Kerodon

Proof of Theorem 9.5.1.1 from Theorems 9.5.1.8 and 9.5.1.9. Let $\operatorname{\mathcal{C}}$ be an accessible $\infty $-category. Assume first that $\operatorname{\mathcal{C}}$ is cocomplete; we will show that $\operatorname{\mathcal{C}}$ is complete. Using Theorem 8.3.3.13, we can identify $\operatorname{\mathcal{C}}$ with the full subcategory $\operatorname{Fun}^{ \mathrm{rep}}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}) \subseteq \operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}})$ spanned by the representable functors $\mathscr {F}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{S}}$. Since the $\infty $-category of spaces $\operatorname{\mathcal{S}}$ is complete (Corollary 7.4.1.3), the functor $\infty $-category $\operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}})$ is also complete (Proposition 7.1.8.2). It will therefore suffice to show that the full subcategory $\operatorname{Fun}^{\mathrm{rep}}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}})$ is closed under small limits in $\operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}})$. This follows from the characterization of representable functors given by Theorem 9.5.1.8, together with Remark 7.6.6.23.

We now prove the converse. Assume that $\operatorname{\mathcal{C}}$ is complete; we wish to show that $\operatorname{\mathcal{C}}$ is cocomplete, or equivalently that the opposite $\infty $-category $\operatorname{\mathcal{C}}^{\operatorname{op}}$ is complete. Using Theorem 8.3.3.13, we can identify $\operatorname{\mathcal{C}}^{\operatorname{op}}$ with the full subcategory $\operatorname{Fun}^{ \mathrm{corep}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}) \subseteq \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{S}})$ spanned by the corepresentable functors $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$. Arguing as above, we see that the $\infty $-category $\operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{S}})$ admits small limits. It will therefore suffice to show that the full subcategory $\operatorname{Fun}^{\mathrm{corep}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{S}})$ is closed under small limits. Suppose we are given a small simplicial set $K$ and a diagram $G: K \rightarrow \operatorname{Fun}^{\mathrm{corep}}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{S}})$; we wish to show that the limit $\varprojlim (G)$ (formed in the $\infty $-category $\operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{S}})$) is also corepresentable. By virtue of Theorem 9.5.1.9, it will suffice to show that $\varprojlim (G)$ is continuous and accessible. As above, the continuity follows from Remark 7.6.6.23. Let us identify $G$ with a functor $g: \operatorname{\mathcal{C}}\rightarrow \operatorname{Fun}(K, \operatorname{\mathcal{S}})$, so that $\varprojlim (G)$ is obtained by composing $g$ with the limit functor $\varprojlim : \operatorname{Fun}(K, \operatorname{\mathcal{S}}) \rightarrow \operatorname{\mathcal{S}}$ (Corollary 7.1.8.4). Note that the functor $\varprojlim $ is accessible: in fact, it preserves small $\kappa $-filtered colimits for any regular cardinal $\kappa $ for which $K$ is $\kappa $-small (Theorem 9.1.5.9). It will therefore suffice to show that the functor $g$ is accessible, which is a special case of Remark 9.4.8.2. $\square$

Proof. The necessity of condition $(1)$ follows from Corollary 7.4.1.23. To prove the necessity of condition $(2)$, we must show that if $C$ and $D$ are objects of $\operatorname{\mathcal{C}}$ where $C$ is $(\kappa ,\lambda )$-compact, then the morphism space $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D)$ is essentially $\lambda $-small. Let us regard the object $C \in \operatorname{\mathcal{C}}$ as fixed, and let $\operatorname{\mathcal{C}}_0 \subseteq \operatorname{\mathcal{C}}$ be the full subcategory spanned by those objects $D \in \operatorname{\mathcal{C}}$ for which $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(C, D)$ is essentially $\lambda $-small. Since $C$ is $(\kappa ,\lambda )$-compact, the full subcategory $\operatorname{\mathcal{C}}_0 \subseteq \operatorname{\mathcal{C}}$ is closed under $\lambda $-small $\kappa $-filtered colimits (Corollary 7.4.3.8). Consequently, to show that $\operatorname{\mathcal{C}}_0 = \operatorname{\mathcal{C}}$, it will suffice to show that it contains every $(\kappa ,\lambda )$-compact object of $\operatorname{\mathcal{C}}$. This follows from our assumption that $\operatorname{\mathcal{C}}_{< \kappa }$ is essentially $\lambda $-small.

