Proposition 16.4.2 (Commutative monoids and groups under adjunctions). Let \[ L\colon C\rightleftarrows D\noloc R \] be an adjunction between presentably symmetric monoidal \(\infty \)-categories, and assume that \(L\) is symmetric monoidal. Postcomposition with \(R\) induces functors on commutative monoids and groups which admit symmetric monoidal left adjoints: \[ \widetilde L\colon \CMon (C)\rightleftarrows \CMon (D)\noloc R_* \qquad \text {and}\qquad \widetilde L\colon \CGrp (C)\rightleftarrows \CGrp (D)\noloc R_*. \] If \(L\) preserves finite products, then the two functors \(\widetilde L\) are given by postcomposition with \(L\).
Proof. The right adjoint \(R\) preserves finite products, so postcomposition with \(R\) preserves commutative monoids. It also preserves grouplike commutative monoids, since it carries shear isomorphisms to shear isomorphisms. This gives the two functors denoted by \(R_*\) in the statement.
Let \(E_C\) and \(E_D\) denote either the commutative-monoid categories or the commutative-group categories of \(C\) and \(D\), respectively. Write \(Q_C\) and \(Q_D\) for the corresponding reflections of Proposition 16.4.1, and \(i_C\) and \(i_D\) for their fully faithful right adjoints. Postcomposition gives an adjunction \[ L^{\Span }\colon \Fun (\Span (\Fin ),C) \rightleftarrows \Fun (\Span (\Fin ),D)\noloc R^{\Span }. \] The left adjoint of \(R_*\) is the composite \[ \widetilde L:=Q_DL^{\Span }i_C. \] The adjunctions \(Q_D\dashv i_D\) and \(L^{\Span }\dashv R^{\Span }\), together with the full faithfulness of \(i_C\), immediately give \(\widetilde L\dashv R_*\).
Since \(L\) is a left adjoint, it preserves small colimits, so Proposition 16.2.10 shows that \(L^{\Span }\) is symmetric monoidal for the ambient Day convolution structures. Moreover, the two functors \[ \widetilde LQ_C \qquad \text {and}\qquad Q_DL^{\Span } \] are left adjoint to the same functor \(R^{\Span }i_D=i_CR_*\). They are therefore naturally isomorphic. Since \(Q_DL^{\Span }\) is symmetric monoidal, the universal property of the symmetric monoidal Bousfield localization \(Q_C\) supplies \(\widetilde L\) with a unique symmetric monoidal refinement. Its right adjoint \(R_*\) acquires the canonical lax symmetric monoidal structure of Proposition 14.3.6.
If \(L\) preserves finite products, postcomposition with \(L\) preserves commutative monoids and groups. The object \(L^{\Span }i_C(A)\) is then already local for every \(A\), so the localization map \[ L^{\Span }i_C(A)\longrightarrow i_DQ_DL^{\Span }i_C(A) \] is an isomorphism. Thus \(\widetilde L\) is given by postcomposition with \(L\), as claimed. □
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