Lemma 9.1.2.12 (02Q0)

Lemma 9.1.2.12. An $\infty $-category $\operatorname{\mathcal{C}}$ is filtered if and only if it satisfies the following condition for every integer $n \geq 0$:

$(\ast '_ n)$: Every morphism of simplicial sets $\operatorname{\partial \Delta }^ n \rightarrow \operatorname{\mathcal{C}}$ can be extended to a morphism $( \operatorname{\partial \Delta }^ n )^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$.

Proof. The necessity of condition $(\ast '_ n)$ is clear. For the converse, suppose that $\operatorname{\mathcal{C}}$ satisfies $(\ast '_ n)$ for each $n \geq 0$. We wish to prove that $\operatorname{\mathcal{C}}$ is filtered. Let $f: K \rightarrow \operatorname{\mathcal{C}}$ be a diagram where $K$ is a finite simplicial set; we wish to show that the $\infty $-category $\operatorname{\mathcal{C}}_{f/}$ is nonempty. If $K = \emptyset $, then this follows immediately from assumption $(\ast '_{0} )$. Otherwise, the simplicial set $K$ has dimension $m$ for some integer $m \geq 0$. We proceed by induction on $m$ and on the number of nondegenerate $m$-simplices of $K$. Choose a nondegenerate $m$-simplex $\sigma : \Delta ^ m \rightarrow K$. Using Proposition 1.1.4.12, we can choose a pushout diagram

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\partial \Delta }^{m} \ar [r] \ar [d] & \Delta ^{m} \ar [d]^{\sigma } \\ K' \ar [r] & K } \]

where $K' \subseteq K$ is a simplicial subset having a smaller number of nondegenerate $m$-simplices. Set $f' = f|_{K'}$, $f_{0} = f \circ \sigma $, and $f'_0 = f \circ \sigma |_{ \operatorname{\partial \Delta }^ m}$, so that we have a pullback diagram of $\infty $-categories

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}_{f/} \ar [r] \ar [d] & \operatorname{\mathcal{C}}_{ f' / } \ar [d]^{\Phi } \\ \operatorname{\mathcal{C}}_{ f_0 / } \ar [r]^-{\Psi } & \operatorname{\mathcal{C}}_{ f'_0 / }. } \]

Applying our inductive hypothesis, we deduce that the $\infty $-category $\operatorname{\mathcal{C}}_{f'/}$ is nonempty. Choose an object $X$ of $\operatorname{\mathcal{C}}_{f'/}$, so that $\Phi (X) \in \operatorname{\mathcal{C}}_{ f_{\operatorname{\partial \Delta }} / }$ can be identified with a morphism of simplicial sets $g: (\operatorname{\partial \Delta }^ m)^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$. Amalgamating $f \circ \sigma $ with $g$, we obtain a morphism of simplicial sets

\[ \overline{g}: \operatorname{\partial \Delta }^{m+1} \simeq ( \operatorname{\partial \Delta }^{m} )^{\triangleright } {\coprod }_{ \operatorname{\partial \Delta }^{m} } \Delta ^ m \rightarrow \operatorname{\mathcal{C}}. \]

Invoking $(\ast _{m+1})$, we conclude that $\overline{g}$ can be extended to a morphism of simplicial sets $( \operatorname{\partial \Delta }^{m+1 } )^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$. Unwinding the definitions, we see that this extension supplies an object $Y \in \operatorname{\mathcal{C}}_{ f_0 / }$ together with a morphism $u: \Phi (X) \rightarrow \Psi (Y)$ in the $\infty $-category $\operatorname{\mathcal{C}}_{ f'_0 / }$.

Note that the projection maps $\operatorname{\mathcal{C}}_{f' / } \rightarrow \operatorname{\mathcal{C}}\leftarrow \operatorname{\mathcal{C}}_{ f'_0 / }$ are left fibrations (Proposition 4.3.6.1). Let $\overline{X}$ denote the image of $X$ in the $\infty $-category $\operatorname{\mathcal{C}}$, so that Corollary 4.3.7.13 guarantees that the vertical maps in the diagram

\[ \xymatrix@R =50pt@C=50pt{ (\operatorname{\mathcal{C}}_{f'/})_{X/} \ar [rr]^{ \Phi _{X/} } \ar [dr] & & (\operatorname{\mathcal{C}}_{ f'_0 / } )_{U(X) / } \ar [dl] \\ & \operatorname{\mathcal{C}}_{ \overline{X} / } & } \]

are trivial Kan fibrations. In particular, they are equivalences of $\infty $-categories, so that the functor $\Phi _{X/}$ is also an equivalence of $\infty $-categories. It follows that we can choose a morphism $w: X \rightarrow Z$ in the $\infty $-category $\operatorname{\mathcal{C}}_{f'/}$ and a $2$-simplex

\[ \xymatrix@R =50pt@C=50pt{ & \Psi (Y) \ar [dr]^{v} & \\ \Phi (X) \ar [ur]^{u} \ar [rr]^{ \Phi (w) } & & \Phi (Z) } \]

in the $\infty $-category $\operatorname{\mathcal{C}}_{ f'_0 / }$, where $v$ is an isomorphism. Since $\Psi $ is a left fibration (Corollary 4.3.6.12), we can lift $v$ to a morphism $\widetilde{v}: Y \rightarrow \widetilde{Z}$ of the $\infty $-category $\operatorname{\mathcal{C}}_{ f_0 / }$. The pair $(Z, \widetilde{Z})$ can then be regarded as an object of the $\infty $-category $\operatorname{\mathcal{C}}_{f/} = \operatorname{\mathcal{C}}_{ f' / } \times _{ \operatorname{\mathcal{C}}_{ f'_0 / } } \operatorname{\mathcal{C}}_{ f_0 / }$. $\square$