Kerodon

Proof. Using Proposition 8.4.6.3, we can choose a functor $h': \operatorname{\mathcal{C}}\rightarrow \widehat{\operatorname{\mathcal{C}}}'$ which exhibits $\widehat{\operatorname{\mathcal{C}}}'$ as a $\lambda $-cocompletion of $\operatorname{\mathcal{C}}$. Without loss of generality, we may assume that $\widehat{\operatorname{\mathcal{C}}}$ is the smallest full subcategory of $\widehat{\operatorname{\mathcal{C}}}'$ which contains the essential image of $h'$ and is closed under $\kappa $-small colimits. Let $T: \operatorname{Ind}_{\kappa }^{\lambda }( \widehat{\operatorname{\mathcal{C}}} ) \rightarrow \widehat{\operatorname{\mathcal{C}}}'$ be an $\operatorname{Ind}_{\kappa }^{\lambda }$-extension of the inclusion functor $\widehat{\operatorname{\mathcal{C}}} \hookrightarrow \widehat{\operatorname{\mathcal{C}}}'$ (Definition 9.3.1.12). It follows from Proposition 9.2.5.24 that every object of $\widehat{\operatorname{\mathcal{C}}}$ is $(\kappa ,\lambda )$-compact when viewed as an object of $\widehat{\operatorname{\mathcal{C}}}'$, so the functor $T$ is fully faithful (Proposition 9.3.2.1). To complete the proof, it will suffice to show that $T$ is essentially surjective. Without loss of generality, we may assume that $\lambda $ is uncountable (otherwise, the result is immediate from Example 9.3.1.10). In this case, Variant 9.3.4.18 guarantees that every object $X \in \widehat{\operatorname{\mathcal{C}}}'$ can be realized as the colimit of a diagram $K \xrightarrow {F} \operatorname{\mathcal{C}}\xrightarrow {h'} \widehat{\operatorname{\mathcal{C}}}'$, where $K$ is a $\lambda $-small simplicial set. Applying Lemma 9.1.7.18 we can realize $K$ as the colimit of a diagram

where $A$ is a $\lambda $-small $\kappa $-directed partially ordered set and each $K_{\alpha }$ is a $\kappa $-small simplicial set. For each $\alpha \in A$, the diagram $(h' \circ F)|_{ K_{\alpha } }$ has some colimit $X_{\alpha } \in \widehat{\operatorname{\mathcal{C}}}$. Since $K$ is also a categorical colimit of the diagram $\{ K_{\alpha } \} _{\alpha \in A}$ (Proposition 9.1.6.1), we can promote the construction $\alpha \mapsto X_{\alpha }$ to a $\lambda $-small $\kappa $-filtered diagram $\operatorname{N}_{\bullet }(A) \rightarrow \widehat{\operatorname{\mathcal{C}}}$ having colimit $X$ (see Proposition 7.5.8.12), so that $X$ belongs to the essential image of $T$. $\square$

Proof. Using Proposition 8.4.6.3, we can choose a functor $h: \operatorname{\mathcal{C}}\rightarrow \widehat{\operatorname{\mathcal{C}}}$ which exhibits $\widehat{\operatorname{\mathcal{C}}}$ as a $\kappa $-cocompletion of $\operatorname{\mathcal{C}}$. Moreover, $h$ is fully faithful, so the functor $H = \operatorname{Ind}_{\kappa }^{\lambda }(h): \operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}}) \rightarrow \operatorname{Ind}_{\kappa }^{\lambda }( \widehat{\operatorname{\mathcal{C}}} )$ is also fully faithful (Corollary 9.3.2.2). If $\operatorname{\mathcal{C}}$ is $\kappa $-cocomplete, then the functor $h$ admits a left adjoint (Proposition 8.4.6.13). It follows that $H$ also admits a left adjoint (Exercise 9.3.3.11), and therefore induces an equivalence from $\operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}})$ to a reflective localization of $\operatorname{Ind}_{\kappa }^{\lambda }( \widehat{\operatorname{\mathcal{C}}} )$. By virtue of Corollary 7.1.4.29, it will suffice to show that the $\infty $-category $\operatorname{Ind}_{\kappa }^{\lambda }( \widehat{\operatorname{\mathcal{C}}} )$ is $\lambda $-cocomplete, which follows from Proposition 9.5.5.5. $\square$

Proof. The implication $(1) \Rightarrow (2)$ is immediate, the implication $(2) \Rightarrow (3)$ follows from Proposition 9.2.5.24, and the implication $(3) \Rightarrow (1)$ follows from Corollary 9.5.5.6. $\square$

Proof. Apply Corollary 9.5.5.7 in the special case where $\lambda = \Omega $ is a strongly inaccessible cardinal. $\square$

Proof of Proposition 9.5.5.3. Combine Proposition 9.4.7.2 with Corollaries 9.5.5.7 and 9.5.5.8. $\square$

Proof. Fix an object $D \in \operatorname{\mathcal{D}}$. Using the criterion of Theorem 9.3.5.15, it will suffice to prove the equivalence of the following conditions:

$(1_ D)$

The $\infty $-category $\operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}_{/D}$ is $\kappa $-filtered.

