Monoidal localization of derivable functors

Proof of Theorem 20.1.6. Let us write \[ q\colon C^{\otimes }[(W^{\otimes })^{-1}] \to \Span (\Fin ) \] for the functor from Construction 20.1.5. By Proposition 20.1.15, the cocartesian fibration \(p_C\colon C^{\otimes } \to \Span (\Fin )\) is left derivable with respect to the classes \(\{W^I\}_{I \in \Fin }\) on its fibers. In particular, it follows from Proposition 20.1.14 that \(q\) is a cocartesian fibration. Moreover, for every finite set \(I\), the induced map on fibers \[ C_I^{\otimes } = C^I \to C^{\otimes }[(W^{\otimes })^{-1}]_I \] exhibits the target as the localization \(C^I[(W^I)^{-1}]\). Using Exercise 1.5.24, we obtain natural equivalences \[ C^{\otimes }[(W^{\otimes })^{-1}]_I \simeq C^I[(W^I)^{-1}] \simeq (C[W^{-1}])^I. \] For \(i\in I\), let \[ \rho _i\colon I\hookleftarrow \{i\}\xrightarrow {=}\{i\} \] be the corresponding inert span. Cocartesian transport along \(\rho _i\) in \(C^\otimes \) is the projection \(\pr _i\colon C^I\to C\), which preserves componentwise weak equivalences. Its total left derived functor is therefore the induced projection \[ C^I[(W^I)^{-1}]\longrightarrow C[W^{-1}]. \] Part (3) of Proposition 20.1.14 identifies this with cocartesian transport along \(\rho _i\) for \(q\). Consequently, under the equivalences above, the Segal map for the straightening of \(q\) is the equivalence \[ C^I[(W^I)^{-1}]\iso (C[W^{-1}])^I. \] The same statement for \(I=\emptyset \) identifies the fiber with the terminal category. Thus the cocartesian straightening of \(q\) preserves finite products, and Lemma 14.2.4 shows that \(q\) defines a symmetric monoidal \(\infty \)-category. This proves (1).

For (2), the structure transformation for each projection is invertible because the projection itself preserves weak equivalences. Thus \(\gamma ^\otimes \) carries cocartesian lifts of the spans \(\rho _i\) to cocartesian lifts. By part (iii) of Proposition 14.1.9, these lifts exhibit the product decompositions in the source and target total categories. It follows that \(\gamma ^\otimes \) preserves finite products. Thus it is a morphism of \(\infty \)-operads, in the sense of Definition 14.1.1, and hence encodes a lax symmetric monoidal functor. This proves (2).

For (3), we evaluate part (2) of Proposition 20.1.14 on the one-point set \(\lra {1}\). This identifies the underlying \(\infty \)-category of the symmetric monoidal \(\infty \)-category from (1) with the localization \(C[W^{-1}]\), and under this equivalence the restriction of \(\gamma ^{\otimes }\) to the fiber over \(\lra {1}\) is precisely the localization functor \(\gamma \colon C \to C[W^{-1}]\). Thus the symmetric monoidal structure from (1) may be regarded as a symmetric monoidal structure on \(C[W^{-1}]\) itself.

Finally, let \(D\) be a symmetric monoidal \(\infty \)-category. Since \(\gamma ^{\otimes }\colon C^{\otimes } \to C^{\otimes }[(W^{\otimes })^{-1}]\) is the localization of the total category at \(W^{\otimes }\), precomposition with \(\gamma ^{\otimes }\) induces a fully faithful functor \[ \Fun _{/\Span (\Fin )}(C^{\otimes }[(W^{\otimes })^{-1}], D^{\otimes }) \hookrightarrow \Fun _{/\Span (\Fin )}(C^{\otimes }, D^{\otimes }) \] whose essential image consists of those functors over \(\Span (\Fin )\) that invert the morphisms in \(W^{\otimes }\). In the sense of Definition 14.1.1, the lax symmetric monoidal functors are precisely the morphisms of \(\infty \)-operads, i.e. the functors over \(\Span (\Fin )\) that preserve finite products. Since \(\gamma ^{\otimes }\) preserves finite products, precomposition with \(\gamma ^{\otimes }\) certainly sends lax symmetric monoidal functors to lax symmetric monoidal functors. Conversely, let \(F\colon C^{\otimes }[(W^{\otimes })^{-1}] \to D^{\otimes }\) be a functor over \(\Span (\Fin )\) such that \(F \circ \gamma ^{\otimes }\) preserves finite products. By part (2) of Proposition 20.1.14, the functor \(\gamma ^{\otimes }\) is essentially surjective on each fiber, hence essentially surjective on the total category. Since it also preserves finite products, it follows that \(F\) preserves finite products as well. We conclude that the displayed fully faithful functor restricts to a fully faithful functor \[ \Fun ^{\otimes \textup {-lax}}(C[W^{-1}],D) \hookrightarrow \Fun ^{\otimes \textup {-lax}}(C,D) \] whose essential image consists of those lax symmetric monoidal functors that invert the morphisms in \(W\). Indeed, for a lax symmetric monoidal functor, inverting \(W\) on the fiber over \(\lra {1}\) is equivalent to inverting \(W^{\otimes }\) on every fiber, since the induced functor on the fiber over \(I\) is the product of the underlying functor on the fiber over \(\lra {1}\). This proves (4). □