Section 9.3 (06FR): Ind-Completions of $\infty $-Categories

9.3 Ind-Completions of $\infty $-Categories

Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. In §9.3.1, we show that $\operatorname{\mathcal{C}}$ admits an $\operatorname{Ind}$-completion $\operatorname{Ind}(\operatorname{\mathcal{C}})$, which is characterized (up to equivalence) by the existence of a functor $H: \operatorname{\mathcal{C}}\rightarrow \operatorname{Ind}(\operatorname{\mathcal{C}})$ satisfying the following conditions:

$(a)$

The $\infty $-category $\operatorname{Ind}(\operatorname{\mathcal{C}})$ admits small filtered colimits.

$(b)$

For every $\infty $-category $\operatorname{\mathcal{D}}$ which admits small filtered colimits, precomposition with $H$ induces an equivalence of $\infty $-categories

\[ \operatorname{Fun}^{\operatorname{fin}}( \operatorname{Ind}(\operatorname{\mathcal{C}}), \operatorname{\mathcal{D}}) \rightarrow \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}). \]

Here $\operatorname{Fun}^{\operatorname{fin}}( \operatorname{Ind}(\operatorname{\mathcal{C}}), \operatorname{\mathcal{D}})$ denotes the full subcategory of $\operatorname{Fun}( \operatorname{Ind}(\operatorname{\mathcal{C}}), \operatorname{\mathcal{D}})$ spanned by those functors which preserve small filtered colimits (see Definition 9.2.2.1).

Stated more informally, the $\infty $-category $\operatorname{Ind}(\operatorname{\mathcal{C}})$ is obtained from $\operatorname{\mathcal{C}}$ by “freely” adjoining colimits of small filtered diagrams.

It follows immediately from the definitions that if an $\infty $-category $\operatorname{\mathcal{C}}$ admits an $\operatorname{Ind}$-completion, then $\operatorname{Ind}(\operatorname{\mathcal{C}})$ is uniquely determined up to equivalence. The existence of $\operatorname{Ind}(\operatorname{\mathcal{C}})$ follows from the general results of §8.4.5. Assume for simplicity that the $\infty $-category $\operatorname{\mathcal{C}}$ is small. Using Proposition 8.4.5.8, we can identify $\operatorname{Ind}(\operatorname{\mathcal{C}})$ with the smallest full subcategory of $\operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{S}})$ which contains all representable functors and is closed under small filtered colimits. This subcategory can be described more intrinsically:

In §9.3.4, we show that a functor $\mathscr {F}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{S}}$ belongs to $\operatorname{Ind}(\operatorname{\mathcal{C}})$ if and only if the $\infty $-category of elements $\int _{\operatorname{\mathcal{C}}^{\operatorname{op}} } \mathscr {F}$ is cofiltered (Proposition 9.3.4.5). If this condition is satisfied, we say that the functor $\mathscr {F}$ is flat (Definition 9.3.4.1).
Assume that the $\infty $-category $\operatorname{\mathcal{C}}$ admits finite colimits (so that $\operatorname{\mathcal{C}}^{\operatorname{op}}$ admits finite limits). In §9.3.5, we show that a functor $\mathscr {F}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{S}}$ belongs to $\operatorname{Ind}(\operatorname{\mathcal{C}})$ if and only if it preserves finite limits (Corollary 9.3.5.29). If this condition is satisfied, we say that $\mathscr {F}$ is left exact (see Definition 9.3.5.5 and Theorem 9.3.5.8).

For some applications, it will be useful to have a recognition principle for $\operatorname{Ind}$-completions which is does not explicitly reference their universal property. Suppose we are given a functor of $\infty $-categories $H: \operatorname{\mathcal{C}}\rightarrow \widehat{\operatorname{\mathcal{C}}}$, where $\widehat{\operatorname{\mathcal{C}}}$ admits small filtered colimits. In §9.3.2, we show that $H$ exhibits $\widehat{\operatorname{\mathcal{C}}}$ as an $\operatorname{Ind}$-completion of $\operatorname{\mathcal{C}}$ if and only if its essential image consist of compact objects which generate $\widehat{\operatorname{\mathcal{C}}}$ under small filtered colimits (Corollary 9.3.2.6). If this condition is satisfied, then $\widehat{\operatorname{\mathcal{C}}}$ is generated by $\operatorname{\mathcal{C}}$ in a very strong sense: every object of $\widehat{\operatorname{\mathcal{C}}}$ can be realized as the colimit of a small filtered diagram which factors through $\operatorname{\mathcal{C}}$ (Corollary 9.3.4.17).