$(2_ D)$

The $\infty $-category $\operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}}) \times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}_{/D}$ is $\kappa $-filtered.

$(3_ D)$

The $\infty $-category $\operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}}) \times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}_{/D}$ is $\lambda $-filtered.

Note that the $\infty $-category $\operatorname{\mathcal{E}}= \operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}}) \times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}_{/D}$ admits $\lambda $-small $\kappa $-filtered colimits which are preserved by the right fibration $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}})$ (Proposition 7.1.9.8). Applying Proposition 9.3.1.16, we can identify $\operatorname{\mathcal{E}}$ with the $\operatorname{Ind}_{\kappa }^{\lambda }$-completion of the $\infty $-category $\operatorname{\mathcal{C}}\times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}_{/D}$. The equivalence of $(1_ D)$, $(2_ D)$, and $(3_ D)$ now follows from Corollary 9.3.7.6. $\square$

Proof. Combine Proposition 9.5.5.9 with Corollary 9.3.5.25. $\square$

Proof. By virtue of the universal property of $\operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}})$, this is a reformulation of Corollary 9.5.5.10. $\square$

Proof of Corollary 9.5.5.12. Fix an uncountable regular cardinal $\lambda $ (for example, we can take $\lambda = \aleph _1$). We first treat the case where $\operatorname{\mathcal{D}}$ is $\lambda $-cocomplete. By virtue of Variant 9.4.3.21, we can identify $\operatorname{\mathcal{S}}^{\leq n}_{< \lambda }$ with an $\operatorname{Ind}_{ \aleph _0}^{\lambda }$-completion of $\operatorname{\mathcal{S}}^{\leq n}_{\mathrm{fin} }$. Recall that the $\infty $-category $\operatorname{\mathcal{S}}^{\leq n}_{\mathrm{fin}}$ admits finite colimits (Corollary 9.4.3.16). Applying Corollary 9.5.5.11 (in the special case $\kappa = \aleph _0$), we conclude that the restriction functor

is an equivalence of $\infty $-categories. It will therefore suffice to show that that the evaluation functor

is fully faithful and that its essential image is spanned by the collection of $n$-cotruncated objects of $\operatorname{\mathcal{D}}$. This follows from Corollary 8.4.4.3.

We now treat the general case. By virtue of Corollary 8.3.3.17, we may assume that $\operatorname{\mathcal{D}}$ is a replete full subcategory of a $\lambda $-cocomplete $\infty $-category $\widehat{\operatorname{\mathcal{D}}}$, which is closed under finite colimits (for example, we can take $\widehat{\operatorname{\mathcal{D}}}$ to be an $\operatorname{Ind}_{\aleph _0}^{\lambda }$-completion of $\operatorname{\mathcal{D}}$). We then have a commutative diagram

where the vertical maps are inclusions of full subcategories. Since the $\infty $-category $\operatorname{\mathcal{S}}^{\leq n}_{\mathrm{fin}}$ is generated under finite colimits by $\Delta ^0$ (Corollary 9.4.3.17), this diagram is a categorical square. It follows from the first part of the proof that the upper horizontal map is fully faithful, and its essential image consists of those objects $X \in \operatorname{\mathcal{D}}$ which are $n$-cotruncated when viewed as objects of $\widehat{\operatorname{\mathcal{D}}}$. We complete the proof by observing that this is equivalent to the requirement that $X$ is $n$-cotruncated when viewed as an object of $\operatorname{\mathcal{D}}$, by virtue of our assumption that the inclusion $\operatorname{\mathcal{D}}\hookrightarrow \widehat{\operatorname{\mathcal{D}}}$ preserves finite colimits (Remark 7.1.4.10). $\square$

Proof. The necessity of conditions $(a)$ and $(b^{-})$ is clear. The sufficiency follows from Corollary 9.5.5.10, since $\operatorname{\mathcal{C}}$ can be identified with the $\operatorname{Ind}_{\kappa }^{\lambda }$-completion of $\operatorname{\mathcal{C}}_{< \kappa }$ (Proposition 9.4.1.11). $\square$

