Proposition 16.6.4. The lax symmetric monoidal functors \[ \Sp \xrightarrow {\Omega ^{\infty }} \CGrp (\An ) \hookrightarrow \CMon (\An ) \xrightarrow {\fgt } \An _* \xrightarrow {\fgt } \An \] admit strongly monoidal left adjoints: \[ (\An ,\times ) \xrightarrow {(-)_+} (\An _*,\wedge ) \xrightarrow {F^{\CMon }} (\CMon (\An ),\otimes ) \xrightarrow {(-)^{\grp }} (\CGrp (\An ),\otimes ) \xhookrightarrow {\bB ^{\infty }} (\Sp ,\otimes ). \] Here, \((-)_+\) adjoins a disjoint basepoint, \(F^{\CMon }\) is the free commutative monoid functor, \((-)^{\grp }\) is group completion, and \(\bB ^{\infty }\) is infinite delooping.
Proof. The functor \[ (\An ,\times )\longrightarrow \bigl (\Ar (\An ),\square \bigr ), \qquad X\longmapsto (\emptyset \to X), \] is symmetric monoidal: the pushout-product formula identifies its tensor comparison with \(\emptyset \to X\times Y\), and its unit is \(\emptyset \to *\). Its composite with the symmetric monoidal cofiber functor of Lemma 16.3.4 is \((-)_+\), so adjoining a basepoint is strongly monoidal.
The same conclusion for \(F^{\CMon }\) follows from the bilinear universal property of Lemma 16.4.3. Indeed, for pointed animae \(X,Y\) and a commutative monoid \(M\), there are natural equivalences \[ \begin {aligned} \Hom _{\CMon (\An )}(F^{\CMon }(X)\otimes F^{\CMon }(Y),M) &\simeq \Nat _{\Span (\Fin )^2} \bigl (F^{\CMon }(X)(-)\times F^{\CMon }(Y)(-),M(-\times -)\bigr ) \\ &\simeq \Hom _{\An _*}(X\wedge Y,\fgt (M)) \\ &\simeq \Hom _{\CMon (\An )}(F^{\CMon }(X\wedge Y),M). \end {aligned} \] The middle equivalence is obtained by applying the free commutative monoid adjunction in each variable; the functors \(S\mapsto M(S\times T)\) remain commutative monoids because \(-\times T\) preserves the finite products of \(\Span (\Fin )\). The analogous nullary argument identifies the monoidal units. The group completion functor is strongly monoidal because it is the symmetric monoidal Bousfield localization of Proposition 16.4.1.
It remains to consider infinite delooping. Write \(F^{\CGrp }:=(-)^{\grp }F^{\CMon }\). The objects \(F^{\CGrp }(X)\) generate \(\CGrp (\An )\) under colimits, and infinite delooping preserves colimits because it identifies \(\CGrp (\An )\) with the full subcategory \(\Sp _{\geq 0}\subseteq \Sp \) of connective spectra. Under this identification, \[ \bB ^\infty F^{\CGrp }(X)\simeq \Sigma ^\infty X. \] Here \(\Sigma ^\infty X\) denotes the reduced suspension spectrum of the pointed anima \(X\). Both functors \[ (G,H)\longmapsto \bB ^\infty (G\otimes H) \qquadtext {and}\qquad (G,H)\longmapsto \bB ^\infty G\otimes \bB ^\infty H \] preserve colimits separately in \(G\) and \(H\). On free generators, the canonical comparison is the strong monoidal comparison for reduced suspension spectra. This follows from Proposition 16.6.2: using the cofiber description of reduced suspension spectra from Remark 4.4.3, the pushout-product formula identifies this comparison with the isomorphism \[ \Sigma ^\infty X\otimes \Sigma ^\infty Y \simeq \Sigma ^\infty (X\wedge Y). \] The comparison is therefore an equivalence for all \(G\) and \(H\). The same argument treats the unit and the coherence maps, proving that \(\bB ^\infty \) is strongly monoidal. Finally, the right adjoints acquire the displayed lax symmetric monoidal structures by Proposition 14.3.6. □
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