We first make precise what it means to impose a small collection of relations on a presheaf category. Let \(E\) be an \(\infty \)-category and let \(S\) be a collection of morphisms in \(E\). An object \(X\in E\) is called \(S\)-local if for every morphism \(f\colon A\to B\) in \(S\), precomposition with \(f\) induces an equivalence \[ \Hom _E(B,X)\iso \Hom _E(A,X). \] We write \(E^{S\text {-}\mathrm {loc}}\subseteq E\) for the full subcategory of \(S\)-local objects.
Definition 22.2.1. An \(\infty \)-category \(D\) is called presentable if there exist a small \(\infty \)-category \(C\) and a small collection \(S\) of morphisms in \(\PSh (C)\) such that the inclusion \[ \PSh (C)^{S\text {-}\mathrm {loc}}\hookrightarrow \PSh (C) \] admits a left adjoint and \(D\) is equivalent to \(\PSh (C)^{S\text {-}\mathrm {loc}}\).
Thus a presentation of \(D\) consists of a small collection of generators, encoded by \(C\), and a small collection of relations, encoded by \(S\). The presentation is not part of the structure on \(D\), and is very far from unique. The existence of the reflector may in fact be omitted from the definition: any small collection of morphisms in a presheaf category generates a Bousfield localization. We include it to keep the definition directly tied to the localization formalism of Section 21.8.
Theorem 22.2.2 (Basic constructions on presentable categories, [Lurie (2009), Theorem 5.5.1.1 and Propositions 5.5.3.6, 5.5.3.10--5.5.3.11 and 5.5.4.15]). The following statements hold.
- (1)
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For every small \(\infty \)-category \(C\), the presheaf category \(\PSh (C)\) is presentable. In particular, \(\An \simeq \PSh (*)\) is presentable.
- (2)
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If \(C\) is presentable and \(K\) is small, then the functor category \(\Fun (K,C)\) is presentable.
- (3)
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If \(C\) is presentable and \(X\in C\), then the slice categories \(C_{/X}\) and \(C_{X/}\) are presentable.
- (4)
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If \(C\) is presentable and \(S\) is a small collection of morphisms in \(C\), then the full subcategory \(C^{S\text {-}\mathrm {loc}}\) is presentable and its inclusion into \(C\) admits a left adjoint.
We will also use the following immediate consequence of the definition: every presentable \(\infty \)-category \(C\) admits a small collection \(\Gg \) of objects such that the functors \(\Hom _C(G,-)\), for \(G\in \Gg \), jointly detect isomorphisms. Indeed, in any presentation of \(C\) as a localization of a presheaf category, one may take the images under the reflector of the representable presheaves. The adjunction and the Yoneda lemma then reduce detection to the fact that isomorphisms of presheaves are detected objectwise.
We write \(\PrL \) for the \(\infty \)-category whose objects are presentable \(\infty \)-categories and whose morphisms are the colimit-preserving functors. By the adjoint functor theorem below, these are precisely the left adjoint functors. Similarly, we write \(\PrR \) for the \(\infty \)-category of presentable \(\infty \)-categories and right adjoint functors. Passing between left and right adjoints gives an equivalence \[ (\PrL )\catop \simeq \PrR . \] Their objects are large \(\infty \)-categories, so all statements about \(\PrL \) and \(\PrR \) are interpreted one universe higher, in the enlarged universe \(\widehat {\Cat }_{\infty }\) from Remark 1.8.1.
Theorem 22.2.3 (Limits of presentable categories, [Lurie (2009), Proposition 5.5.3.13 and Theorem 5.5.3.18]). The \(\infty \)-categories \(\PrL \) and \(\PrR \) admit small limits. Moreover, the inclusion functors \[ \PrL \longrightarrow \widehat {\Cat }_{\infty } \qquad \text {and}\qquad \PrR \longrightarrow \widehat {\Cat }_{\infty } \] preserve small limits.
Corollary 22.2.4 (Presentability of kernels). Let \(F\colon C\to D\) be a colimit-preserving functor between pointed presentable \(\infty \)-categories. Then the full subcategory \[ \ker (F):=\{X\in C\mid F(X)\simeq 0\}\subseteq C \] is presentable, and its inclusion into \(C\) preserves small colimits.
Proof. The functor \(*\to D\) selecting the zero object is a left adjoint, since the zero object is initial. The pullback
may therefore be formed in \(\PrL \). By Theorem 22.2.3, its underlying \(\infty \)-category is the pullback in \(\widehat {\Cat }_{\infty }\), which identifies it with the displayed full subcategory. The projection \(\ker (F)\to C\) is a morphism in \(\PrL \), so it preserves small colimits. □
The main reason presentable categories are useful is that functors out of them have adjoints as soon as they preserve colimits. This is the form of the adjoint functor theorem used throughout the book.
Theorem 22.2.5 (Adjoint functor theorem, [Lurie (2009), Corollary 5.5.2.9]). Let \(C\) and \(D\) be presentable \(\infty \)-categories. A functor \(F\colon C\to D\) admits a right adjoint if and only if it preserves small colimits.
There is a dual-looking criterion for the existence of a left adjoint, but preservation of limits by itself is not enough: one must also impose a size condition on the functor. We will not need its general formulation. The special cases involving structured objects that occur later are included in Proposition 22.2.7 below.
Corollary 22.2.6 (Representability, [Lurie (2009), Proposition 5.5.2.2]). Let \(C\) be a presentable \(\infty \)-category. A functor \(F\colon C\catop \to \An \) is representable if and only if it preserves small limits.
Proposition 22.2.7 (Presentability of structured objects). Let \(C\) be a presentable \(\infty \)-category.
- (1)
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The \(\infty \)-categories \(\CMon (C)\) and \(\CGrp (C)\) are presentable, and the inclusions \[ \CGrp (C)\hookrightarrow \CMon (C)\hookrightarrow \Fun (\Span (\Fin ),C) \] admit left adjoints.
- (2)
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If \(C\) is stable and \(\Gg \) is a small collection of objects of \(C\), then the smallest full stable subcategory \(\Loc (\Gg )\subseteq C\) that contains \(\Gg \) and is closed under small colimits is presentable. Its inclusion into \(C\) preserves small colimits and consequently admits a right adjoint.
Proof. For (1), choose a small collection \(\Gg \) of objects of \(C\) whose corepresentable functors jointly detect isomorphisms. Evaluation at a finite set \(S\) has a left adjoint \[ \gamma _S\colon C\longrightarrow \Fun (\Span (\Fin ),C). \] Localizing the presentable functor category \(\Fun (\Span (\Fin ),C)\) at the maps \[ \emptyset \longrightarrow \gamma _{\emptyset }(G) \qquad \text {and}\qquad \gamma _S(G)\sqcup \gamma _T(G)\longrightarrow \gamma _{S\sqcup T}(G) \] for \(G\in \Gg \) cuts out precisely the product-preserving functors, hence \(\CMon (C)\). This is an instance of the accessible-localization theorem [Lurie (2009), Proposition 5.5.4.15]. Localizing in addition at the endomorphisms induced by the shear \(\left (\begin {smallmatrix}1&0\\1&1\end {smallmatrix}\right )\) cuts out precisely the grouplike objects. This proves the assertion for \(\CGrp (C)\) and gives both reflectors.
For (2), apply [Lurie (2017), Proposition 1.4.4.11(2)] to the small collection of all shifts \(G[n]\) with \(G\in \Gg \) and \(n\in \Z \). The resulting presentable subcategory is \(\Loc (\Gg )\). Its inclusion preserves small colimits by definition, so it admits a right adjoint by Theorem 22.2.5. □
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