Remark 9.3.0.1. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let us abuse notation by identifying $\operatorname{\mathcal{C}}$ with a full subcategory of its $\operatorname{Ind}$-completion $\operatorname{Ind}(\operatorname{\mathcal{C}})$. Then every object $X \in \operatorname{Ind}(\operatorname{\mathcal{C}})$ can be realized as the colimit of a small filtered diagram $\{ X_ i \} _{i \in \operatorname{\mathcal{I}}}$ in the $\infty $-category $\operatorname{\mathcal{C}}$. If $Y \in \operatorname{Ind}(\operatorname{\mathcal{C}})$ is the colimit of another small filtered diagram $\{ Y_ j \} _{j \in \operatorname{\mathcal{J}}}$ in $\operatorname{\mathcal{C}}$, then we have canonical homotopy equivalences

\begin{eqnarray*} \operatorname{Hom}_{ \operatorname{Ind}(\operatorname{\mathcal{C}}) }( X, Y) & = & \operatorname{Hom}_{ \operatorname{Ind}(\operatorname{\mathcal{C}}) }( \varinjlim _{i \in \operatorname{\mathcal{I}}} X_ i, Y ) \\ & \simeq & \varprojlim _{i \in \operatorname{\mathcal{I}}} \operatorname{Hom}_{\operatorname{Ind}(\operatorname{\mathcal{C}})}( X_ i, Y) \\ & = & \varprojlim _{i \in \operatorname{\mathcal{I}}} \operatorname{Hom}_{\operatorname{Ind}(\operatorname{\mathcal{C}})}( X_ i, \varinjlim _{j} Y_ j ) \\ & \simeq & \varprojlim _{i \in \operatorname{\mathcal{I}}} \varinjlim _{j \in \operatorname{\mathcal{J}}} \operatorname{Hom}_{\operatorname{Ind}(\operatorname{\mathcal{C}})}( X_ i, Y_ j) \\ & = & \varprojlim _{i \in \operatorname{\mathcal{I}}} \varinjlim _{j \in \operatorname{\mathcal{J}}} \operatorname{Hom}_{\operatorname{\mathcal{C}}}( X_ i, Y_ j). \end{eqnarray*}

Here the limits and colimits are computed in the $\infty $-category of spaces $\operatorname{\mathcal{S}}$, and the equivalence on the fourth line comes from the observation that each $X_ i$ is compact as an object of $\operatorname{\mathcal{C}}$.

We therefore obtain the following more informal description of the $\infty $-category $\operatorname{Ind}(\operatorname{\mathcal{C}})$:

Objects of $\operatorname{Ind}(\operatorname{\mathcal{C}})$ are represented by small filtered diagrams $\{ X_ i \} _{i \in \operatorname{\mathcal{I}}}$ in the $\infty $-category $\operatorname{\mathcal{C}}_0$.
Morphism spaces in $\operatorname{Ind}(\operatorname{\mathcal{C}})$ are given by the formula

\[ \operatorname{Hom}_{ \operatorname{Ind}(\operatorname{\mathcal{C}}) }( \{ X_ i \} _{i \in \operatorname{\mathcal{I}}}, \{ Y_ j \} _{j \in \operatorname{\mathcal{J}}} ) = \varprojlim _{i \in \operatorname{\mathcal{I}}} \varinjlim _{j \in \operatorname{\mathcal{J}}} \operatorname{Hom}_{\operatorname{\mathcal{C}}}( X_ i, Y_ j). \]

Remark 9.3.0.2. For any $\infty $-category $\operatorname{\mathcal{C}}$, the $\operatorname{Ind}$-completion $\operatorname{Ind}(\operatorname{\mathcal{C}})$ is obtained from $\operatorname{\mathcal{C}}$ by (freely) adjoining colimits of diagrams $\operatorname{\mathcal{K}}\rightarrow \operatorname{\mathcal{C}}$, where $\operatorname{\mathcal{K}}$ is a small filtered $\infty $-category. One can consider many variants of this construction, where we impose additional conditions on the $\infty $-category $\operatorname{\mathcal{K}}$. We will be particularly interested in the special case where we demand that $\operatorname{\mathcal{K}}$ is $\kappa $-filtered and $\lambda $-small, for some pair of regular cardinals $\kappa \leq \lambda $; in this case, we denote the corresponding enlargement of $\operatorname{\mathcal{C}}$ by $\operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}})$. Most of the pleasant properties of the construction $\operatorname{\mathcal{C}}\mapsto \operatorname{Ind}(\operatorname{\mathcal{C}})$ hold also for the construction $\operatorname{\mathcal{C}}\mapsto \operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}})$, at least when the cardinals $\kappa $ and $\lambda $ satisfy the condition $\kappa \trianglelefteq \lambda $ (see Definition 9.1.7.5). In §9.3.6, we show that the formation of restricted $\operatorname{Ind}$-completions is transitive in the following sense:

For every triple of regular cardinals $\kappa \leq \lambda \leq \mu $, there is a fully faithful functor

\[ T: \operatorname{Ind}_{\lambda }^{\mu }( \operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}}) ) \rightarrow \operatorname{Ind}_{\kappa }^{\mu }(\operatorname{\mathcal{C}}); \]

see Construction 9.3.6.2 and Proposition 9.3.6.3.
If $\kappa \trianglelefteq \lambda \trianglelefteq \mu $, then $T$ is an equivalence of $\infty $-categories (Theorem 9.3.6.4).

Structure

Subsection 9.3.1: Ind-Completion
Subsection 9.3.2: Recognition of Ind-Completions
Subsection 9.3.3: Functoriality of Ind-Completion
Subsection 9.3.4: Flat Functors
Subsection 9.3.5: Exact Functors
Subsection 9.3.6: Transitivity of Ind-Completion
Subsection 9.3.7: Final Objects of Ind-Completions