We now prove the converse. Assume that conditions $(1)$ and $(2)$ are satisfied; we wish to show that the functor $\mathscr {F}$ is representable. By virtue of Proposition 8.6.8.3, we may assume that $\mathscr {F}$ is the contravariant transport representation of a right fibration $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$. It follows from assumption $(1)$ that the $\infty $-category $\operatorname{\mathcal{E}}$ is $\lambda $-cocomplete and that the functor $U$ is $\lambda $-cocontinuous (Corollary 7.4.1.21). Let $\operatorname{\mathcal{E}}_{< \kappa } \subseteq \operatorname{\mathcal{E}}$ denote the inverse image of $\operatorname{\mathcal{C}}_{< \kappa }$, so that $\operatorname{\mathcal{E}}$ is an $\operatorname{Ind}_{\kappa }^{\lambda }$-completion of $\operatorname{\mathcal{E}}_{< \kappa }$ (Proposition 9.3.1.16). The $\infty $-category $\operatorname{\mathcal{E}}_{< \kappa }$ is closed under $\kappa $-small colimits in $\operatorname{\mathcal{E}}$; in particular, it is $\kappa $-filtered (Example 9.1.1.7). Assumption $(2)$ guarantees that the right fibration $U_{< \kappa }: \operatorname{\mathcal{E}}_{< \kappa } \rightarrow \operatorname{\mathcal{C}}_{< \kappa }$ is essentially $\lambda $-small, so that the $\infty $-category $\operatorname{\mathcal{E}}_{< \kappa }$ is also essentially $\lambda $-small (Corollary 4.9.8.12). Applying Proposition 9.3.7.2, we deduce that the $\infty $-category $\operatorname{\mathcal{E}}$ has a final object (given by the colimit of the inclusion map $\operatorname{\mathcal{E}}_{< \kappa } \hookrightarrow \operatorname{\mathcal{E}}$), so that the functor $\mathscr {F}$ is representable as desired (Remark 8.6.8.9). $\square$

Proof of Theorem 9.5.1.8. Let $\operatorname{\mathcal{C}}$ be an accessible cocomplete $\infty $-category and let $\mathscr {F}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{S}}$ be a continuous functor; we wish to show that $\mathscr {F}$ is representable. Choose a small regular cardinal $\kappa $ such that $\operatorname{\mathcal{C}}$ is $\kappa $-compactly generated. The desired result now follows by applying Proposition 9.5.1.11 in the special case $\lambda = \mu = \Omega $, where $ \Omega $ is a strongly inaccessible cardinal. $\square$

Proof. Fix a regular cardinal $\mu $ such that $\operatorname{\mathcal{C}}$ is locally $\mu $-small and $\operatorname{\mathcal{S}}_{< \mu }$ admits $K$-indexed limits. As in the proof of Theorem 9.5.1.1, we can identify $\operatorname{\mathcal{C}}$ with the $\infty $-category $\operatorname{Fun}^{\mathrm{rep}}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}_{< \mu } )$ spanned by the representable functors $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}_{< \mu }$. Since $\operatorname{\mathcal{S}}_{< \mu }$ admits $K$-indexed limits, the $\infty $-category $\operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}_{< \mu } )$ has the same property (Proposition 7.1.8.2). It will therefore suffice to show that the full subcategory $\operatorname{Fun}^{\mathrm{rep}}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}}_{< \mu } )$ is closed under the formation of $K$-indexed limits. This follows from representability criterion of Proposition 9.5.1.11 (together with Remark 7.6.6.23). $\square$

Proof of Proposition 9.5.1.13. Suppose first that $\mathscr {F}$ is corepresentable by an object $C \in \operatorname{\mathcal{C}}_{< \kappa }$. In this case, condition $(a)$ follows from the definition of $\operatorname{\mathcal{C}}_{< \kappa }$, condition $(b)$ from Corollary 7.4.1.23, and condition $(c)$ from our assumption that $\operatorname{\mathcal{C}}_{< \kappa }$ is essentially $\lambda $-small.

We now prove the converse. Assume that the functor $\mathscr {F}$ satisfies conditions $(a)$, $(b)$, and $(c)$, and let $\mathscr {F}_{< \kappa }$ denote the restriction of $\mathscr {F}$ to $\operatorname{\mathcal{C}}_{< \kappa }$. We will show that $\mathscr {F}_{< \kappa }$ is corepresentable by an object $C \in \operatorname{\mathcal{C}}_{< \kappa }$. Assuming that this condition is satisfied, let $h^{C}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}_{< \mu }$ be the functor corepresented by $C$ on the entire $\infty $-category $\operatorname{\mathcal{C}}$. Our assumption that $C$ is $(\kappa ,\mu )$-compact then guarantees that $h^{C}$ is $(\kappa ,\mu )$-finitary. Then $\mathscr {F}$ and $h^{C}$ are $(\kappa ,\mu )$-finitary functors which become isomorphic when restricted to $\operatorname{\mathcal{C}}_{< \kappa }$, and are therefore isomorphic (since $\operatorname{\mathcal{C}}$ is an $\operatorname{Ind}_{\kappa }^{\mu }$-completion of $\operatorname{\mathcal{C}}_{< \kappa }$), which proves that the functor $\mathscr {F}$ is also corepresentable by the object $C$.

