Kerodon

Proof. Let $\operatorname{\mathcal{E}}$ denote the oriented fiber product $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{ \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}_{< \lambda } ) } \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}_{< \lambda } )$, and let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ and $V: \operatorname{\mathcal{E}}\rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}_{< \lambda } )$ be the projection maps. For each object $\mathscr {F} \in \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}_{< \lambda } )$, let $\operatorname{\mathcal{E}}_{\mathscr {F}}$ denote the fiber $V^{-1} \{ \mathscr {F} \} $, and let $U_{\mathscr {F}}: \operatorname{\mathcal{E}}_{\mathscr {F}} \rightarrow \operatorname{\mathcal{C}}$ be the restriction of $U$. Then $U_{\mathscr {F}}$ is a right fibration with contravariant transport representation $\mathscr {F}$ (Corollary 8.4.2.7). In particular, the right fibration $U_{\mathscr {F}}$ is essentially $\lambda $-small, so the $\infty $-category $\operatorname{\mathcal{E}}_{\mathscr {F}}$ is essentially $\lambda $-small (Proposition 4.7.9.10). It follows that $V$ is a cocartesian fibration (Corollary 5.3.7.3) which admits a covariant transport representation

We will complete the proof by showing that the functor $T$ is $\lambda $-cocontinuous.

Fix a $\lambda $-small $\infty $-category $\operatorname{\mathcal{K}}$ and a (levelwise) colimit diagram $\overline{G}: \operatorname{\mathcal{K}}^{\triangleright } \rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}_{< \lambda } )$; we wish to show that $T \circ \overline{G}$ is a colimit diagram in $\operatorname{\mathcal{QC}}_{< \lambda }$. Set $\operatorname{\mathcal{E}}' = \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{ \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}_{< \lambda } ) } \operatorname{\mathcal{K}}^{\triangleright }$, so that $T \circ \overline{G}$ is a covariant transport representation for the projection map $V': \operatorname{\mathcal{E}}' \rightarrow \operatorname{\mathcal{K}}^{\triangleright }$. Let $z$ denote the cone point of $\operatorname{\mathcal{K}}^{\triangleright } = \operatorname{\mathcal{K}}\star \{ z\} $ and define

so that the cocartesian fibration $V'$ admits a covariant refraction diagram $\mathrm{Rf}: \operatorname{\mathcal{E}}'_0 \rightarrow \operatorname{\mathcal{E}}'_{z}$ (see Definition 7.4.5.8). Let $U': \operatorname{\mathcal{E}}' \rightarrow \operatorname{\mathcal{C}}$ be projection onto the first factor and let $W$ be the collection of all morphisms $e$ of $\operatorname{\mathcal{E}}'_0$ such that $U'(e)$ is an identity morphism in $\operatorname{\mathcal{C}}$. It follows from Corollary 5.3.7.3 that a morphism $e$ of $\operatorname{\mathcal{E}}'_0$ is $V'$-cocartesian if and only if $U(e)$ is an isomorphism in $\operatorname{\mathcal{C}}$: that is, if and only if $e$ is isomorphic (as an object of $\operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{E}}'_0 )$) to an element of $W$. By virtue of Theorem 7.4.5.13, it will suffice to show that the functor $\mathrm{Rf}$ exhibits the $\infty $-category $\operatorname{\mathcal{E}}'_{v}$ as a localization of $\operatorname{\mathcal{E}}'_0$ with respect to $W$.

Let $h: \Delta ^1 \times \operatorname{\mathcal{E}}'_0 \rightarrow \operatorname{\mathcal{E}}'$ be a natural transformation which exhibits $\mathrm{Rf}$ as a covariant refraction diagram for $V'$ (see Definition 7.4.5.8). Then $h$ carries each object $E \in \operatorname{\mathcal{E}}'_0$ to a $V'$-cocartesian morphism $h_{E}: E \rightarrow \mathrm{Rf}(E)$, so that $U'( h_{E} )$ is an isomorphism in $\operatorname{\mathcal{C}}$. We may therefore assume (replacing $\mathrm{Rf}$ by an isomorphic functor if necessary) that the composition $(U' \circ h): \Delta ^1 \times \operatorname{\mathcal{E}}'_0 \rightarrow \operatorname{\mathcal{C}}$ factors through $\operatorname{\mathcal{E}}'_0$ (so that each $U'(h_{E})$ is an identity morphism in $\operatorname{\mathcal{C}}$). Then, for each $C \in \operatorname{\mathcal{C}}$, $\mathrm{Rf}$ restricts to a functor

