Let $R$ be an associative ring. Recall that a (left) $R$-module $M$ is finitely presented if there is an exact sequence of $R$-modules
where $m$ and $n$ are nonnegative integers. Equivalently, $M$ is finitely presented if it can be realized as a quotient $R^{n} / K$, where $K \subseteq R^ n$ is a finitely generated submodule. This condition admits a purely categorical reformulation:
Proposition 9.2.0.1. Let $R$ be an associative ring and let $M$ be a left $R$-module. Then $M$ is finitely presented if and only if the functor preserves small filtered colimits.
\[ \{ \textnormal{$R$-modules} \} \rightarrow \operatorname{Set}\quad \quad N \mapsto \operatorname{Hom}_{R}(M,N) \]
Proof. Assume first that $R$ is finitely presented: that is, there exists an exact sequence of left $R$-modules
For any filtered diagram of $R$-modules $\{ N_ i \} $ having colimit $N$, we have a commutative diagram of exact sequences
Since the middle and right vertical maps are bijective, the left vertical map is also bijective.
We now prove the converse. Let $\{ M_{\alpha } \} $ be the collection of all finitely generated submodules of $M$, partially ordered by inclusion. Then $M$ is the colimit of the filtered diagram $\{ M_{\alpha } \} $. If the functor $\operatorname{Hom}_{R}(M, \bullet )$ preserves small filtered colimits, then the tautological map
is a bijection. In particular, the identity map $\operatorname{id}_{M}: M \rightarrow M$ factors through a finitely generated submodule of $M$, so $M$ is generated by a finite collection of elements $x_1, x_2, \cdots , x_ n$. Let $R^{n}$ be the free $R$-module on generators $\widetilde{x}_1, \cdots , \widetilde{x}_{n}$, so there is surjective $R$-module homomorphism $f: R^{n} \twoheadrightarrow M$ satisfying $f( \widetilde{x}_ i ) = x_ i$ for $1 \leq i \leq n$. Let $K$ be the kernel of $f$, and let $\{ K_{\beta } \} $ be the collection of all finitely generated submodules of $K$, partially ordered by inclusion. If the functor $\operatorname{Hom}_{R}(M, \bullet )$ preserves small filtered colimits, then the tautological map
is bijective. In particular, the identity map $\operatorname{id}_{M}$ factors as a composition
where $K'$ is some finitely generated submodule of $K$. Enlarging $K'$ if necessary, we may assume that $g$ carries each $x_ i$ to the image of $\widetilde{x}_ i$ in $R^{n} / K'$: that is, the composite map $g \circ f$ coincides with the tautological map from $R^{n}$ to $R^{n}/K'$. We then have $K = \ker (f) \subseteq \ker (g \circ f) = K'$, so that $K = K'$ is finitely generated and $M = R^{n} / K$ is finitely presented, as desired. $\square$
Proposition 9.2.0.1 motivates the following:
Definition 9.2.0.2. Let $\operatorname{\mathcal{C}}$ be a category which admits small filtered colimits. We say that an object $C \in \operatorname{\mathcal{C}}$ is compact if the functor $D \mapsto \operatorname{Hom}_{\operatorname{\mathcal{C}}}(C,D)$ preserves small filtered colimits.
Remark 9.2.0.3. Let $R$ be an associative ring and let $\operatorname{\mathcal{C}}$ be the category of left $R$-modules. Proposition 9.2.0.1 asserts that an object $M \in \operatorname{\mathcal{C}}$ is compact if and only if it is finitely presented as an $R$-module. This result has counterparts for many other classes of mathematical structures, which can be proved by the same argument:
Let $\operatorname{\mathcal{C}}= \operatorname{Group}$ be the category of groups. An object $G \in \operatorname{\mathcal{C}}$ is compact if and only if it is finitely presented as a group: that is, it can be realized as the quotient of a finitely generated free group by the normal subgroup generated by finitely many elements.
Let $\operatorname{\mathcal{C}}= \mathrm{Ring}$ be the category of associative rings. Then an object $R \in \operatorname{\mathcal{C}}$ is compact if and only if it is finitely presented as a ring: that is, it can be realized as the quotient of a noncommutative polynomial ring $\operatorname{\mathbf{Z}}\langle x_1, x_2, \cdots , x_ n \rangle $ by a finitely generated two-sided ideal.