Let $\operatorname{\mathcal{E}}= \int _{\operatorname{\mathcal{C}}} \mathscr {F}$ be the $\infty $-category of elements of $\mathscr {F}$ (Definition 5.6.2.1) and let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be the projection map, so that $U$ is a left fibration with covariant transport representation $\mathscr {F}$. Since $\operatorname{\mathcal{C}}$ is $\lambda $-complete and $\mathscr {F}$ is $\lambda $-continuous, the $\infty $-category $\operatorname{\mathcal{E}}$ is also $\lambda $-complete (Corollary 7.4.1.21). In particular, $\operatorname{\mathcal{E}}$ is idempotent-complete (Proposition 8.5.4.10).

Set $\operatorname{\mathcal{E}}_{< \kappa } = \operatorname{\mathcal{C}}_{<\kappa } \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$. It follows from assumption $(c)$ that the projection map $\operatorname{\mathcal{E}}_{< \kappa } \rightarrow \operatorname{\mathcal{C}}_{< \kappa }$ is an essentially $\lambda $-small left fibration. Since $\operatorname{\mathcal{C}}_{< \kappa }$ is essentially $\lambda $-small, the $\infty $-category $\operatorname{\mathcal{E}}_{< \kappa }$ is also essentially $\lambda $-small (Corollary 4.9.8.12). It follows that the inclusion functor $\iota : \operatorname{\mathcal{E}}_{<\kappa } \hookrightarrow \operatorname{\mathcal{E}}$ admits a limit in $\operatorname{\mathcal{E}}$. In particular, we can choose an object $X \in \operatorname{\mathcal{E}}$ and a natural transformation $u: \underline{X} \rightarrow \iota $, where $\underline{X}$ denotes the constant functor $\operatorname{\mathcal{E}}_{<\kappa } \rightarrow \{ X\} \hookrightarrow \operatorname{\mathcal{E}}$. Since $\operatorname{\mathcal{C}}$ is $(\kappa ,\mu )$-compactly generated, the object $U(X)$ can be realized as the colimit of a $\mu $-small $\kappa $-filtered diagram in $\operatorname{\mathcal{C}}_{< \kappa }$. That is, there exists a $\mu $-small $\kappa $-filtered $\infty $-category $\operatorname{\mathcal{K}}$ and a colimit diagram $Q: \operatorname{\mathcal{K}}^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$ carrying $\operatorname{\mathcal{K}}$ into $\operatorname{\mathcal{C}}_{< \kappa }$ and the cone point of $\operatorname{\mathcal{K}}^{\triangleright }$ to the object $U(X)$. Our assumption that $\mathscr {F}$ is $(\kappa ,\mu )$-finitary guarantees that $\mathscr {F} \circ Q$ is a colimit diagram in the $\infty $-category $\operatorname{\mathcal{S}}_{< \mu }$. Applying Corollary 7.4.3.14, we deduce that the inclusion map

is left cofinal. In particular, there exists a morphism $v: Y \rightarrow X$ of $\operatorname{\mathcal{E}}$, where $Y$ belongs to $\operatorname{\mathcal{E}}_{< \kappa }$. Let $\underline{Y}: \operatorname{\mathcal{E}}_{< \kappa } \rightarrow \operatorname{\mathcal{E}}_{< \kappa }$ be the constant functor taking the value $Y$, so that $v$ determines a natural transformation $\underline{v}: \underline{Y} \rightarrow \underline{X}$. We can then regard $(u \circ \underline{v})$ as a natural transformation from the constant functor $\underline{Y}$ to the identity functor $\operatorname{id}: \operatorname{\mathcal{E}}_{< \kappa } \rightarrow \operatorname{\mathcal{E}}_{< \kappa }$. Applying Proposition 9.1.8.11, we conclude that the idempotent-completion of $\operatorname{\mathcal{E}}_{< \kappa }$ has an initial object. Since $\operatorname{\mathcal{C}}_{< \kappa }$ is closed under retracts in $\operatorname{\mathcal{C}}$ (Remark 9.2.5.18), $\operatorname{\mathcal{E}}_{< \kappa }$ is closed under retracts in $\operatorname{\mathcal{E}}$, and is therefore also idempotent-complete. It follows that the $\infty $-category $\operatorname{\mathcal{E}}_{< \kappa }$ has an initial object. Applying Proposition 5.6.6.21, we conclude that the covariant transport representation $\mathscr {F}_{< \kappa }: \operatorname{\mathcal{C}}_{< \kappa } \rightarrow \operatorname{\mathcal{S}}_{< \mu }$ is corepresentable, as desired. $\square$

Proof of Theorem 9.5.1.9. Let $\operatorname{\mathcal{C}}$ be an accessible complete $\infty $-category and let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}$ be a functor which is continuous and accessible; we wish to show that $\mathscr {F}$ is corepresentable by an object of $\operatorname{\mathcal{C}}$ (the reverse implication follows from Corollary 7.4.1.23 and Remark 9.4.6.8). Fix a small regular cardinal $\kappa $ such that $\operatorname{\mathcal{C}}$ is $\kappa $-accessible and the functor $\mathscr {F}$ is $\kappa $-finitary. The desired result now follows by applying Proposition 9.5.1.13 in the special case where $\lambda = \mu = \Omega $ is a strongly inaccessible cardinal. $\square$