Remark 9.2.9.1. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $\kappa \leq \lambda $ be regular cardinals, and let $h: \operatorname{\mathcal{C}}\rightarrow \widehat{\operatorname{\mathcal{C}}}$ be a functor which exhibits $\widehat{\operatorname{\mathcal{C}}}$ as an $\operatorname{Ind}_{\kappa }^{\lambda }$-completion of $\operatorname{\mathcal{C}}$ (Variant 9.2.1.7). Suppose we are given regular cardinals $\kappa '$ and $\lambda '$ satisfying $\kappa \leq \kappa ' \leq \lambda ' \leq \lambda $, and let $\widehat{\operatorname{\mathcal{C}}}_0$ be the smallest full subcategory of $\widehat{\operatorname{\mathcal{C}}}$ which contains the image of $h$ and is closed under $\lambda '$-small $\kappa '$-filtered colimits. It follows from Proposition 9.2.3.3 that the functor $h: \operatorname{\mathcal{C}}\rightarrow \widehat{\operatorname{\mathcal{C}}}_0$ exhibits $\widehat{\operatorname{\mathcal{C}}}_0$ as an $\operatorname{Ind}_{\kappa '}^{\lambda '}$-completion of $\operatorname{\mathcal{C}}$. Stated more informally, we can identify $\operatorname{Ind}_{\kappa '}^{\lambda '}(\operatorname{\mathcal{C}})$ with the full subcategory of $\operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}})$ generated by $\operatorname{\mathcal{C}}$ under $\lambda '$-small $\kappa '$-filtered colimits.
9.2.9 Transitivity of $\operatorname{Ind}$-Completion
We now study the dependence of the construction $\operatorname{\mathcal{C}}\mapsto \operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}})$ on the cardinals $\kappa $ and $\lambda $.
Construction 9.2.9.2. Let $\kappa \leq \lambda \leq \mu $ be regular cardinals, let $\operatorname{\mathcal{C}}$ be an $\infty $-category, and let us abuse notation by identifying $\operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}})$ with a full subcategory of $\operatorname{Ind}_{\kappa }^{\mu }(\operatorname{\mathcal{C}})$ (Remark 9.2.9.1). We let denote the $\operatorname{Ind}_{\lambda }^{\mu }$-extension of the inclusion functor $\operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}}) \hookrightarrow \operatorname{Ind}_{\kappa }^{\mu }(\operatorname{\mathcal{C}})$ (see Definition 9.2.1.13). We will refer to $T$ as the transitivity functor.
Proposition 9.2.9.3. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. For every triple of regular cardinals $\kappa \leq \lambda \leq \mu $, the transitivity functor $T: \operatorname{Ind}_{\lambda }^{\mu }( \operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}}) ) \rightarrow \operatorname{Ind}_{\kappa }^{\mu }(\operatorname{\mathcal{C}})$ is fully faithful.
Proof. By virtue of Proposition 9.2.3.1, it will suffice to show that every object of $\operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}})$ is $(\lambda ,\mu )$-compact when viewed as an object of $\operatorname{Ind}_{\kappa }^{\mu }(\operatorname{\mathcal{C}})$. This follows from Proposition 9.2.3.1. $\square$
Under some additional assumptions, we have the following stronger result:
Theorem 9.2.9.4. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. For every triple of regular cardinals $\kappa \trianglelefteq \lambda \trianglelefteq \mu $, the transitivity functor $T: \operatorname{Ind}_{\lambda }^{\mu }( \operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}}) ) \rightarrow \operatorname{Ind}_{\kappa }^{\mu }(\operatorname{\mathcal{C}})$ is an equivalence of $\infty $-categories.