which is a covariant refraction diagram for the left fibration $\{ C\} \operatorname{\vec{\times }}_{\operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}_{< \lambda } )} \operatorname{\mathcal{K}}^{\triangleright } \rightarrow \operatorname{\mathcal{K}}^{\triangleright }$. Our assumption that $\overline{G}$ is a (levelwise) colimit diagram then guarantees that $\mathrm{Rf}_{C}$ is a weak homotopy equivalence: that is, it exhibits the Kan complex $\{ C\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}'_{z}$ as a localization of $\{ C\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}'_0$ with respect to the collection $W_{C}$ of all morphisms of $\{ C\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}'_0$ (Theorem 7.4.5.13). Applying Theorem 6.3.4.1, we conclude that $\mathrm{Rf}$ exhibits $\operatorname{\mathcal{E}}'_{v}$ as a localization of $\operatorname{\mathcal{E}}'_0$ with respect to the collection of morphisms $W = \bigcup _{C \in \operatorname{\mathcal{C}}} W_{C}$, as desired. $\square$

Proof. Enlarging $\lambda $ if necessary, we may assume that $\operatorname{\mathcal{C}}$ is essentially $\lambda $-small. Let $T: \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}_{< \lambda }) \rightarrow \operatorname{\mathcal{QC}}_{< \lambda }$ be as in Lemma 9.2.4.15. Then a functor $\mathscr {F}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{S}}_{< \lambda }$ belongs to $\operatorname{Fun}^{\kappa -\flat }( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}_{< \lambda } )$ if and only if the $\infty $-category $T( \mathscr {F} )$ is $\kappa $-filtered (Remark 9.2.4.13). Since $T$ is $\lambda $-cocontinuous, the collection of objects which satisfy this condition is closed under $\lambda $-small, $\kappa $-filtered colimits (Corollary 9.1.5.13). $\square$

Proof. The “if” direction follows by combining Lemma 9.2.4.16 with Example 9.2.4.11. For the converse, assume that $\mathscr {F}$ is $\kappa $-flat. Then the $\infty $-category

is $\kappa $-filtered (Remark 9.2.4.13). Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be the projection map. Since the covariant Yoneda embedding $h_{\bullet }$ is dense (Variant 8.4.2.4), $\mathscr {F}$ is a colimit of the diagram $(h_{\bullet } \circ U)$. To complete the proof, it will suffice to show that the $\infty $-category $\operatorname{\mathcal{E}}$ is essentially $\lambda $-small. This follows from Corollary 4.7.9.12, since $\operatorname{\mathcal{C}}$ is essentially $\lambda $-small and the right fibration $U$ is essentially $\lambda $-small. $\square$

Proof. Enlarging $\lambda $ if necessary, we may assume that $\lambda $ is regular, that $\operatorname{\mathcal{D}}$ is essentially $\lambda $-small, and that $\lambda $ has exponential cofinality $\geq \kappa $ (so that the $\infty $-category $\operatorname{\mathcal{S}}_{< \lambda }$ admits $\kappa $-small limits). Using Lemma 9.2.4.17, we can realize $\mathscr {F}$ as the colimit of a $\lambda $-small $\kappa $-filtered diagram

where each $\mathscr {F}_{\alpha }$ is corepresentable by an object of $\operatorname{\mathcal{D}}$ and therefore preserves all limits which exist in $\operatorname{\mathcal{D}}$ (Proposition 7.4.1.18). Since $\operatorname{\mathcal{K}}$-indexed colimits commute with $\kappa $-small limits in the $\infty $-category $\operatorname{\mathcal{S}}_{< \lambda }$ (Proposition 9.1.5.8), it follows that the functor $\mathscr {F}$ preserves $\kappa $-small limits. $\square$

