Kerodon

Proof. We wish to show that, for every object $Y \in \widehat{\operatorname{\mathcal{C}}}$, the morphism space $\operatorname{Hom}_{\widehat{\operatorname{\mathcal{C}}}}(Y, X)$ is contractible. Using Proposition 7.4.1.22, we see that the collection of objects $Y \in \widehat{\operatorname{\mathcal{C}}}$ which satisfy this condition is closed under colimits; in particular, it is closed under $\lambda $-small $\kappa $-filtered colimits. We may therefore assume without loss of generality that $Y = F(C)$ for some object $C \in \operatorname{\mathcal{C}}$. Fix a regular cardinal $\mu \geq \lambda $ such that $\widehat{\operatorname{\mathcal{C}}}$ is locally $\mu $-small, so that $F(C)$ corepresents a functor

We wish to show that $G(X)$ is contractible. Since $F(C)$ is a $(\kappa ,\lambda )$-compact object of $\widehat{\operatorname{\mathcal{C}}}$ (Proposition 9.3.2.3), we can identify $G(X)$ with the colimit of the diagram $(G \circ F): \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}_{< \mu }$. Since $F$ is fully faithful, we can identify $G \circ F$ with the corepresentable functor $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(C, \bullet )$, so that $\varinjlim (G \circ F)$ is contractible by virtue of Example 7.4.3.7. $\square$

Proof. The implication $(4) \Rightarrow (3)$ follows from Example 9.1.1.6 and the implication $(3) \Rightarrow (2)$ from Remark 9.1.1.11. If $\lambda $ is uncountable and $\operatorname{\mathcal{C}}$ is essentially $\lambda $-small, then the implication $(4) \Rightarrow (1)$ follows from Proposition 9.3.7.2. We will complete the proof by showing that $(2)$ implies $(1)$. In what follows, let us abuse notation by identifying $\operatorname{\mathcal{C}}$ with a full subcategory of $\widehat{\operatorname{\mathcal{C}}} = \operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}})$. Assume that $\widehat{\operatorname{\mathcal{C}}}$ is $\kappa $-filtered; we will show that $\operatorname{\mathcal{C}}$ is $\kappa $-filtered. Fix a $\kappa $-small diagram $F: K \rightarrow \operatorname{\mathcal{C}}$; we wish to show that there exists a natural transformation from $F$ to the constant diagram $\underline{X}: K \rightarrow \operatorname{\mathcal{C}}$, for some $X \in \operatorname{\mathcal{C}}$. Note that each object $C \in \operatorname{\mathcal{C}}$ is $(\kappa ,\lambda )$-compact when viewed as an object of $\widehat{\operatorname{\mathcal{C}}}$ (Proposition 9.3.2.3). Applying Proposition 9.2.8.9, we conclude that $F$ is $(\kappa ,\lambda )$-compact when viewed as an object of the diagram $\infty $-category $\operatorname{Fun}(K, \widehat{\operatorname{\mathcal{C}}} )$. Fix a regular cardinal $\mu \geq \lambda $ such that $\operatorname{Fun}(K, \widehat{\operatorname{\mathcal{C}}} )$ is locally $\mu $-small, so that the functor

is $(\kappa ,\lambda )$-finitary. Since $\widehat{\operatorname{\mathcal{C}}}$ is $\kappa $-filtered, there exists a natural transformation from $F$ to a constant diagram in $\widehat{\operatorname{\mathcal{C}}}$: that is, the Kan complex $Q(X)$ is nonempty for some $X \in \widehat{\operatorname{\mathcal{C}}}$. Invoking the universal property of $\widehat{\operatorname{\mathcal{C}}}$, we see that the restriction functor $\operatorname{Fun}^{(\kappa ,\lambda )-\operatorname{fin}}( \widehat{\operatorname{\mathcal{C}}}, \operatorname{\mathcal{S}}_{< \mu } ) \rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}_{< \mu } )$ is an equivalence of $\infty $-categories. It follows that $Q(X)$ must be also be nonempty for some $X \in \operatorname{\mathcal{C}}$, as desired. $\square$

Proof. Apply Proposition 9.3.7.4 in the case where $\lambda = \operatorname{\Omega }$ is a strongly inaccessible cardinal. $\square$

Proof of Theorem 9.3.7.1. Apply Corollary 9.3.7.5 in the special case $\kappa = \aleph _0$. $\square$

Proof. The implications $(3) \Rightarrow (2) \Rightarrow (1)$ follow from Proposition 9.3.7.4 (and do not require the assumption that $\kappa \trianglelefteq \lambda $). We will show that $(1)$ implies $(3)$. Choose a regular cardinal $\mu $ satisfying $\lambda \trianglelefteq \mu $ such that $\operatorname{\mathcal{C}}$ is essentially $\mu $-small. If $\operatorname{\mathcal{C}}$ is $\kappa $-filtered, then Proposition 9.3.7.4 implies that the $\infty $-category $\operatorname{Ind}_{\kappa }^{\mu }(\operatorname{\mathcal{C}})$ has a final object. Theorem 9.3.6.4 guarantees that $\operatorname{Ind}_{\kappa }^{\mu }(\operatorname{\mathcal{C}})$ can be viewed as an $\operatorname{Ind}_{\lambda }^{\mu }$-completion of $\operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}})$. Applying Proposition 9.3.7.4 again, we conclude that $\operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}})$ is $\lambda $-filtered. $\square$

