9.2.3 Recognition of Ind-Completions
We now use the theory of compact objects developed in ยง9.2.2 to formulate a recognition principle for $\operatorname{Ind}$-completions. Our starting point is the following special case of Lemma 8.4.6.8:
Proposition 9.2.3.1. Let $\kappa \leq \lambda $ be regular cardinals and let $f: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories satisfying the following conditions:
- $(0)$
The $\infty $-category $\operatorname{\mathcal{D}}$ admits $\lambda $-small, $\kappa $-filtered colimits.
- $(1)$
The functor $f$ is fully faithful.
- $(2)$
For each object $C \in \operatorname{\mathcal{C}}$, the image $f(C)$ is a $(\kappa ,\lambda )$-compact object of $\operatorname{\mathcal{D}}$.
Then the $\operatorname{Ind}_{\kappa }^{\lambda }$-extension of $f$ is a fully faithful functor $F: \operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{D}}$.
Corollary 9.2.3.2. Let $\kappa \leq \lambda $ be regular cardinals and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. If $F$ is fully faithful, then the functor $\operatorname{Ind}_{\kappa }^{\lambda }(F): \operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}}) \rightarrow \operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{D}})$ is fully faithful.
Proof.
Combine Proposition 9.2.3.1 with Remark 9.2.1.20.
$\square$
Applying Variant 8.4.6.9, we obtain the following recognition principle for $\operatorname{Ind}$-completions:
Proposition 9.2.3.3. Let $\kappa \leq \lambda $ be regular cardinals and let $f: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. Then $f$ exhibits $\operatorname{\mathcal{D}}$ as an $\operatorname{Ind}_{\kappa }^{\lambda }$-completion of $\operatorname{\mathcal{C}}$ if and only if it satisfies conditions $(0)$, $(1)$, and $(2)$ of Proposition 9.2.3.1, together with the following additional condition:
- $(3)$
The $\infty $-category $\operatorname{\mathcal{D}}$ is generated under $\lambda $-small, $\kappa $-filtered colimits by the image of $f$. That is, if $\operatorname{\mathcal{D}}_0 \subseteq \operatorname{\mathcal{D}}$ is a full subcategory which contains $f(\operatorname{\mathcal{C}})$ and is closed under $\lambda $-small, $\kappa $-filtered colimits, then $\operatorname{\mathcal{D}}_0 = \operatorname{\mathcal{D}}$.
In other words, every object of $\operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}})$ can be built in a single step as a $\lambda $-small $\kappa $-filtered colimit of objects of $\operatorname{\mathcal{C}}$ (rather than iteratively). See Corollary 9.2.4.21.
Corollary 9.2.3.5. Let $\kappa $ be a small regular cardinal and $f: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. Then $f$ exhibits $\operatorname{\mathcal{D}}$ as an $\operatorname{Ind}_{\kappa }$-completion of $\operatorname{\mathcal{C}}$ if and only if the following conditions are satisfied:
- $(0)$
The $\infty $-category $\operatorname{\mathcal{D}}$ admits small $\kappa $-filtered colimits.
- $(1)$
The functor $f$ is fully faithful.
- $(2)$
For each object $X \in \operatorname{\mathcal{C}}$, the image $f(X)$ is a $\kappa $-compact object of $\operatorname{\mathcal{D}}$.
- $(3)$
The $\infty $-category $\operatorname{\mathcal{D}}$ is generated under small $\kappa $-filtered colimits by the image of $f$.
Moreover, if $f$ is assumed only to satisfy conditions $(0)$, $(1)$, and $(2)$, then its $\operatorname{Ind}_{\kappa }$-extension $F: \operatorname{Ind}_{\kappa }(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{D}}$ is fully faithful.
Corollary 9.2.3.6. Let $f: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. Then $f$ exhibits $\operatorname{\mathcal{D}}$ as an $\operatorname{Ind}$-completion of $\operatorname{\mathcal{C}}$ if and only if the following conditions are satisfied:
- $(0)$
The $\infty $-category $\operatorname{\mathcal{D}}$ admits small filtered colimits.
- $(1)$
The functor $f$ is fully faithful.
- $(2)$
For each object $X \in \operatorname{\mathcal{C}}$, the image $f(X)$ is a compact object of $\operatorname{\mathcal{D}}$.
- $(3)$
The $\infty $-category $\operatorname{\mathcal{D}}$ is generated under small filtered colimits by the image of $f$.