Let $\operatorname{\mathcal{C}}= \mathrm{CRing}$ be the category of commutative rings. Then an object $R \in \operatorname{\mathcal{C}}$ is compact if and only if it is finitely presented as a commutative ring: that is, it can be realized as the quotient of a finitely generated polynomial ring $\operatorname{\mathbf{Z}}[x_1, \cdots , x_ n]$ by an ideal $I$ (here $I$ is automatically finitely generated, by virtue of the Hilbert basis theorem).
Example 9.2.0.4. Let $\operatorname{\mathcal{C}}= \operatorname{Set}$ be the category of sets. An object $S \in \operatorname{\mathcal{C}}$ is compact if and only if is is a finite set. See Example 9.2.2.9 for a more general statement.
Variant 9.2.0.5. Let $\operatorname{\mathcal{C}}= \operatorname{Set_{\Delta }}$ be the category of simplicial sets. An object $S \in \operatorname{\mathcal{C}}$ is compact if and only if it is a finite simplicial set, in the sense of Definition 3.6.1.1. See Proposition 3.6.1.7.
Example 9.2.0.6. Let $X$ be a topological space and let $\operatorname{\mathcal{C}}$ be the category of open sets of $X$ (with morphisms given by inclusions). Then an object $U \in \operatorname{\mathcal{C}}$ is compact if and only if $U$ is quasi-compact when viewed as a topological space: that is, every open cover of $U$ admits a finite subcover.
Definition 9.2.0.2 has an $\infty $-categorical counterpart. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category which admits small filtered colimits, and assume for simplicity that $\operatorname{\mathcal{C}}$ is locally small. By virtue of Theorem 5.6.6.13, every object $C \in \operatorname{\mathcal{C}}$ determines a corepresentable functor
We say that the object $C$ is compact if the functor $h^{C}$ is finitary: that is, if it preserves small filtered colimits (Definition 9.2.2.1). In §9.2.2, we study several variants of this condition and summarize their formal properties.
Our primary attention in this section is on $\infty $-categories which contain “enough” compact objects. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category which admits small filtered colimits, and let $\operatorname{\mathcal{C}}_{< \aleph _0}$ denote the full subcategory of $\operatorname{\mathcal{C}}$ spanned by its compact objects. We will say that $\operatorname{\mathcal{C}}$ is compact generated if every object of $\operatorname{\mathcal{C}}$ can be realized as the colimit of a small filtered diagram in $\operatorname{\mathcal{C}}_{< \aleph _0}$ (Definition 9.2.6.1). In §9.2.6 we show that, if this condition is satisfied, then the $\infty $-categories $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{C}}_{< \aleph _0}$ are essentially interchangeable data. More precisely, Corollary 9.2.6.18 implies that the construction $\operatorname{\mathcal{C}}\mapsto \operatorname{\mathcal{C}}_{< \aleph _0}$ induces a bijection
The bulk of our focus will be in understanding the inverse of the preceding bijection. In §9.2.1, we show that every $\infty $-category $\operatorname{\mathcal{C}}_0$ admits an $\operatorname{Ind}$-completion $\operatorname{Ind}(\operatorname{\mathcal{C}}_0)$, which is characterized (up to equivalence) by the existence of a functor $H: \operatorname{\mathcal{C}}_0 \rightarrow \operatorname{Ind}(\operatorname{\mathcal{C}}_0)$ with the following properties:
The $\infty $-category $\operatorname{Ind}(\operatorname{\mathcal{C}}_0)$ admits small filtered colimits.
For every $\infty $-category $\operatorname{\mathcal{D}}$ which admits small filtered colimits, precomposition with $H$ induces an equivalence of $\infty $-categories
Here $\operatorname{Fun}^{\operatorname{fin}}( \operatorname{Ind}(\operatorname{\mathcal{C}}_0), \operatorname{\mathcal{D}})$ denotes the full subcategory of $\operatorname{Fun}( \operatorname{Ind}(\operatorname{\mathcal{C}}_0), \operatorname{\mathcal{D}})$ spanned by those functors which preserve small filtered colimits.
Stated more informally, the $\infty $-category $\operatorname{Ind}(\operatorname{\mathcal{C}}_0)$ is obtained from $\operatorname{\mathcal{C}}_0$ by “freely” adjoining colimits of small filtered diagrams. In §9.2.3, we show that condition $(b)$ is satisfied if and only if the functor $H$ is fully faithful, and its essential image consists of compact objects which generate $\operatorname{Ind}(\operatorname{\mathcal{C}}_0)$ under small filtered colimits (Corollary 9.2.3.6). From this recognition principle we deduce the following (Corollary 9.2.6.10):
For every $\infty $-category $\operatorname{\mathcal{C}}_0$, the $\operatorname{Ind}$-completion $\operatorname{Ind}(\operatorname{\mathcal{C}}_0)$ is compactly generated (Corollary 9.2.6.10).