Proof. Let us abuse notation by identifying $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ with full subcategories of the $\infty $-categories $\widehat{\operatorname{\mathcal{C}}} = \operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}})$ and $\widehat{\operatorname{\mathcal{D}}} = \operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{D}})$, respectively. Setting $\widehat{F} = \operatorname{Ind}_{\kappa }^{\lambda }(F)$, we can assume without loss of generality that $F = \widehat{F}|_{\operatorname{\mathcal{C}}}$. Fix a regular cardinal $\mu \geq \lambda $ such that $\widehat{\operatorname{\mathcal{C}}}$ and $\widehat{\operatorname{\mathcal{D}}}$ are $\mu $-small. By virtue of Proposition 7.4.3.11, it will suffice to show that for every functor $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{S}}_{< \mu }$ which is corepresentable by an object $D \in \operatorname{\mathcal{D}}$, the composite functor $(G \circ F): \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}_{< \mu }$ has a contractible colimit. Let $\widehat{G}: \widehat{\operatorname{\mathcal{D}}} \rightarrow \operatorname{\mathcal{S}}_{< \mu }$ be the functor corepresented by $D$. Since $D$ is $(\kappa ,\lambda )$-compact when viewed as an object of $\widehat{\operatorname{\mathcal{D}}}$, the functor $\widehat{G}$ is $(\kappa ,\lambda )$-finitary. It follows that the composite functor $( \widehat{G} \circ \widehat{F} ): \widehat{\operatorname{\mathcal{C}}} \rightarrow \operatorname{\mathcal{S}}_{< \mu }$ is also $(\kappa ,\lambda )$-finitary, and is therefore left Kan extended from $\operatorname{\mathcal{C}}$ (Remark 9.3.1.14). Applying Corollary 7.3.8.3, we are reduced to showing that the colimit $\varinjlim ( \widehat{G} \circ \widehat{F} )$ a contractible Kan complex, which follows from our assumption that $\widehat{F}$ is right cofinal (Proposition 7.4.3.11). $\square$

Proof. Since $\operatorname{\mathcal{C}}$ is $\lambda $-small and $\kappa $-filtered, the $\infty $-category $\operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}})$ has a final object (Proposition 9.3.7.2). The equivalence $(2) \Leftrightarrow (3)$ now follows from Corollary 7.2.1.9, and the implication $(2) \Rightarrow (1)$ is a special case of Lemma 9.3.7.10. We complete the proof by showing that $(1) \Rightarrow (3)$. Assume that $F$ is right cofinal and let $X$ be a final object of $\operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}})$; we wish to show that $\widehat{F} = \operatorname{Ind}_{\kappa }^{\lambda }(F)$ carries $X$ to a final object of $\operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{D}})$. Using Proposition 9.3.7.2, we can identify $X$ with a colimit of the tautological map $h: \operatorname{\mathcal{C}}\rightarrow \operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}})$. Since the functor $\widehat{F}$ is $(\kappa ,\lambda )$-finitary, it follows that $\widehat{F}(X)$ is a colimit of the diagram $\widehat{F} \circ h$, which is isomorphic to the composition $\operatorname{\mathcal{C}}\xrightarrow {F} \operatorname{\mathcal{D}}\rightarrow \operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{D}})$. If $F$ is right cofinal, then this is also a colimit of the tautological map $\operatorname{\mathcal{D}}\rightarrow \operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{D}})$ (Corollary 7.2.2.3), and is therefore a final object of $\operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{D}})$ (Proposition 9.3.7.2). $\square$

Proof. Apply Proposition 9.3.7.11 in the special case where $\lambda = \operatorname{\Omega }$ is a strongly inaccessible cardinal. $\square$

Proof of Theorem 9.3.7.9. Apply Corollary 9.3.7.12 in the special case $\kappa = \aleph _0$. $\square$

Proof. Assume that $F$ is right cofinal; we will show that $\widehat{F} = \operatorname{Ind}_{\kappa }^{\lambda }(F)$ is right cofinal (the converse is a special case of Lemma 9.3.7.10, and requires weaker assumptions). Choose a regular cardinal $\mu $ satisfying $\lambda \trianglelefteq \mu $ such that $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ are essentially $\mu $-small. By virtue of Lemma 9.3.7.10, it will suffice to show that $\operatorname{Ind}_{\lambda }^{\mu }( \widehat{F} )$ is right cofinal. This follows from Proposition 9.3.7.11, because we can identify $\operatorname{Ind}_{\lambda }^{\mu }( \widehat{F} )$ with the functor $\operatorname{Ind}_{\kappa }^{\mu }(F): \operatorname{Ind}_{\kappa }^{\mu }(\operatorname{\mathcal{C}}) \rightarrow \operatorname{Ind}_{\kappa }^{\mu }(\operatorname{\mathcal{D}})$ (see Theorem 9.3.6.4 and Proposition 9.2.2.29). $\square$