which are isomorphic to $K^{\triangleright }$, for some finite 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 7.2.7.3 can be restated as follows:

  • Every finite 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 finite subset of $A$, it follows from $(\ast ')$ that we have $\bigcup _{\alpha \in A_0} L_{\alpha } = K_{\beta } \subset L_{\beta }$ for some element $\beta \in A$. In particular, we have $\alpha < \beta $ for each $\alpha \in A_0$. Allowing $A_0$ to vary, we conclude that the partially ordered set $A$ is 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 $n=0$, so that $\sigma $ can be identified with an element $\alpha \in A$, then $F(\sigma )$ is the object $C_{\alpha } \in \operatorname{\mathcal{C}}$.

  • Suppose that $n > 0$, and let $\sigma ' = d_ n(\sigma )$ 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 filtered (this is sufficient, by virtue of Proposition 7.2.4.9). To prove this, let $S$ be any finite 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

\[ g_0: S \rightarrow \operatorname{N}_{\bullet }(A) \quad \quad g_1: S^{\triangleleft } \rightarrow \operatorname{\mathcal{C}} \]

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 finite simplicial subset of $\operatorname{\mathcal{C}}$. Since $\operatorname{\mathcal{C}}$ satisfies condition $(\ast )$ of Lemma 7.2.7.3, 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

\[ \overline{g}_{1}: ( S^{\triangleleft } )^{\triangleright } \rightarrow K_{\alpha }^{\triangleright } = L_{\alpha } \subseteq \operatorname{\mathcal{C}}. \]

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 $\overline{g}|_{S} = g$. $\square$