Proof. Apply Corollary 9.5.5.13 in the special case where $\lambda = \Omega $ is a strongly inaccessible cardinal. $\square$

Proof. Note that the functor $\infty $-category $\operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ admits small colimits, which are computed levelwise (Proposition 7.1.8.2). The full subcategory $\operatorname{Fun}^{\operatorname{\mathrm{cocont}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \subseteq \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ is closed under small colimits (Remark 7.6.6.23), and is therefore cocomplete. It will therefore suffice to show that the $\infty $-category $\operatorname{Fun}^{\operatorname{\mathrm{cocont}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$ is accessible. Fix a small regular cardinal $\kappa $ such that $\operatorname{\mathcal{C}}$ is $\kappa $-presentable, and let $\operatorname{\mathcal{C}}_{< \kappa }$ denote the full subcategory of $\operatorname{\mathcal{C}}$ spanned by the $\kappa $-compact objects. It follows from Corollary 9.5.5.14 that the restriction functor $\operatorname{Fun}^{\operatorname{\mathrm{cocont}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \rightarrow \operatorname{Fun}^{\kappa -\mathrm{cocont}}( \operatorname{\mathcal{C}}_{< \kappa }, \operatorname{\mathcal{D}})$ is an equivalence of $\infty $-categories. We will show that the $\infty $-category $\operatorname{Fun}^{\kappa -\mathrm{cocont}}( \operatorname{\mathcal{C}}_{< \kappa }, \operatorname{\mathcal{D}})$ is accessible. Since $\operatorname{\mathcal{C}}$ is presentable, the $\infty $-category $\operatorname{\mathcal{C}}_{< \kappa }$ is essentially small, so that $\operatorname{Fun}( \operatorname{\mathcal{C}}_{< \kappa }, \operatorname{\mathcal{D}})$ is accessible (Proposition 9.4.9.1). We will complete the proof by showing that the full subcategory $\operatorname{Fun}^{\kappa -\mathrm{cocont}}( \operatorname{\mathcal{C}}_{< \kappa }, \operatorname{\mathcal{D}}) \subseteq \operatorname{Fun}( \operatorname{\mathcal{C}}_{< \kappa }, \operatorname{\mathcal{D}})$ is accessibly embedded.

For every diagram $\overline{q}: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}_{< \kappa }$, let $\operatorname{Fun}^{ \overline{q} }( \operatorname{\mathcal{C}}_{< \kappa }, \operatorname{\mathcal{D}})$ denote the full subcategory of $\operatorname{Fun}( \operatorname{\mathcal{C}}_{< \kappa }, \operatorname{\mathcal{D}})$ spanned by those functors $F$ such that $F \circ \overline{q}$ is a colimit diagram in $\operatorname{\mathcal{D}}$. Then $\operatorname{Fun}^{\kappa -\mathrm{cocont}}( \operatorname{\mathcal{C}}_{< \kappa }, \operatorname{\mathcal{D}})$ can be identified with the intersection $\bigcap _{ \overline{q} } \operatorname{Fun}^{ \overline{q} }( \operatorname{\mathcal{C}}_{< \kappa }, \operatorname{\mathcal{D}})$, where $\overline{q}$ ranges over a set of representatives for all isomorphism classes of $\kappa $-small colimit diagrams in $\operatorname{\mathcal{C}}_{< \kappa }$. By virtue of Corollary 9.4.9.13, it will suffice to show that each $\operatorname{Fun}^{ \overline{q} }( \operatorname{\mathcal{C}}_{< \kappa }, \operatorname{\mathcal{D}})$ is an accessibly embedded subcategory of $\operatorname{Fun}( \operatorname{\mathcal{C}}_{< \kappa }, \operatorname{\mathcal{D}})$. By construction, $\operatorname{Fun}^{ \overline{q} }( \operatorname{\mathcal{C}}_{< \kappa }, \operatorname{\mathcal{D}})$ is the inverse image of $\operatorname{Fun}'( K^{\triangleright }, \operatorname{\mathcal{D}})$ under the functor given by precomposition with $\overline{q}$, where $\operatorname{Fun}'( K^{\triangleright }, \operatorname{\mathcal{D}}) \subseteq \operatorname{Fun}( K^{\triangleright }, \operatorname{\mathcal{D}})$ is the full subcategory spanned by the colimit diagrams. Using Corollary 9.4.9.12, we are reduced to proving that $\operatorname{Fun}'( K^{\triangleright }, \operatorname{\mathcal{D}})$ is an accessibly embedded subcategory of $\operatorname{Fun}( K^{\triangleright }, \operatorname{\mathcal{D}})$. This is clear: since $\operatorname{\mathcal{D}}$ admits $K$-indexed colimits, the $\infty $-category $\operatorname{Fun}'( K^{\triangleright }, \operatorname{\mathcal{D}})$ is equivalent to $\operatorname{Fun}(K, \operatorname{\mathcal{D}})$ (and therefore accessible by virtue of Proposition 9.4.9.1). Moreover, since $\operatorname{Fun}'( K^{\triangleright }, \operatorname{\mathcal{D}})$ is closed under colimits in $\operatorname{Fun}( K^{\triangleright }, \operatorname{\mathcal{D}})$ (Remark 7.6.6.23), so the inclusion $\operatorname{Fun}'(K^{\triangleright }, \operatorname{\mathcal{D}}) \hookrightarrow \operatorname{Fun}(K^{\triangleright }, \operatorname{\mathcal{C}})$ is automatically an accessible functor. $\square$

