Proposition 20.2.2 (The symmetric monoidal derived category). Let \(R\) be a commutative ring. The derived \(\infty \)-category \[ \D (R):=\Ch (R)[\{\textup {quasi-isomorphisms}\}^{-1}] \] admits a canonical symmetric monoidal structure whose tensor product is the derived tensor product and whose unit is \(R[0]\). The localization functor \[ \gamma \colon \Ch (R)\longrightarrow \D (R) \] admits a canonical lax symmetric monoidal refinement. Moreover, the derived tensor product preserves small colimits separately in both variables.
Proof. Equip \(\Ch (R)\) with the weak equivalences and projective cofibrations of Proposition 20.2.1. Properties (2) and (3) of that proposition show that the ordinary tensor product is left derivable. The first two assertions therefore follow from Theorem 20.1.6. The description of cocartesian transport in part (3) of Proposition 20.1.14 identifies tensor product in the localization with the total left derived functor of ordinary tensor product, so this is the derived tensor product. Applying the same description to the unique span from \(\emptyset \) to \(\lra {1}\) identifies the monoidal unit with the image of the ordinary unit \(R[0]\).
It remains to prove the assertion about colimits. By part (1) of Corollary 2.2.9, we may represent an object \(X\in \D (R)\) by a cofibrant complex \(P\). For a complex \(Y\), choose a cofibrant replacement \(Q\xrightarrow {\sim }Y\). The preceding identification and part (3) of Proposition 20.2.1 give \[ X\otimes _R^{\bL }\gamma (Y) \simeq \gamma (P\otimes _R Q) \simeq \gamma (P\otimes _R Y). \] Thus \(X\otimes _R^{\bL }-\) is the functor on localizations induced by \(P\otimes _R-\). The latter is homotopy cocontinuous by part (3) of Proposition 20.2.1. The dual form of Theorem 2.2.11 therefore shows that \(X\otimes _R^{\bL }-\) preserves small colimits. By symmetry, the same holds in the other variable. □
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