If $\operatorname{\mathcal{C}}$ is a compactly generated $\infty $-category, then it is an $\operatorname{Ind}$-completion of the full subcategory $\operatorname{\mathcal{C}}_{< \aleph _0}$.
Let $\operatorname{\mathcal{C}}_0$ be an $\infty $-category. It follows immediately from the definitions that the $\operatorname{Ind}$-completion $\operatorname{Ind}(\operatorname{\mathcal{C}}_0)$ is unique (up to equivalence) provided that it exists. The existence of $\operatorname{Ind}(\operatorname{\mathcal{C}}_0)$ follows from general results of §8.4.5. Assume for simplicity that the $\infty $-category $\operatorname{\mathcal{C}}_0$ is small. Using Proposition 8.4.5.8, we can identify $\operatorname{Ind}(\operatorname{\mathcal{C}}_0)$ with the smallest full subcategory of $\operatorname{Fun}( \operatorname{\mathcal{C}}_0^{\operatorname{op}}, \operatorname{\mathcal{S}})$ which contains all representable functors and is closed under small filtered colimits. This subcategory can be described more intrinsically:
In §9.2.4, we show that a functor $\mathscr {F}: \operatorname{\mathcal{C}}_0^{\operatorname{op}} \rightarrow \operatorname{\mathcal{S}}$ belongs to $\operatorname{Ind}(\operatorname{\mathcal{C}})$ if and only if the $\infty $-category of elements $\int _{\operatorname{\mathcal{C}}_0^{\operatorname{op}} } \mathscr {F}$ is cofiltered (Proposition 9.2.4.5). If this condition is satisfied, we say that the functor $\mathscr {F}$ is flat (Definition 9.2.4.1).
Assume that the $\infty $-category $\operatorname{\mathcal{C}}_0$ admits finite colimits (so that $\operatorname{\mathcal{C}}_0^{\operatorname{op}}$ admits finite limits). In §9.2.5, we show that a functor $\mathscr {F}: \operatorname{\mathcal{C}}_0^{\operatorname{op}} \rightarrow \operatorname{\mathcal{S}}$ belongs to $\operatorname{Ind}(\operatorname{\mathcal{C}})$ if and only if it preserves finite limits (Corollary 9.2.5.27). If this condition is satisfied, we say that $\mathscr {F}$ is left exact (see Definition 9.2.5.5 and Theorem 9.2.5.8).
Remark 9.2.0.7. Let $\operatorname{\mathcal{C}}_0$ be an $\infty $-category and let us abuse notation by identifying $\operatorname{\mathcal{C}}_0$ with a full subcategory of its $\operatorname{Ind}$-completion $\operatorname{\mathcal{C}}= \operatorname{Ind}(\operatorname{\mathcal{C}}_0)$. Using the results of §9.2.4, we will see that every object $C \in \operatorname{\mathcal{C}}$ can be realized the colimit of a small filtered diagram $\{ C_ i \} _{i \in \operatorname{\mathcal{I}}}$ in the $\infty $-category $\operatorname{\mathcal{C}}_0$ (Corollary 9.2.4.21). If $D$ is the colimit of a small filtered diagram $\{ D_ j \} _{j \in \operatorname{\mathcal{J}}}$ in $\operatorname{\mathcal{C}}_0$, then we have canonical homotopy equivalences 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 $C_ i$ is compact as an object of $\operatorname{\mathcal{C}}$. We therefore obtain the following more informal description of the $\operatorname{Ind}$-completion:
Objects of $\operatorname{Ind}(\operatorname{\mathcal{C}}_0)$ are represented by small filtered diagrams $\{ C_ i \} _{i \in \operatorname{\mathcal{I}}}$ in the $\infty $-category $\operatorname{\mathcal{C}}_0$.
