We now specialize to finiteness controlled by ordinary filtered colimits. This is sufficient for all applications in the book, so we will not need the more general notion of \(\kappa \)-compactness.
Definition 22.3.1 (Compact object). Let \(C\) be an \(\infty \)-category with filtered colimits. An object \(X\in C\) is called compact if the functor \(\Hom _C(X,-)\colon C\to \An \) preserves filtered colimits. We write \(C^\omega \subseteq C\) for the full subcategory of compact objects.
Corollary 22.3.2. If \(C\) is a stable \(\infty \)-category with filtered colimits, then \(C^\omega \) is a thick subcategory of \(C\).
Definition 22.3.3. An \(\infty \)-category \(C\) is called compactly generated if it admits small colimits, the subcategory \(C^\omega \) is small, and every object of \(C\) is a filtered colimit of compact objects.
If \(C\) is stable, a small collection \(\Gg \) of compact objects is called a collection of compact generators if the smallest localizing subcategory of \(C\) containing \(\Gg \) is \(C\) itself. Here a localizing subcategory is a full stable subcategory closed under small colimits.
Proposition 22.3.4 (Compact generation criterion, cf. Hovey et al. (1997), Corollary 2.3.12). Let \(C\) be a stable \(\infty \)-category with small colimits and let \(\Gg \) be a small collection of compact objects. The following conditions are equivalent:
- (1)
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The collection \(\Gg \) generates \(C\).
- (2)
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The functors \[ \Hom _C(G[n],-)\colon C\to \An , \qquad G\in \Gg ,\ n\in \Z , \] are jointly conservative.
- (3)
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Every object of \(C\) is a filtered colimit of objects in the smallest thick subcategory of \(C\) containing \(\Gg \).
If these conditions hold, then \(C\) is compactly generated and its compact objects form the smallest thick subcategory of \(C\) containing \(\Gg \).
Proof. See Reference ? of [Cisinski et al. (2026)]. □
The preceding statement is often the most convenient way to prove compact generation. Notice the distinction between the two closure operations: the generators produce all objects under small colimits, but the compact objects are obtained from the generators using only finite colimits, shifts, and retracts.
Definition 22.3.5 (Ind-completion). Let \(C\) be a small \(\infty \)-category. Its Ind-completion is the full subcategory \[ \Ind (C)\subseteq \PSh (C) \] spanned by the filtered colimits of representable presheaves. The Yoneda embedding factors through a fully faithful functor \(C\hookrightarrow \Ind (C)\).
Thus \(\Ind (C)\) is the partial free cocompletion from Proposition 22.1.2 obtained by freely adjoining filtered colimits. In particular, if \(D\) admits filtered colimits, restriction along the Yoneda embedding induces an equivalence \[ \Fun _\omega (\Ind (C),D)\iso \Fun (C,D), \] where the left-hand side denotes the full subcategory of functors preserving filtered colimits.
Proposition 22.3.6 ([Lurie (2009), Corollary 4.2.3.11, Propositions 5.3.2.9, 5.3.5.10 and 5.3.5.14, and Theorem 5.5.1.1]). If \(C\) is a small \(\infty \)-category with finite colimits, then \(\Ind (C)\) admits small colimits and is presentable. Moreover, the inclusion \(C\hookrightarrow \Ind (C)\) preserves finite colimits, and the inclusion \[ \Ind (C)\hookrightarrow \PSh (C) \] identifies \(\Ind (C)\) with the full subcategory spanned by the presheaves \(C\catop \to \An \) which send finite colimits in \(C\) to limits in \(\An \).
Proposition 22.3.7 (Compactly generated categories as Ind-completions, [Lurie (2009), Propositions 5.5.7.8 and 5.5.7.10]). Let \(D\) be an \(\infty \)-category with small colimits and assume that \(D^\omega \) is small. The following are equivalent:
- (1)
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Every object of \(D\) is a filtered colimit of compact objects.
- (2)
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The restricted Yoneda functor induces an equivalence \[ D\iso \Ind (D^\omega ). \]
Under these conditions, a colimit-preserving functor out of \(D\) is uniquely determined by its restriction to \(D^\omega \).
Remark 22.3.8 (Compact presentations). A presentable \(\infty \)-category \(D\) is compactly generated if and only if it admits a presentation \[ D\simeq \PSh (C)^{S\text {-}\mathrm {loc}} \] in which \(C\) is small and \(S\) is a small collection of morphisms between compact objects of \(\PSh (C)\). We will not use this characterization; see [Lurie (2009), Corollary 5.5.7.3 and Proposition 5.5.7.8].
Lemma 22.3.9 (Animae as Ind-completions). The restricted Yoneda functors induce equivalences \[ \An \iso \Ind (\An ^{\fin }) \qquad \text {and}\qquad \An _*\iso \Ind (\An _*^{\fin }). \]
Proof. The first equivalence follows immediately from Proposition 4.3.11, Proposition 22.3.6. For the second, Lemma 4.1.13 gives \[ \Fun ^{\lex }((\An _*^{\fin })\catop ,\An ) \iso \Fun ^{\lex }((\An _*^{\fin })\catop ,\An _*), \] and the right-hand side is equivalent to \(\An _*\) by Lemma 4.3.12. In both cases the inverse to evaluation is the restricted Yoneda functor, so the result follows from Proposition 22.3.6. □
Example 22.3.10. The categories \(\An \), \(\An _*\), and \(\Sp \) are compactly generated. For animae and pointed animae this follows from Lemma 22.3.9. The sphere spectrum is a compact generator of \(\Sp \): compactness follows because \(\Hom _{\Sp }(\S ,-)\simeq \Omega ^\infty \) preserves filtered colimits by Lemma 4.3.21, while its shifts jointly detect isomorphisms by Corollary 4.4.27; now apply Proposition 22.3.4.
For every associative ring spectrum \(A\), the free module \(A\) is a compact generator of \(\LMod _A\) by Lemma 8.1.8; for every commutative ring \(R\), the complex \(R[0]\) is a compact generator of \(\D (R)\) by Corollary 20.2.5. Consequently, \[ \Sp \simeq \Ind (\Sp ^\omega ), \qquad \LMod _A\simeq \Ind (\Perf (A)), \qquad \D (R)\simeq \Ind (\Perf (R)). \] The Eilenberg–MacLane comparison of Corollary 8.3.4 identifies the last presentation with the corresponding presentation of \(\Mod _{HR}\).
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