Proof. In what follows, we will abuse notation by identifying $\operatorname{\mathcal{C}}$ and $\operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}})$ with full subcategories of $\operatorname{Ind}_{\kappa }^{\mu }(\operatorname{\mathcal{C}})$. By virtue of Proposition 9.2.3.3, it will suffice to show that every object $X \in \operatorname{Ind}_{\kappa }^{\mu }(\operatorname{\mathcal{C}})$ can be realized as the colimit of a $\mu $-small $\lambda $-filtered diagram in $\operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}})$. Using Corollary 9.2.4.21, we can realize $X$ as the colimit of a diagram $F: \operatorname{\mathcal{K}}\rightarrow \operatorname{\mathcal{C}}$, where $\operatorname{\mathcal{K}}$ is a $\mu $-small $\kappa $-filtered $\infty $-category. If $\kappa = \lambda $, then $\operatorname{\mathcal{K}}$ is $\lambda $-filtered and there is nothing to prove. Otherwise, we can use Proposition 9.1.7.20 to realize $\operatorname{\mathcal{K}}$ as the colimit of a diagram
where $A$ is a $\mu $-small $\lambda $-directed partially ordered set and each $\operatorname{\mathcal{K}}_{\alpha }$ is a $\lambda $-small $\kappa $-filtered $\infty $-category. For each $\alpha \in A$, let $X_{\alpha }$ be a colimit of the diagram $F|_{ \operatorname{\mathcal{K}}_{\alpha } }$. Since $\operatorname{\mathcal{K}}$ is also a categorical colimit of the diagram $\{ \operatorname{\mathcal{K}}_{\alpha } \} _{\alpha \in A}$ (Proposition 9.1.6.1), we can promote the construction $\alpha \mapsto X_{\alpha }$ to a diagram $\operatorname{N}_{\bullet }(A) \rightarrow \operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}})$ having colimit $X$ (see Proposition 7.5.8.12). $\square$
Warning 9.2.9.5. In the situation of Theorem 9.2.9.4, the assumption $\lambda \trianglelefteq \mu $ cannot be omitted. For example, let $S$ be set of cardinality $\geq \mu $, let $P_{< \kappa }(S)$ denote the nerve of the partially ordered set of all $\kappa $-small subsets of $S$, and define $P_{< \lambda }(S)$ and $P_{< \mu }(S)$ similarly. Assume that $\kappa \trianglelefteq \lambda $ and $\kappa \trianglelefteq \mu $ (this condition is always satisfied, for example, if $\kappa = \aleph _0$: see Example 9.1.7.10). Then Example 9.2.3.11 supplies equivalences In this case, the conclusion of Theorem 9.2.9.4 is satisfied if and only if the comparison map $\operatorname{Ind}_{\lambda }^{\mu }( P_{< \lambda }(S) ) \hookrightarrow P_{< \mu }(S)$ is an equivalence, which is equivalent to the requirement $\lambda \trianglelefteq \mu $ (Example 9.2.3.11).
Warning 9.2.9.6. In the situation of Theorem 9.2.9.4, the assumption $\kappa \trianglelefteq \lambda $ cannot be omitted. For example, suppose that $\mu $ satisfies $\kappa \trianglelefteq \mu $ and $\lambda \trianglelefteq \mu $ (these condition are satisfied, for example, if $\mu = (2^{\lambda })^{+}$: see Corollary 9.1.7.9). If $S$ is a set of cardinality $\geq \mu $, then Example 9.2.3.11 supplies equivalences In this case, the conclusion of Theorem 9.2.9.4 is satisfied if and only if the comparison map $\iota : \operatorname{Ind}_{\kappa }^{\lambda }( P_{< \kappa }(S) ) \hookrightarrow P_{< \lambda }(S)$ induces an equivalence of $\operatorname{Ind}_{\lambda }^{\mu }$-completions. If this condition is satisfied, then $\iota $ itself is an equivalence (see Exercise 9.2.6.14), so that $\kappa \trianglelefteq \lambda $ (Example 9.2.3.11).
Let us now record some consequences of Theorem 9.2.9.4.