Moreover, if $f$ satisfies conditions $(0)$, $(1)$, and $(2)$, then its $\operatorname{Ind}$-extension $F: \operatorname{Ind}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{D}}$ is fully faithful.
Example 9.2.3.7. Let $\operatorname{Set}$ be the category of sets, and let $\operatorname{Set}_{\mathrm{fin}}$ be the full subcategory spanned by the finite sets. Then the inclusion functor $\operatorname{Set}_{\mathrm{fin}} \hookrightarrow \operatorname{Set}$ exhibits (the nerve of) $\operatorname{Set}$ as an $\operatorname{Ind}$-completion of the $\operatorname{Set}_{\mathrm{fin}}$. This follows from Corollary 9.2.3.6, since every finite set is a compact object of $\operatorname{Set}$ (Example 9.2.0.4) and every set can be realized as a filtered colimit of finite sets.
Example 9.2.3.8. Let $R$ be an associative ring, let $\operatorname{\mathcal{C}}$ be (the nerve of) the category of left $R$-modules, and let $\operatorname{\mathcal{C}}_{\mathrm{fin}}$ denote the full subcategory of $\operatorname{\mathcal{C}}$ spanned by the finitely presented left $R$-modules. Then the inclusion functor $\operatorname{\mathcal{C}}_{\mathrm{fin}} \hookrightarrow \operatorname{\mathcal{C}}$ exhibit $\operatorname{\mathcal{C}}$ as an $\operatorname{Ind}$-completion of $\operatorname{\mathcal{C}}_{\mathrm{fin}}$. This follows from Corollary 9.2.3.6, since every finitely presented $R$-module $M$ is a compact object of $\operatorname{\mathcal{C}}$ (Proposition 9.2.0.1), and every $R$-module can be realized as a filtered colimit of finitely presented $R$-modules. This assertion has counterparts for many other mathematical structures studied in algebra; see Remark 9.2.0.3.
Variant 9.2.3.9 (Lazard's Theorem). Let $R$ be an associative ring. Recall that a (left) $R$-module $M$ is flat if the tensor product functor
\[ \{ \textnormal{Right $R$-modules} \} \xrightarrow { \otimes _{R} M} \{ \textnormal{Abelian Groups} \} \]
is exact. It is easy to see that the collection of flat $R$-modules is closed under the formation of filtered colimits. Conversely, a theorem of Lazard ([MR0168625]) asserts that every flat $R$-module $M$ can be realized as a filtered colimit of free $R$-modules of finite rank (see 
for a more general statement). It follows that the category of flat $R$-modules is an $\operatorname{Ind}$-completion of the full subcategory spanned by the free $R$-modules of finite rank.
Example 9.2.3.10. Let $X$ be a Kan complex. Then the identity map $\operatorname{id}_{X}: X \rightarrow X$ exhibits $X$ as an $\operatorname{Ind}$-completion of itself. This follows from the criterion of Corollary 9.2.3.6, since every vertex $x \in X$ is compact (see Example 9.2.2.5). More generally, for every pair of regular cardinals $\kappa \leq \lambda $, the identity map $\operatorname{id}_{X}$ exhibits $X$ as an $\operatorname{Ind}_{\kappa }^{\lambda }$-completion of itself.
Example 9.2.3.11. Let $S$ be a set. For every regular cardinal $\kappa $, let $P_{< \kappa }(S)$ denote the nerve of the partially ordered set of $\kappa $-small subsets of $S$. For every regular cardinal $\lambda \geq \kappa $, Proposition 9.2.3.1 and Variant 9.2.2.11 supplies a fully faithful functor
\[ \operatorname{Ind}_{\kappa }^{\lambda }( P_{< \kappa }(S) ) \hookrightarrow P_{ < \lambda }(S). \]
This functor is an equivalence if and only if every $\lambda $-small subset $S' \subseteq S$ can be realized as a $\lambda $-small $\kappa $-filtered union of $\kappa $-small subsets of $S'$ (Remark 9.2.3.4). Assuming that $S$ has cardinality $\geq \lambda $, this is equivalent to the requirement that $\kappa \trianglelefteq \lambda $ (see Definition 9.1.7.5).
Exercise 9.2.3.12 ($\operatorname{Ind}$-Completion of Partially Ordered Sets). Let $(A, \leq )$ be a (small) partially ordered set. Recall that an ideal of $A$ is a subset $J \subseteq A$ satisfying the following conditions:
The set $J$ is downward-closed: that is, for elements $a \leq b$ of $A$, if $b$ belongs to $J$, then $a$ also belongs to $J$.
