Let $R$ be an associative ring. Recall that a (left) $R$-module $M$ 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 $M$ 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. $\square$
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 (all conjugates of) 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 it is a finite set. See Example 9.2.5.9 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.
In this section, we study an $\infty $-categorical counterpart of Definition 9.2.0.2. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category which admits small filtered colimits. Assume for simplicity that $\operatorname{\mathcal{C}}$ is locally small, so that every object $C \in \operatorname{\mathcal{C}}$ corepresents a functor
We say that the object $C$ is compact if the functor $h^{C}$ preserves small filtered colimits (Definition 9.2.5.1). In §9.2.5, we introduce several variants of this condition and study their formal properties.
In practice, one can often characterize the compact objects of an ordinary category $\operatorname{\mathcal{C}}$ as those which can be built, using finite colimits, from very simple “generators” of $\operatorname{\mathcal{C}}$ (Remark 9.2.0.3). For this reason, many authors use the term finitely presentable object for what we refer to as a compact object. Beware that, in the setting of $\infty $-categories, this terminology can be misleading. The collection of compact objects of an $\infty $-category $\operatorname{\mathcal{C}}$ is closed under both finite colimits (Corollary 9.2.5.26) and the formation of retracts (Remark 9.2.5.18). In classical category theory, the latter is a special case of the former: any retract of an object $X \in \operatorname{\mathcal{C}}$ can be realized as the coequalizer of a pair of morphisms $(e, \operatorname{id}_{X}): X \rightarrow X$ (see Corollary 8.5.2.5). In the $\infty $-categorical setting, this is no longer true. Consequently, to construct a general compact object $C \in \operatorname{\mathcal{C}}$ out of “generators”, it is often necessary to allow both finite colimits and retracts. In §9.2.6, we study this phenomenon in the case where $\operatorname{\mathcal{C}}= \operatorname{\mathcal{S}}$ is the $\infty $-category of spaces. Let us say that a Kan complex $X$ is essentially finite if there exists a weak homotopy equivalence $K \rightarrow X$, where $K$ is a finite simplicial set (Definition 9.2.6.1). Then:
A Kan complex $X$ is essentially finite if and only if it it belongs to the smallest full subcategory $\operatorname{\mathcal{S}}_{\mathrm{fin}} \subseteq \operatorname{\mathcal{S}}$ which contains $\Delta ^0$ and is closed under finite colimits (Proposition 9.2.6.3).
Every essentially finite Kan complex is compact when viewed as an object of $\operatorname{\mathcal{S}}$ (Corollary 9.2.6.5).
In general, an object $X \in \operatorname{\mathcal{S}}$ is compact if and only if it can be realized as a retract of an essentially finite Kan complex (Proposition 9.2.6.7). If this condition is satisfied, we say that $X$ is finitely dominated (Definition 9.2.6.6). Beware that there exist finitely dominated Kan complexes which are not essentially finite: this is the subject of the classical Wall finiteness obstruction (see [MR0211402]).
All of these results have counterparts where we replace $\operatorname{\mathcal{S}}$ by the $\infty $-category $\operatorname{\mathcal{QC}}$ of (small) $\infty $-categories, which we formulate and prove in §9.2.7.
Our final goal in this section is to establish some stability properties of compactness under various categorical constructions. Suppose we are given a finite diagram of $\infty $-categories $\{ \operatorname{\mathcal{C}}_{\alpha } \} $, where each $\operatorname{\mathcal{C}}_{\alpha }$ admits small filtered colimits and each of the transition functors $\operatorname{\mathcal{C}}_{\alpha } \rightarrow \operatorname{\mathcal{C}}_{\beta }$ is finitary. In §9.2.9, we show that an object $C = \{ C_{\alpha } \} $ of the limit $\infty $-category $\varprojlim ( \operatorname{\mathcal{C}}_{\alpha } )$ is compact provided that each $C_{\alpha }$ is compact as an object of $\operatorname{\mathcal{C}}_{\alpha }$ (Corollary 9.2.9.2; beware that the converse is false in general). Using formal arguments, the proof reduces to the special case of diagrams of the form
in which case it can be regarded as an assertion about compact objects of the homotopy fiber product $\operatorname{\mathcal{C}}_{-} \times ^{\mathrm{h}}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{+}$. We deduce this assertion in §9.2.8 from a more general result about compact objects of an oriented fiber product $\operatorname{\mathcal{C}}_{-} \vec{\times }_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{+}$ (see Proposition 9.2.8.3 and 9.2.8.5), which has a number of other useful consequences.
Remark 9.2.0.7. With an eye toward future applications, it will be useful to formulate the definitions and results of this section in greater generality. Suppose we are given a pair of regular cardinals $\kappa \leq \lambda $. We will say that an $\infty $-category $\operatorname{\mathcal{C}}$ is $(\kappa ,\lambda )$-cocomplete if it admits colimits indexed by diagrams which are $\lambda $-small and $\kappa $-filtered (Definition 9.2.1.3). If this condition is satisfied, we say that a functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is $(\kappa ,\lambda )$-finitary if it preserves such colimits (Definition 9.2.2.6). We say that an object $C \in \operatorname{\mathcal{C}}$ is $(\kappa ,\lambda )$-compact if the corepresentable functor $\operatorname{Hom}_{\operatorname{\mathcal{C}}}( C, \bullet )$ is $(\kappa ,\lambda )$-finitary (Definition 9.2.5.12). Following the convention of Remark 4.9.0.4, we recover the usual notion of compact object by taking $\kappa = \aleph _0$ and $\lambda = \Omega $ to be a strongly inaccessible cardinal. However, there are some other special cases worthy of note: for example, an $\infty $-category $\operatorname{\mathcal{C}}$ is $(\aleph _0, \aleph _1)$-cocomplete if and only if it is sequentially cocomplete, in the sense of Definition 9.2.1.3 (see Corollary 9.2.3.8). In §9.2.3, we prove more generally that an $\infty $-category is $(\kappa , \kappa ^{+})$-cocomplete if and only if it admits colimits indexed by linearly ordered sets of order type $\kappa $ (and a functor is $(\kappa , \kappa ^{+})$-finitary if and only if it preserves such colimits); see Proposition 9.2.3.6.