Morphism spaces in $\operatorname{Ind}(\operatorname{\mathcal{C}}_0)$ are given by the formula
\begin{eqnarray*} \operatorname{Hom}_{ \operatorname{\mathcal{C}}}( C, D) & = & \operatorname{Hom}_{ \operatorname{\mathcal{C}}}( \varinjlim _{i \in \operatorname{\mathcal{I}}} C_ i, D ) \\ & \simeq & \varprojlim _{i \in \operatorname{\mathcal{I}}} \operatorname{Hom}_{\operatorname{\mathcal{C}}}( C_ i, D) \\ & = & \varprojlim _{i \in \operatorname{\mathcal{I}}} \operatorname{Hom}_{\operatorname{\mathcal{C}}}( C_ i, \varinjlim _{j} D_ j ) \\ & \simeq & \varprojlim _{i \in \operatorname{\mathcal{I}}} \varinjlim _{j \in \operatorname{\mathcal{J}}} \operatorname{Hom}_{\operatorname{\mathcal{C}}}( C_ i, D_ j) \\ & = & \varprojlim _{i \in \operatorname{\mathcal{I}}} \varinjlim _{j \in \operatorname{\mathcal{J}}} \operatorname{Hom}_{\operatorname{\mathcal{C}}_0}( C_ i, D_ j). \end{eqnarray*}
\[ \operatorname{Hom}_{ \operatorname{Ind}(\operatorname{\mathcal{C}}_0) }( \{ C_ i \} _{i \in \operatorname{\mathcal{I}}}, \{ D_ j \} _{j \in \operatorname{\mathcal{J}}} ) = \varprojlim _{i \in \operatorname{\mathcal{I}}} \varinjlim _{j \in \operatorname{\mathcal{J}}} \operatorname{Hom}_{\operatorname{\mathcal{C}}_0}( C_ i, D_ j). \]
Corollary 9.2.6.10 asserts that an $\infty $-category $\operatorname{\mathcal{C}}$ is compactly generated if and only if can be realized as an $\operatorname{Ind}$-completion $\operatorname{Ind}(\operatorname{\mathcal{C}}_0)$, for some $\infty $-category $\operatorname{\mathcal{C}}_0$. Beware that, in this case, the $\infty $-category $\operatorname{\mathcal{C}}_0$ is not unique up to equivalence: it is determined only up to Morita equivalence (Exercise 9.2.6.14). One can eliminate this ambiguity by requiring that $\operatorname{\mathcal{C}}_0$ is idempotent complete: in this case, one can identify $\operatorname{\mathcal{C}}_0$ with the $\infty $-category $\operatorname{\mathcal{C}}_{< \aleph _0}$ of compact objects of $\operatorname{\mathcal{C}}$ (Proposition 9.2.6.13). However, there are often other candidates for $\operatorname{\mathcal{C}}_0$ which arise more naturally in practice. This occurs already in the special case where $\operatorname{\mathcal{C}}= \operatorname{\mathcal{S}}$ is the $\infty $-category of spaces. In §9.2.7, we show that $\operatorname{\mathcal{S}}$ can be realized as the idempotent completion of the full subcategory $\operatorname{\mathcal{S}}_{\mathrm{fin}}$ spanned by those Kan complexes $X$ essentially finite: that is, which admit a weak homotopy equivalence $K \rightarrow X$, where $K$ is a finite simplicial set (Corollary 9.2.7.6). Beware that $\operatorname{\mathcal{S}}_{\mathrm{fin}}$ is not closed under retracts, and is therefore strictly smaller than the $\operatorname{\mathcal{S}}_{< \aleph _0}$ of compact objects of $\operatorname{\mathcal{S}}$ (Warning 9.2.7.9). A similar phenomenon occurs when $\operatorname{\mathcal{C}}= \operatorname{\mathcal{QC}}$ is the $\infty $-category of small $\infty $-categories, which we discuss in §9.2.8.
Remark 9.2.0.8. For any $\infty $-category $\operatorname{\mathcal{C}}_0$, the $\operatorname{Ind}$-completion $\operatorname{Ind}(\operatorname{\mathcal{C}}_0)$ is obtained from $\operatorname{\mathcal{C}}_0$ by (freely) adjoining colimits of diagrams $\operatorname{\mathcal{I}}\rightarrow \operatorname{\mathcal{C}}_0$, where $\operatorname{\mathcal{I}}$ is a small filtered $\infty $-category. For some purposes, it is convenient to consider variants of this construction where we impose additional conditions on the $\infty $-category $\operatorname{\mathcal{I}}$: for example, we can allow only $\infty $-categories $\operatorname{\mathcal{I}}$ which are $\kappa $-filtered and $\lambda $-small, for some pair of regular cardinals $\kappa \leq \lambda $ (Variant 9.2.1.7). With an eye toward future applications, we formulate many of our results in a way that applies to this more general notion of $\operatorname{Ind}$-completion.