The set $J$ is directed: that is, every finite subset of $J$ has an upper bound (which is also contained in $J$).
Let $\widehat{A}$ denote the collection of all ideals $J \subseteq A$, which we regard as partially ordered by inclusion. Show that the nondecreasing function
\[ A \rightarrow \widehat{A} \quad \quad (b \in A) \mapsto (A_{\leq b} = \{ a \in A: a \leq b \} ) \]
exhibits the $\infty $-category $\operatorname{N}_{\bullet }( \widehat{A} )$ as an $\operatorname{Ind}$-completion of $\operatorname{N}_{\bullet }(A)$.
It follows from Exercise 9.2.3.12 that if an $\infty $-category $\operatorname{\mathcal{C}}$ is equivalent to the nerve of a partially ordered set, then the $\operatorname{Ind}$-completion $\operatorname{Ind}(\operatorname{\mathcal{C}})$ has the same property. This is a special case of the following more generally assertion:
Proposition 9.2.3.13. Let $\kappa \leq \lambda $ be regular cardinals, let $\operatorname{\mathcal{C}}$ be an $\infty $-category, and let $n$ be an integer. Then $\operatorname{\mathcal{C}}$ is locally $n$-truncated (see Definition 4.8.2.1) if and only if $\operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}})$ is locally $n$-truncated.
Proof.
We proceed as in the proof of Proposition 8.7.3.11. Choose a functor $H: \operatorname{\mathcal{C}}\rightarrow \operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}})$ which exhibits $\operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}})$ as an $\operatorname{Ind}_{\kappa }^{\lambda }$-completion of $\operatorname{\mathcal{C}}$. Assume that $\operatorname{\mathcal{C}}$ is locally $n$-truncated; we will show that $\operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}})$ has the same property (the converse is immediate, since the functor $H$ is fully faithful). Fix objects $\widehat{X},\widehat{Y} \in \operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}})$; we wish to show that the morphism space $\operatorname{Hom}_{ \operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}}) }( \widehat{X}, \widehat{Y})$ is an $n$-truncated Kan complex. Let us first regard the object $\widehat{Y}$ as fixed, and let $\operatorname{\mathcal{D}}$ denote the full subcategory of $\operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}})$ spanned by those objects $\widehat{X}$ for which $\operatorname{Hom}_{ \operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}})}( \widehat{X}, \widehat{Y})$ is $n$-truncated. Since the collection of $n$-truncated spaces is closed under limits (Remark 7.4.1.5), we see that $\operatorname{\mathcal{D}}$ is closed under colimits in $\operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}})$ (see Proposition 7.4.1.18). In particular, it is closed under $\lambda $-small $\kappa $-filtered colimits. Consequently, to show that $\operatorname{\mathcal{D}}= \operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}})$, it will suffice to show that the mapping space $\operatorname{Hom}_{ \operatorname{Ind}(\operatorname{\mathcal{C}})}( \widehat{X}, \widehat{Y})$ is $n$-truncated in the special case $\widehat{X} = H(X)$ for some object $X \in \operatorname{\mathcal{C}}$.
Let us now regard the object $\widehat{X} = H(X)$ as fixed, and let $\operatorname{\mathcal{D}}' \subseteq \operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}})$ denote the full subcategory spanned by those objects $\widehat{Y}$ for which the morphism space $\operatorname{Hom}_{ \operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}})}( \widehat{X}, \widehat{Y})$ is an $n$-truncated Kan complex. Since $\widehat{X}$ is a $(\kappa ,\lambda )$-compact object of $\operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}})$, the subcategory $\operatorname{\mathcal{D}}'$ is closed under $\lambda $-small $\kappa $-filtered colimits (Variant 9.1.10.3). Consequently, to prove that $\operatorname{\mathcal{D}}' = \operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}})$, it will suffice to show that it contains $H(Y)$, for each object $Y \in \operatorname{\mathcal{C}}$. Since $H$ is fully faithful, we are reduced to proving that the morphism space $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$ is $n$-truncated, which follows from our assumption that the $\infty $-category $\operatorname{\mathcal{C}}$ is locally $n$-truncated.
$\square$
Corollary 9.2.3.14. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. If $\operatorname{\mathcal{C}}$ is equivalent to the nerve of an ordinary category, then $\operatorname{Ind}(\operatorname{\mathcal{C}})$ has the same property.
Proof.
