Monoidal Dwyer–Kan localizations and derived categories

In Section 14.5 we treated localizations that are reflective: the local objects form a full subcategory of \(C\), and the monoidal structure is transported along the reflector. Derived categories are not presented in this way. The \(\infty \)-category \(\D (R)\) arises from the \(1\)-category \(\Ch (R)\) of chain complexes by inverting the quasi-isomorphisms, which is a Dwyer–Kan localization, and the reflective criteria do not apply to it. Supplying the missing monoidal statement is the purpose of this chapter.

The difficulty is that a symmetric monoidal structure on \(C\) is not extra structure on \(C\) alone, but a functor \(C^{\otimes } \to \Span (\Fin )\), so inverting a class \(W\) of morphisms of \(C\) is not by itself enough to say what should happen to the tensor product. The remedy is to invert morphisms upstairs as well: we localize \(C^{\otimes }\) at a class \(W^{\otimes }\) built from \(W\), and give conditions under which the result is again a symmetric monoidal \(\infty \)-category with the expected universal property.

Section 20.1 carries this out. The case where tensoring with any object preserves \(W\) is straightforward (Proposition 20.1.1); the substantial statement is Theorem 20.1.6, which only requires the monoidal structure to be left derivable, and which follows [Nikolaus and Scholze (2018), Theorem A.7]. In Section 20.2 we apply it to unbounded chain complexes, obtaining both the derived tensor product on \(\D (R)\) and a lax symmetric monoidal refinement of the localization \(\Ch (R) \to \D (R)\). We show that this tensor product preserves colimits, and record the compact presentation of \(\D (R)\) needed for the comparison with Eilenberg–MacLane modules in Corollary 8.3.4.

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