Corollary 9.2.9.7. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $\kappa \trianglelefteq \lambda \trianglelefteq \mu $ be regular cardinals, where $\lambda $ is uncountable. Then an object $X \in \operatorname{Ind}^{\mu }_{\kappa }(\operatorname{\mathcal{C}})$ is $(\lambda ,\mu )$-compact if and only if it belongs to the full subcategory $\operatorname{Ind}^{\lambda }_{\kappa }(\operatorname{\mathcal{C}})$ (see Remark 9.2.9.1).
Proof. Combine Corollary 9.2.6.15 with Theorem 9.2.9.4. $\square$
Corollary 9.2.9.8. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $\kappa \trianglelefteq \lambda $ be small regular cardinals, where $\lambda $ is uncountable. Then an object $X \in \operatorname{Ind}_{\kappa }(\operatorname{\mathcal{C}})$ is $\lambda $-compact if and only if it belongs to the full subcategory $\operatorname{Ind}^{\lambda }_{\kappa }(\operatorname{\mathcal{C}})$.
Proof. Apply Corollary 9.2.9.7 in the special case where $\mu = \operatorname{\textnormal{\cjRL {t}}}$ is strongly inaccessible. $\square$
Warning 9.2.9.9. Corollary 9.2.9.8 is not quite true in the case $\kappa = \lambda = \aleph _0$: in this case, an object of $\operatorname{Ind}(\operatorname{\mathcal{C}})$ is compact if and only if can be obtained as a retract of an object of $\operatorname{Ind}^{\lambda }_{\kappa }(\operatorname{\mathcal{C}}) \simeq \operatorname{\mathcal{C}}$ (see Proposition 9.2.6.13).
Corollary 9.2.9.10. Let $\kappa \trianglelefteq \lambda \trianglelefteq \mu $ be regular cardinals and let $\operatorname{\mathcal{C}}$ be an $\infty $-category which admits $\lambda $-small $\kappa $-filtered colimits. Then the $\infty $-category $\operatorname{Ind}_{\lambda }^{\mu }(\operatorname{\mathcal{C}})$ admits $\mu $-small $\kappa $-filtered colimits.
Proof. Choose a functor $g: \operatorname{\mathcal{C}}\rightarrow \widehat{\operatorname{\mathcal{C}}}$ which exhibits $\widehat{\operatorname{\mathcal{C}}}$ as an $\operatorname{Ind}_{\kappa }^{\lambda }$-completion of $\operatorname{\mathcal{C}}$. Since $\operatorname{\mathcal{C}}$ admits $\lambda $-small $\kappa $-filtered colimits, the functor $g$ admits a left adjoint $f: \widehat{\operatorname{\mathcal{C}}} \rightarrow \operatorname{\mathcal{C}}$. Then the functor $F = \operatorname{Ind}_{\lambda }^{\mu }(f)$ is left adjoint to $G = \operatorname{Ind}_{\lambda }^{\mu }(g)$ (Exercise 9.2.1.22). Since $g$ is fully faithful, the functor $G$ is also fully faithful (Corollary 9.2.3.2), and therefore induces an equivalence from $\operatorname{Ind}_{\lambda }^{\mu }(\operatorname{\mathcal{C}})$ to a reflective subcategory of $\operatorname{Ind}_{\lambda }^{\mu }( \widehat{\operatorname{\mathcal{C}}} )$ (Remark 6.3.3.4). By virtue of Corollary 7.1.4.23, it will suffice to show that the $\infty $-category $\operatorname{Ind}_{\lambda }^{\mu }( \widehat{\operatorname{\mathcal{C}}} )$ admits $\mu $-small $\kappa $-filtered colimits. This follows immediately from the equivalence $\operatorname{Ind}_{\lambda }^{\mu }( \widehat{\operatorname{\mathcal{C}}}) \simeq \operatorname{Ind}_{\kappa }^{\mu }(\operatorname{\mathcal{C}})$ supplied by Theorem 9.2.9.4. $\square$
Corollary 9.2.9.11. Let $\kappa \trianglelefteq \lambda \trianglelefteq \mu $ be regular cardinals. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category which admits $\lambda $-small $\kappa $-filtered colimits, let $\operatorname{\mathcal{D}}$ be an $\infty $-category which admits $\mu $-small $\kappa $-filtered colimits, and let $h: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor with $\operatorname{Ind}_{\lambda }^{\mu }$-extension $H: \operatorname{Ind}_{\lambda }^{\mu }(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{D}}$. The following conditions are equivalent:
The functor $H$ is $(\kappa ,\mu )$-cocontinuous.
