Tensor-stable monoidal localization

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. □