Monoidal Dwyer–Kan localizations

Proof. We may first replace \(W\) by the smallest class of morphisms in \(C\) that contains \(W\) and is closed under isomorphisms, composition, and homotopy. This does not change the localization, since a functor out of \(C\) inverts the original class \(W\) if and only if it inverts this larger class. The enlarged class is still preserved by tensoring with an arbitrary object, since tensor products preserve isomorphisms, composition, and homotopies. Thus we may assume that \(W\) is a wide subcategory of \(C\).

For (1), consider the full subcategory \(\RelCat _{\infty } \subseteq \Ar (\Cat _{\infty })\) spanned by the relative \(\infty \)-categories: those functors \(W \hookrightarrow C\) which are simultaneously monomorphisms in \(\Cat _{\infty }\) and essentially surjective. (Such a choice of \(W\) corresponds to a wide subcategory of \(C\).) We will generally denote objects of \(\RelCat _{\infty }\) as pairs \((C,W)\).

Observe that there is a fully faithful inclusion functor \[ i_{\mathrm {rel}}\colon \Cat _{\infty } \hookrightarrow \RelCat _{\infty }, \qquad D \mapsto (D, D^{\simeq }). \] Observe that for a relative \(\infty \)-category \((C,W)\), the localization functor \(\gamma \colon C \to C[W^{-1}]\) refines to a morphism of relative \(\infty \)-categories \((C,W) \to i_{\mathrm {rel}}(C[W^{-1}])\). We claim that this map exhibits \(C[W^{-1}]\) as a left adjoint object to \((C,W)\) under \(i_{\mathrm {rel}}\). For every other \(\infty \)-category \(D\), precomposition with \(\gamma \) by definition induces a fully faithful inclusion of animae \[ \Hom _{\Cat _{\infty }}(C[W^{-1}],D) \to \Hom _{\Cat _{\infty }}(C,D) \] whose image consists of the functors \(C \to D\) that send \(W\) to isomorphisms. This image is precisely the image of \(\Hom _{\RelCat _{\infty }}((C,W), i_{\mathrm {rel}}(D))\), proving the adjointness.

By the pointwise criterion for left adjoints (Lemma 21.1.4), the assignment \((C,W) \mapsto C[W^{-1}]\) assembles into a functor \(\Ll \colon \RelCat _{\infty } \to \Cat _{\infty }\). Moreover, by Exercise 1.5.24, this functor preserves finite products, hence induces a functor on commutative monoids: \[ \Ll \colon \CMon (\RelCat _{\infty }) \to \CMon (\Cat _{\infty }). \] Since also the right adjoint \(i_{\mathrm {rel}}\) preserves finite products, the entire adjunction lifts to commutative monoids. Unwinding definitions, the condition on the pair \((C,W)\) in the statement of the proposition precisely guarantees that the symmetric monoidal structure on \(C\) refines to an object \((C,W) \in \CMon (\RelCat _{\infty })\). The unit of the adjunction now takes the form of a symmetric monoidal functor \(\gamma \colon C \to C[W^{-1}]\) refining the localization functor.

The adjunction on commutative monoids gives the claimed universal property on mapping animae. Applying the same argument with the pointwise symmetric monoidal categories \(\Fun ([n],D)\) for every \(n\geq 0\), as in Definition 14.2.3, gives the asserted fully faithful functor between symmetric monoidal functor \(\infty \)-categories. The uniqueness statement follows immediately. □

Proof sketch. We refer to [Nikolaus and Scholze (2018), Proposition A.14] for the proof and only indicate its structure. By Example 20.1.13, one may first restrict to the bases \([0]\) and \([1]\). The result is immediate over \([0]\). Over \([1]\), the cocartesian fibration is classified by a functor \(F\colon C_0\to C_1\). Its absolute left derived functor unstraightens to a cocartesian fibration with the desired fibers. The substantive point is that absoluteness identifies the mapping simplex of this derived functor with the localization of the mapping simplex of \(F\), by the universal property of right Kan extension and the description in Example 23.2.6.