Proof. Proposition 9.3.2.1 guarantees that $H$ is fully faithful, and therefore restricts to an equivalence of $\operatorname{Ind}_{\kappa }^{\lambda }( \operatorname{\mathcal{C}})$ with a replete full subcategory $\widehat{\operatorname{\mathcal{C}}}' \subseteq \widehat{\operatorname{\mathcal{C}}}$. Since $h$ is $\kappa $-cocontinuous, Corollary 9.5.5.10 guarantees that $H$ is $\lambda $-cocontinuous: that is, the full subcategory $\widehat{\operatorname{\mathcal{C}}}' \subseteq \widehat{\operatorname{\mathcal{C}}}$ is closed under $\lambda $-small colimits. We conclude by observing that every object of $\widehat{\operatorname{\mathcal{C}}}'$ can be realized as the colimit of a $\lambda $-small diagram in $\operatorname{\mathcal{C}}$ (which we can even take to be $\kappa $-filtered; see Corollary 9.3.4.17). $\square$

Proof. The sufficiency of conditions $(0)$, $(1)$, $(2)$, and $(3)$ follows from Lemma 9.5.5.18. Necessity follows by combining Proposition 9.3.2.3 with Corollaries 9.5.5.6 and 9.3.5.27. $\square$

Proof. If $\operatorname{\mathcal{C}}$ is $\kappa $-presentable, then assertions $(b)$ and $(c_{\kappa })$ are immediate and $(a)$ follows from Remark 9.4.7.14. Conversely, suppose that conditions $(a)$, $(b)$, and $(c_{\kappa })$ are satisfied. Let $\{ C_ i \} _{i \in I}$ be a small collection of $\kappa $-compact objects which generates $\operatorname{\mathcal{C}}$ under small colimits, and let $\operatorname{\mathcal{C}}_0 \subseteq \operatorname{\mathcal{C}}$ be the full subcategory spanned by the objects $C_ i$. Enlarging $\operatorname{\mathcal{C}}_0$ if necessary, we can assume that it is closed under $\kappa $-small colimits (Proposition 9.2.5.24), so that $\operatorname{\mathcal{C}}_0$ is $\kappa $-cocomplete and the inclusion functor $\iota : \operatorname{\mathcal{C}}_0 \hookrightarrow \operatorname{\mathcal{C}}$ is $\kappa $-cocontinuous. It follows from assumption $(a)$ that $\operatorname{\mathcal{C}}_0$ is essentially small (see Proposition 4.9.7.7). Applying Lemma 9.5.5.19, we conclude that $\iota $ exhibits $\operatorname{\mathcal{C}}$ as an $\operatorname{Ind}_{\kappa }$-completion of $\operatorname{\mathcal{C}}_0$, and is therefore $\kappa $-presentable. $\square$

Proof of Proposition 9.5.5.17. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category which satisfies the hypotheses of Proposition 9.5.5.17; we wish to show that $\operatorname{\mathcal{C}}$ is presentable (the converse follows from Remark 9.4.7.14 and 9.4.7.8). By assumption, $\operatorname{\mathcal{C}}$ is generated under small colimits by a small collection of objects $\{ C_ i \} _{i \in I}$. Moreover, each $C_ i$ is $\kappa _ i$-compact for some small regular cardinal $\kappa _ i$. Choose a small regular cardinal $\kappa $ which is an upper bound for the collection $\{ \kappa _ i \} _{i \in I}$. Then each of the objects $C_ i$ is $\kappa $-compact. Applying Proposition 9.5.5.21, we conclude that $\operatorname{\mathcal{C}}$ is $\kappa $-presentable. In particular, it is presentable. $\square$