The functor $H$ is $(\kappa ,\lambda )$-cocontinuous.
The functor $h$ is $(\kappa ,\lambda )$-cocontinuous.
Proof. The implication $(1) \Rightarrow (2)$ is trivial (Remark 9.1.9.18) and the implication $(2) \Rightarrow (3)$ follows from the observation that the functor $\operatorname{\mathcal{C}}\rightarrow \operatorname{Ind}_{\lambda }^{\mu }(\operatorname{\mathcal{C}})$ is $(\kappa ,\lambda )$-cocontinuous (Proposition 9.2.3.16). We will complete the proof by showing that $(3)$ implies $(1)$ (the implication $(2) \Rightarrow (1)$ is a special case of Proposition 9.1.9.19, but we will not need this). As in the proof of Corollary 9.2.9.10, fix a functor $g: \operatorname{\mathcal{C}}\rightarrow \widehat{\operatorname{\mathcal{C}}}$ which exhibits $\widehat{\operatorname{\mathcal{C}}}$ as an $\operatorname{Ind}_{\kappa }^{\lambda }$-completion of $\operatorname{\mathcal{C}}$ and set $G = \operatorname{Ind}_{\lambda }^{\mu }(g)$, so that $g$ and $G$ admit left adjoints $f$ and $F = \operatorname{Ind}_{\lambda }^{\mu }(f)$, respectively.
Suppose we are given a diagram $Q: \operatorname{\mathcal{K}}\rightarrow \operatorname{Ind}_{\lambda }^{\mu }(\operatorname{\mathcal{C}})$, where $\operatorname{\mathcal{K}}$ is a $\mu $-small $\kappa $-filtered $\infty $-category; we wish to show that the colimit of $Q$ is preserved by the functor $H$. Since $G$ is fully faithful, $Q$ is isomorphic to the composite $F \circ \widehat{Q}$, where $\widehat{Q} = G \circ Q$. The functor $F$ is a left adjoint, and therefore preserves all colimits (Corollary 7.1.4.22). It will therefore suffice to show that the colimit of $\widehat{Q}$ is preserved by the functor $(H \circ F): \operatorname{Ind}_{\lambda }^{\mu }(\widehat{\operatorname{\mathcal{C}}} ) \rightarrow \operatorname{\mathcal{D}}$.
Using Theorem 9.2.9.4, we can identify $\operatorname{Ind}_{\lambda }^{\mu }(\widehat{\operatorname{\mathcal{C}}})$ with an $\operatorname{Ind}_{\kappa }^{\mu }$-completion of $\operatorname{\mathcal{C}}$. Let $H': \operatorname{Ind}_{\lambda }^{\mu }(\widehat{\operatorname{\mathcal{C}}} ) \rightarrow \operatorname{\mathcal{D}}$ be the $\operatorname{Ind}_{\kappa }^{\mu }$-extension of the functor $h$. We will complete the proof by showing that $H'$ is isomorphic to $(H \circ F)$. Since both $H'$ and $H \circ F$ are $(\lambda ,\mu )$-cocontinuous, it will suffice to show that they have the same restriction to $\widehat{\operatorname{\mathcal{C}}}$: that is, that $H'|_{ \widehat{\operatorname{\mathcal{C}}} }$ is isomorphic to the composition $\widehat{\operatorname{\mathcal{C}}} \xrightarrow {f} \operatorname{\mathcal{C}}\xrightarrow {h} \operatorname{\mathcal{D}}$. By construction, they become isomorphic after precomposition with the functor $g$. It will therefore suffice to show that $h \circ f$ is $(\kappa ,\lambda )$-cocontinuous, which follows from assumption $(3)$. $\square$