$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proposition 9.1.5.8. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $\kappa $ be a small infinite cardinal. The following conditions are equivalent:
- $(1)$
The $\infty $-category $\operatorname{\mathcal{C}}$ is $\kappa $-filtered.
- $(2)$
For every $\kappa $-small simplicial set $K$, the diagonal map $\operatorname{\mathcal{C}}\rightarrow \operatorname{Fun}(K, \operatorname{\mathcal{C}})$ is right cofinal.
- $(3)$
There exists an uncountable regular cardinal $\lambda $ of exponential cofinality $\geq \kappa $ such that $\operatorname{\mathcal{C}}$ is essentially $\lambda $-small and the colimit functor $\varinjlim : \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{< \lambda } ) \rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{< \lambda } )$ commutes with $\kappa $-small limits.
- $(4)$
If $\lambda $ is any uncountable regular cardinal of exponential cofinality $\geq \kappa $ such that $\operatorname{\mathcal{C}}$ is essentially $\lambda $-small, then the colimit functor $\varinjlim : \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{< \lambda } ) \rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{< \lambda } )$ preserves $\kappa $-small limits.
- $(5)$
The $\infty $-category $\operatorname{\mathcal{C}}$ is filtered and satisfies condition $(\ast _{I})$ of Lemma 9.1.5.2, for every $\kappa $-small set $I$.
Proof.
As in the proof of Proposition 9.1.5.5, the implications $(4) \Rightarrow (3) \Rightarrow (2) \Leftrightarrow (1)$ follow from Variant 9.1.4.7 and Proposition 9.1.1.14. The implication $(1) \Rightarrow (5)$ follows from Remarks 9.1.1.8 and 9.1.5.3. We will complete the proof by showing that $(5) \Rightarrow (4)$. Assume that condition $(5)$ is satisfied and fix uncountable regular cardinal $\lambda $ of exponential cofinality $\geq \kappa $ such that $\operatorname{\mathcal{C}}$ is essentially $\lambda $-small; we wish to show that the colimit functor $\varinjlim : \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{< \lambda } ) \rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{< \lambda } )$ preserves $\kappa $-small limits. Since $\operatorname{\mathcal{C}}$ is filtered, the functor $\varinjlim $ preserves finite limits (Proposition 9.1.5.5). It will therefore suffice to show that $\varinjlim $ reserves $I$-indexed products, for every $\kappa $-small set $I$ (Exercise 7.6.6.11). Using Variant 9.1.4.4, we can reformulate this statement as follows:
- $(\ast )$
Let $\{ \mathscr {F}_ i: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}^{< \lambda } \} _{i \in I}$ be an $I$-indexed collection of functors from $\operatorname{\mathcal{C}}$ to $\operatorname{\mathcal{S}}^{< \lambda }$, and let $\mathscr {F} = \prod _{i \in I} \mathscr {F}_{i}$ be their product. If each colimit $\varinjlim ( \mathscr {F}_{i} )$ is contractible, then $\varinjlim (\mathscr {F} )$ is also contractible.
For each $i \in I$, we can assume that $\mathscr {F}_{i}$ arises as the covariant transport representation of a left fibration $U_ i: \operatorname{\mathcal{E}}_ i \rightarrow \operatorname{\mathcal{C}}$. Our assumption that $\varinjlim (\mathscr {F}_ i)$ is contractible then guarantees that the $\infty $-category $\operatorname{\mathcal{E}}_{i}$ is weakly contractible (Proposition 7.4.3.6) and therefore filtered (Theorem 9.1.3.2). Applying Lemma 9.1.5.2, we conclude that the $\infty $-category $\operatorname{\mathcal{E}}= \operatorname{\mathcal{C}}\times _{ \operatorname{Fun}(I, \operatorname{\mathcal{C}}) } (\prod _{i \in I} \operatorname{\mathcal{E}}_ i )$ is also filtered, and therefore weakly contractible (Proposition 9.1.1.13). Since $\mathscr {F}$ is a covariant transport representation for the projection map $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$, this is equivalent to the contractibility of $\varinjlim (\mathscr {F} )$ (Proposition 7.4.3.6).
$\square$