Proposition 16.4.1. Let \(C\) be a presentably symmetric monoidal \(\infty \)-category. The inclusions \[ \CGrp (C)\hookrightarrow \CMon (C)\hookrightarrow \Fun (\Span (\Fin ),C) \] admit left adjoints. Moreover, the full subcategories \(\CMon (C)\) and \(\CGrp (C)\) inherit symmetric monoidal structures for which these left adjoints are symmetric monoidal Bousfield localizations of the Day convolution monoidal structure. Their tensor products preserve small colimits separately in both variables.
The free commutative monoid functor \[ F_C^{\CMon }\colon C\longrightarrow \CMon (C), \] left adjoint to evaluation at \(\lra {1}\in \Span (\Fin )\), is symmetric monoidal.
Proof. The existence of the two left adjoints is the structured-object case of presentability recorded in Proposition 22.2.7.
The category \(\Span (\Fin )\) is small, so Corollary 16.2.8, Proposition 22.5.5 equip \(\Fun (\Span (\Fin ),C)\) with its presentably symmetric monoidal Day convolution structure. It remains to apply the internal-hom criterion of Lemma 14.5.6. By Proposition 22.5.3, the category \(C\) is closed, and the Day convolution internal hom is computed by the end formula \[ \iHom _{\Day }(F,G)(S) \simeq \int _{T\in \Span (\Fin )} \iHom _C\bigl (F(T),G(S\times T)\bigr ). \] For every finite set \(T\), the functor \(-\times T\) preserves finite coproducts of finite sets, which are the finite products in \(\Span (\Fin )\). Consequently, if \(G\) preserves finite products, then so does the functor \(S\mapsto G(S\times T)\). Internal homs and ends preserve limits, so \(\iHom _{\Day }(F,G)\) again lies in \(\CMon (C)\). If \(G\) is grouplike, the shear map of \(\iHom _{\Day }(F,G)\) is obtained by applying the limit-preserving functor \(\iHom _{\Day }(F,-)\) to the shear isomorphism of \(G\), so \(\iHom _{\Day }(F,G)\) is grouplike as well. Thus both local subcategories are closed under internal hom with arbitrary \(F\), and Lemma 14.5.6 gives the claimed symmetric monoidal Bousfield localizations.
Part (3) of Proposition 14.5.3 shows that the localized tensor products preserve small colimits separately in both variables.
It remains to consider the free commutative monoid functor. Let \(\iota \colon *\to \Span (\Fin )\) select the monoidal unit \(\lra {1}\) for the cartesian product of finite sets. Left Kan extension along \(\iota \) is left adjoint to evaluation at \(\lra {1}\), \[ \iota _!\colon C\simeq \Fun (*,C)\rightleftarrows \Fun (\Span (\Fin ),C)\noloc \iota ^*. \] The functor \(\iota \) is symmetric monoidal, and Lemma 16.2.11 shows that \(\iota _!\) is symmetric monoidal. The composite of \(\iota _!\) with the symmetric monoidal reflection onto \(\CMon (C)\) is left adjoint to evaluation at \(\lra {1}\) on commutative monoids, hence is \(F_C^{\CMon }\). It is therefore symmetric monoidal. □
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