Apply Proposition 9.2.3.13 in the special case where $\kappa = \aleph _0$, $\lambda = \operatorname{\textnormal{\cjRL {t}}}$ (Example 9.2.1.9) and $n = 0$ (see Corollary 4.8.2.16).
$\square$
These properties determine the category $\widehat{\operatorname{\mathcal{C}}}$ uniquely up to equivalence. To emphasize the uniqueness, we denote $\widehat{\operatorname{\mathcal{C}}}$ by $\operatorname{Ind}( \operatorname{\mathcal{C}})$ and refer to it as the $\operatorname{Ind}$-completion of the category $\operatorname{\mathcal{C}}$. Corollary 9.2.3.14 then supplies an equivalence of $\infty $-categories
\[ \operatorname{Ind}( \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) ) \xrightarrow {\sim } \operatorname{N}_{\bullet }( \operatorname{Ind}(\operatorname{\mathcal{C}}) ). \]
In other words, the category $\operatorname{Ind}(\operatorname{\mathcal{C}})$ automatically satisfies a stronger version of condition $(b)$, where we allow $\operatorname{\mathcal{D}}$ to be an $\infty $-category.
Proposition 9.2.3.16. Let $\kappa \leq \lambda $ be regular cardinals and let $\operatorname{\mathcal{C}}$ be an $\infty $-category. Then the tautological functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}})$ preserves all $\kappa $-small colimits which exist in $\operatorname{\mathcal{C}}$.
Proof.
Let $\mu \geq \lambda $ be an uncountable cardinal of exponential cofinality $\geq \kappa $ such that $\operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}})$ is locally $\mu $-small, and let $h^{\bullet }: \operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}})^{\operatorname{op}} \rightarrow \operatorname{Fun}( \operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}}), \operatorname{\mathcal{S}}_{< \mu } )$ be a contravariant Yoneda embedding for $\operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}})$. By virtue of Proposition 7.4.1.18, it will suffice to show that the composite functor $(h^{\bullet } \circ F^{\operatorname{op}}): \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{Fun}( \operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}}), \operatorname{\mathcal{S}}_{< \mu } )$ preserves $\kappa $-small limits.
Since $F$ carries each object of $\operatorname{\mathcal{C}}$ to a $(\kappa ,\lambda )$-compact object of $\operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}})$ (Proposition 9.2.3.3), the functor $h^{\bullet } \circ F^{\operatorname{op}}$ factors through the full subcategory $\operatorname{Fun}^{(\kappa ,\lambda )-\operatorname{fin}}( \operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}}), \operatorname{\mathcal{S}}_{< \mu })$ of $(\kappa ,\lambda )$-finitary functors from $\operatorname{Ind}(\operatorname{\mathcal{C}})$ to $\operatorname{\mathcal{S}}_{< \mu }$. Since $\lambda $-small $\kappa $-filtered colimits commute with $\kappa $-small limits in $\operatorname{\mathcal{S}}_{< \mu }$ (Proposition 9.1.5.8), the subcategory $\operatorname{Fun}^{(\kappa ,\lambda )-\operatorname{fin}}( \operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}}), \operatorname{\mathcal{S}}_{< \mu })$ is closed under $\kappa $-small limits. It will therefore suffice to show that $(h^{\bullet } \circ F^{\operatorname{op}}): \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{Fun}^{(\kappa ,\lambda )-\operatorname{fin}}( \operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}}), \operatorname{\mathcal{S}}_{< \mu })$ preserves finite limits. Since $F$ exhibits $\operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}})$ as an $\operatorname{Ind}_{\kappa }^{\lambda }$-completion of $\operatorname{\mathcal{C}}$, precomposition with $F$ induces an equivalence of $\infty $-categories $F^{\ast }: \operatorname{Fun}^{(\kappa ,\lambda )-\operatorname{fin}}( \operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}}), \operatorname{\mathcal{S}}_{< \mu } ) \rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}_{< \mu } )$. We are therefore reduced to showing that the composite functor
\[ (F^{\ast } \circ h^{\bullet } \circ F^{\operatorname{op}} ): \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}_{< \mu } ) \]
preserves $\kappa $-small limits. Since $F$ is fully faithful, this composition is a contravariant Yoneda embedding for the $\infty $-category $\operatorname{\mathcal{C}}$, and therefore preserves all limits which exist in $\operatorname{\mathcal{C}}^{\operatorname{op}}$ (Proposition 7.4.1.18).
$\square$