Examples of symmetric monoidal structures

Example 8.1.2. The following are examples of symmetric monoidal \(\infty \)-categories:

(1)

If \(C\) is an \(\infty \)-category and \(D\) is symmetric monoidal, then \(\Fun (C,D)\) admits the pointwise symmetric monoidal structure, with tensor product and unit computed pointwise.

(2)

An \(\infty \)-category \(C\) with finite products admits the cartesian monoidal structure \((C,\times ,*)\). Dually, finite coproducts give the cocartesian monoidal structure \((C,\amalg ,\emptyset )\). These structures are constructed in Section 15.3, Section 15.2.

(3)

Let \((C,\otimes ,\unit )\) be a symmetric monoidal \(\infty \)-category and let \(D\subseteq C\) be a full subcategory which contains the monoidal unit and is closed under tensor products. Then \(D\) inherits a symmetric monoidal structure for which the inclusion \(D\hookrightarrow C\) is symmetric monoidal. See Lemma 14.5.2 for the construction.

(4)

If \(C\) and \(D\) are symmetric monoidal, \(C\) is small, and \(D\) is cocomplete with tensor product preserving colimits in both variables, then \(\Fun (C,D)\) also admits the Day convolution symmetric monoidal structure. We refer to Chapter 16, especially Proposition 16.2.7, Corollary 16.2.8, for its universal property and construction.

(5)

The \(\infty \)-category \(\Sp \) of spectra admits a symmetric monoidal structure \(\otimes \) satisfying the property that for every spectrum \(X\), the functor \(X \otimes -\colon \Sp \to \Sp \) preserves colimits. Furthermore, \(\Sp \) is universal with this structure: if \(D\) is another symmetric monoidal stable \(\infty \)-category which admits colimits and in which the tensor product preserves colimits in both variables, then the \(\infty \)-category of colimit-preserving symmetric monoidal functors \(\Sp \to D\) is contractible. We construct the symmetric monoidal structure in Chapter 16, and discuss its universal property in Section 18.5.

(6)

The \(\infty \)-categories \(\An _*\), \(\CMon (\An )\) and \(\CGrp (\An )\) similarly admit universal symmetric monoidal structures for pointed, semiadditive, and additive \(\infty \)-categories. In particular, the functors \[ \An \xrightarrow {(-)_+} \An _* \to \CMon (\An ) \xrightarrow {(-)^{\grp }} \CGrp (\An ) \xhookrightarrow {\bB ^{\infty }} \Sp \] are colimit-preserving and symmetric monoidal; see Proposition 16.6.4.

(7)

Let \(R\) be a commutative ring. Then the derived \(\infty \)-category of \(R\), defined as the localization \[ \D (R) := \Ch (R)[\{\text {quasi-isomorphisms}\}^{-1}] \] of the category of chain complexes of \(R\)-modules at the quasi-isomorphisms, admits a symmetric monoidal structure \((\D (R),\otimes ,R[0])\) whose monoidal unit is the chain complex \(R[0]\) given by putting \(R\) in degree \(0\). The tensor product corresponds to the derived tensor product of chain complexes. A construction of this symmetric monoidal structure is given in Section 20.2.

(8)

Recall the full subcategory \(\Cat _1 \subseteq \Cat _{\infty }\) from Example 1.8.14(2) spanned by the 1-categories. A commutative monoid in \(\Cat _1\) is precisely that of a symmetric monoidal category as defined classically. Since the inclusion \(\Cat _1 \hookrightarrow \Cat _{\infty }\) preserves products, this implies that any symmetric monoidal category defines a symmetric monoidal \(\infty \)-category.