Finally, the total category of a cocartesian fibration commutes with colimits in the base, and localization commutes with colimits because it is a left adjoint on relative \(\infty \)-categories. The class of bases for which the proposition holds is therefore closed under colimits. Since \(\Cat _{\infty }\) is generated under colimits by \([0]\) and \([1]\) by Proposition 24.1.14, this proves the result for arbitrary \(S\). □

Proof. Fix a span \(\alpha \colon I \xleftarrow {f} K \xrightarrow {g} J\) in \(\Span (\Fin )\). Under the identification \(C_I^{\otimes } \simeq C^I\), the cocartesian transport functor along \(\alpha \) is given by \[ \alpha _!\colon C^I \xrightarrow {f^*} C^K \xrightarrow {g_{\otimes }} C^J, \qquad (x_i)_{i \in I} \mapsto \left (\bigotimes _{k \in g^{-1}(j)} x_{f(k)}\right )_{j \in J}. \] The product category \(C^I\) again carries weak equivalences and cofibrations componentwise, by Lemma 6.6.7. It therefore suffices, by Proposition 6.6.3, to show that \(\alpha _!\) preserves componentwise weak equivalences between componentwise cofibrant objects.

This is clear for the restriction functor \(f^*\). For the functor \(g_{\otimes }\), let \(\{u_k\colon x_k \to y_k\}_{k \in K}\) be a componentwise weak equivalence between componentwise cofibrant tuples. Then every object \(y_k\) is in particular tensor-cofibrant, and the \(j\)-th component of \(g_{\otimes }(u)\) is the map \[ \bigotimes _{k \in g^{-1}(j)} x_k \xrightarrow {\bigotimes _{k \in g^{-1}(j)} u_k} \bigotimes _{k \in g^{-1}(j)} y_k, \] which is a weak equivalence by the left derivability assumption. Thus \(\alpha _!\) admits an absolute left derived functor \[ \bL \alpha _!\colon C^I[(W^I)^{-1}] \to C^J[(W^J)^{-1}]. \]

It remains to verify compatibility with composition. Consider composable spans \[ \alpha \colon I \xleftarrow {f} K \xrightarrow {g} J \qquadtext {and} \qquad \beta \colon J \xleftarrow {h} L \xrightarrow {k} M. \] We show that the canonical comparison map \[ \bL \beta _! \circ \bL \alpha _! \to \bL (\beta \circ \alpha )_! \] is an isomorphism. Let \(X=(X_i)_{i\in I}\) be componentwise cofibrant. Every component of \(\alpha _!(X)\) is a finite tensor product of cofibrant objects, and hence is tensor-cofibrant. Choose any componentwise weak equivalence \[ q\colon P\xrightarrow {\sim }\alpha _!(X) \] with \(P\) componentwise cofibrant. Such a map is obtained by factoring the morphism from the initial object to \(\alpha _!(X)\) in \(C^J\), and no functorial choice is required. By the defining property of total left derived functors, we have \[ (\bL \beta _!\circ \bL \alpha _!)(\gamma _I X) \simeq \bL \beta _!(\gamma _J\alpha _!(X)) \simeq \gamma _M\beta _!(P). \] The structure transformation for \(\bL \beta _!\) at \(\alpha _!(X)\) is represented under this identification by the map \[ \gamma _M\beta _!(q)\colon \gamma _M\beta _!(P)\longrightarrow \gamma _M\beta _!\alpha _!(X). \] This map is an isomorphism. Indeed, restriction along \(h\) preserves weak equivalences, and each component after applying \(k_{\otimes }\) is a tensor product of weak equivalences whose sources are cofibrant and whose targets are tensor-cofibrant; this is a weak equivalence by Definition 20.1.3. The composite structure transformation used to construct the comparison in Observation 20.1.11 is therefore an isomorphism at \(\gamma _I X\). The structure transformation for \(\bL (\beta \circ \alpha )_!\) is also an isomorphism there, since \(X\) is componentwise cofibrant. Hence the comparison map is an isomorphism on the image of every componentwise cofibrant object of \(C^I\). By part (1) of Corollary 2.2.9, every object of \(C^I[(W^I)^{-1}]\) is isomorphic to the image of such an object, so the comparison map is an isomorphism everywhere. We conclude that \(p_C\) is left derivable with respect to \(\{W^I\}_{I \in \Fin }\). □

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). □