Proof of Theorem 9.2.4.14. Let $\kappa \leq \lambda $ be regular cardinals, where $\lambda $ is uncountable, let $\operatorname{\mathcal{C}}$ be an $\infty $-category which is essentially $\lambda $-small. We wish to show that the covariant Yoneda embedding $h_{\bullet }: \operatorname{\mathcal{C}}\rightarrow \operatorname{Fun}^{\kappa -\flat }( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}_{< \lambda })$ exhibits $\operatorname{Fun}^{\kappa -\flat }( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}_{< \lambda })$ as an $\operatorname{Ind}_{\kappa }^{\lambda }$-completion of $\operatorname{\mathcal{C}}$. By virtue of Proposition 8.4.5.8, it will suffice to show that $\operatorname{Fun}^{\kappa -\flat }( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}_{< \lambda })$ is the smallest full subcategory of $\operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}_{< \lambda } )$ which contains all representable functors and is closed under $\lambda $-small $\kappa $-filtered colimits. This follows from Example 9.2.4.11, Lemma 9.2.4.16, and Lemma 9.2.4.17. $\square$

Proof. Without loss of generality, we may assume that $\lambda $ is uncountable (see Example 9.2.1.10). Fix a regular cardinal $\mu > \lambda $ having exponential cofinality $\geq \lambda $ such that $\operatorname{\mathcal{C}}$ is $\mu $-small. Then $\lambda \triangleleft \mu $ (Proposition 9.1.7.8), so $\operatorname{\mathcal{C}}$ can be realized as the colimit of a $\mu $-small, $\lambda $-filtered diagram $\{ \operatorname{\mathcal{C}}_{\alpha } \} $ in the $\infty $-category $\operatorname{\mathcal{QC}}_{< \mu }$ (see Variant 9.1.7.21), where each $\operatorname{\mathcal{C}}_{\alpha }$ is $\lambda $-small. It follows from Proposition 9.2.1.23 that every object of $\operatorname{Ind}_{\kappa }^{\lambda }( \operatorname{\mathcal{C}})$ can be lifted to an object of $\operatorname{Ind}_{\kappa }^{\lambda }( \operatorname{\mathcal{C}}_{\alpha } )$, for some index $\alpha $. We may therefore replace $\operatorname{\mathcal{C}}$ by $\operatorname{\mathcal{C}}_{\alpha }$, and thereby reduce to the case where $\operatorname{\mathcal{C}}$ is $\lambda $-small. In this case, we can use Theorem 9.2.4.14 to identify $\operatorname{Ind}_{\kappa }^{\lambda }( \operatorname{\mathcal{C}})$ with the $\infty $-category $\operatorname{Fun}^{\kappa -\flat }( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}_{< \lambda } )$, and the desired result follows from Lemma 9.2.4.17. $\square$

Proof. Using Proposition 8.4.5.8, we can identify $\operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}})$ with the smallest full subcategory of $\operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}_{< \lambda } )$ which contains all representable functors and is closed under $\lambda $-small $\kappa $-filtered colimits. For every full subcategory $\operatorname{\mathcal{C}}_0$ of $\operatorname{\mathcal{C}}$, the inclusion functor $\iota : \operatorname{\mathcal{C}}_0 \hookrightarrow \operatorname{\mathcal{C}}$ induces a functor

which is given (under the preceding identification) by left Kan extension along $\iota $ (Remark 9.2.1.21). Using Theorem 9.2.4.14, we see that a functor $\mathscr {F}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{S}}_{< \lambda }$ is contained in $\widehat{\operatorname{\mathcal{C}}}$ if and only if it belongs to the essential image of $\operatorname{Ind}_{\kappa }^{\lambda }(\iota )$ for some full subcategory $\operatorname{\mathcal{C}}_0 \subseteq \operatorname{\mathcal{C}}$ which is essentially $\lambda $-small. This follows from Proposition 9.2.1.23, since the collection of such subcategories comprise a $\lambda $-filtered diagram having colimit $\operatorname{\mathcal{C}}$. $\square$

Proof. As in the proof of Proposition 9.2.4.22, we can identify $\operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}})$ with the smallest full subcategory of $\operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}_{< \lambda } )$ which contains all representable functors and is closed under $\lambda $-small $\kappa $-filtered colimits. It follows from Lemma 9.2.4.16 (and Example 9.2.4.11), we see that every functor $\mathscr {F}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{S}}_{< \lambda }$ which belongs to $\operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}})$ is $\kappa $-flat. To complete the proof, it will suffice to show that every $\kappa $-flat functor $\mathscr {F}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{S}}_{< \lambda }$ satisfying condition $(\ast ')$ also satisfies condition $(\ast )$ of Proposition 9.2.4.22.

