Presentably symmetric monoidal categories

Definition 22.5.1. A symmetric monoidal \(\infty \)-category \((C,\otimes )\) is called presentably symmetric monoidal if \(C\) is presentable and the tensor product \(-\otimes -\colon C\times C\to C\) preserves small colimits separately in both variables.

Remark 22.5.2. The suboperad description in Section 22.4 identifies presentably symmetric monoidal \(\infty \)-categories with commutative algebra objects of \((\PrL ,\otimes )\). Indeed, a commutative algebra object is an operad map \[ \Comm \longrightarrow \Mm _{(\PrL ,\otimes )} \subseteq \OpCart _{\widehat {\Cat }_{\infty }}. \] After forgetting the displayed factorization, this is a symmetric monoidal \(\infty \)-category \((C,\otimes _C)\). Its underlying \(\infty \)-category is presentable, and each of its multimorphism functors \(C^n\to C\) preserves small colimits separately in every variable because the operad map lands in \(\Mm _{(\PrL ,\otimes )}\). In particular, its binary tensor product has the property required in Definition 22.5.1.

Conversely, let \((C,\otimes _C)\) be presentably symmetric monoidal. Its symmetric monoidal structure gives an operad map \(\Comm \to \OpCart _{\widehat {\Cat }_{\infty }}\). The binary tensor product preserves small colimits separately by assumption, and hence so do all iterated tensor products. The operad map therefore factors through \(\Mm _{(\PrL ,\otimes )}\). The same factorization criterion for morphisms identifies maps in \(\CAlg (\PrL )\) with colimit-preserving symmetric monoidal functors. This gives the claimed identification of \(\infty \)-categories.

Proposition 22.5.3. Every presentably symmetric monoidal \(\infty \)-category is closed: for every object \(X\in C\), the functor \(X\otimes -\colon C\to C\) admits a right adjoint \(\iHom _C(X,-)\).

Theorem 22.5.4 (Symmetric monoidal Ind-completion, [Lurie (2017), Corollary 4.8.1.14]). Let \(C\) be a small stable symmetric monoidal \(\infty \)-category, and assume that the tensor product on \(C\) is exact separately in both variables. Then \(\Ind (C)\) admits a unique symmetric monoidal structure with the following properties:

(1)

The Yoneda embedding \(C\hookrightarrow \Ind (C)\) is symmetric monoidal.

(2)

The tensor product on \(\Ind (C)\) preserves colimits separately in both variables.

For filtered-colimit presentations \(X\simeq \colim _iX_i\) and \(Y\simeq \colim _jY_j\) with \(X_i,Y_j\in C\), its underlying tensor product is characterized by \[ X\otimes Y\simeq \colim _{(i,j)}(X_i\otimes Y_j). \]

Proposition 22.5.5. The following statements hold.

(1)

If \(C\) is a small symmetric monoidal \(\infty \)-category and \(D\) is presentably symmetric monoidal, then the Day convolution structure on \(\Fun (C,D)\) is presentably symmetric monoidal.

(2)

If \(C\) is presentably symmetric monoidal and \(A\in \Alg (C)\), then \(\LMod _A(C)\) and \(\RMod _A(C)\) are presentable. Their forgetful functors to \(C\) create limits and small colimits.

Exercise 22.1 (Representables as compact generators). Let \(C\) be a small \(\infty \)-category.

(1)

Show that every representable presheaf \(Y(c)\in \PSh (C)\) is compact.

(2)

Show that the representables jointly detect isomorphisms.

(3)

Explain how the density presentation of a presheaf exhibits \(\PSh (C)\) as generated under colimits by the representables.

Exercise 22.2 (A presheaf localization). Let \(C\) be small and let \(S\) be a small collection of morphisms in \(C\). Regard each \(s\colon c\to d\) as a morphism \(Y(c)\to Y(d)\) of representable presheaves.

(1)

Show that a presheaf \(F\) is local with respect to these maps if and only if \(F(s)\colon F(d)\to F(c)\) is an equivalence for every \(s\in S\).

(2)

Identify the local presheaves with presheaves on the localization \(C[S^{-1}]\).

(3)

Conclude that this full subcategory is presentable.

Exercise 22.3 (An Ind-completion). Show that every set is a filtered colimit of its finite subsets and that finite sets are compact in \(\Set \). Prove that \[ \Ind (\Fin )\simeq \Set . \] Explain why the analogous statement for animae requires finite animae rather than finite sets.

Exercise 22.4 (Tensor products of presheaf categories). Let \(C\) and \(D\) be small \(\infty \)-categories. Prove that there is a natural equivalence \[ \PSh (C)\otimes \PSh (D)\simeq \PSh (C\times D) \] in \(\Pr ^L\).