Using Proposition 8.6.8.3, we may assume that $\mathscr {F}$ is the contravariant transport representation of an essentially $\lambda $-small right fibration of $\infty $-categories $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$. Replacing $U$ by a minimal fibration if necessary (Proposition 5.3.8.11). Similarly, we may assume that the $\infty $-category $\operatorname{\mathcal{C}}$ is minimal (Proposition 4.7.6.15). By assumption, the functor $\mathscr {F}$ is left Kan extended from a full subcategory $\operatorname{\mathcal{C}}_0 \subseteq \operatorname{\mathcal{C}}$ which is essentially $\lambda $-small. Since $\operatorname{\mathcal{C}}_0$ is also minimal, it is $\lambda $-small. We extend $\operatorname{\mathcal{C}}_0$ to a transfinite sequence of $\lambda $-small full subcategories $\{ \operatorname{\mathcal{C}}_{\alpha } \subseteq \operatorname{\mathcal{C}}\} _{0 \leq \beta \leq \kappa }$ as follows:

• Suppose that $\beta = \alpha +1$ is a successor ordinal, and that the full subcategory $\operatorname{\mathcal{C}}_{\alpha }$ has been defined. Let $\operatorname{\mathcal{E}}_{\alpha } \subseteq \operatorname{\mathcal{E}}$ be the inverse image of $\operatorname{\mathcal{C}}_{\alpha }$. Since $\operatorname{\mathcal{C}}_{\alpha }$ is $\lambda $-small and $U$ is an essentially $\lambda $-small minimal fibration, the $\infty $-category $\operatorname{\mathcal{E}}_{\alpha }$ is also $\lambda $-small. Our assumption that $\mathscr {F}$ is $\kappa $-flat guarantees that the $\infty $-category $\operatorname{\mathcal{E}}$ is $\kappa $-filtered. Since $\kappa \triangleleft \lambda $, Proposition 9.1.7.14 guarantees that $\operatorname{\mathcal{E}}_{\alpha }$ is contained in a $\lambda $-small simplicial subset $\operatorname{\mathcal{E}}^{+}_{\alpha } \subseteq \operatorname{\mathcal{E}}$ which is also a $\kappa $-filtered $\infty $-category. We define $\operatorname{\mathcal{C}}_{\beta }$ to be the full subcategory of $\operatorname{\mathcal{C}}$ spanned by the objects $U(X)$, where $X$ belongs to $\operatorname{\mathcal{E}}^{+}_{\alpha }$.

• If $\beta $ is a nonzero limit ordinal, we define $\operatorname{\mathcal{C}}_{\beta }$ to be the union $\bigcup _{\alpha < \beta } \operatorname{\mathcal{C}}_{\alpha }$.

We now claim that the full subcategory $\operatorname{\mathcal{C}}_{\kappa } \subseteq \operatorname{\mathcal{C}}$ satisfies condition $(\ast )$ of Proposition 9.2.4.22. By construction, it is a $\lambda $-small subcategory of $\operatorname{\mathcal{C}}$ which contains $\operatorname{\mathcal{C}}_0$, so that $\mathscr {F}$ is left Kan extended from $\operatorname{\mathcal{C}}_{\kappa }^{\operatorname{op}}$ (Proposition 7.3.8.6). It will therefore suffice to show that the functor $\mathscr {F}|_{ \operatorname{\mathcal{C}}_{\kappa }^{\operatorname{op}} }$ is $\kappa $-flat: that is, that the $\infty $-category $\operatorname{\mathcal{E}}_{\kappa } = \operatorname{\mathcal{C}}_{\kappa } \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ is $\kappa $-filtered. This is a special case of Corollary 9.1.5.13, since $\operatorname{\mathcal{E}}_{\kappa }$ can be realized as the colimit of the $\kappa $-filtered diagram $\{ \operatorname{\mathcal{E}}_{\alpha }^{+} \} _{\alpha < \kappa }$ comprised of $\kappa $-filtered $\infty $-categories. $\square$

Proof. Apply Corollary 9.2.4.23 in the special case where $\lambda = \operatorname{\textnormal{\cjRL {t}}}$ is a strongly inaccessible cardinal. $\square$

Proof. Apply Corollary 9.2.4.24 in the special case $\kappa = \aleph _0$. $\square$