Theorem 18.4.12 (Additivization of operads, [Nikolaus (2016), Theorem 5.7]). Let \(\Oo \) be an object in \(\Op _{\infty }^*\) or \(\Op _{\infty }^{\mathrm {prod}}\) respectively. Then the operads \(\Oo _*\), \(\oCMon (\Oo )\) and \(\oCGrp (\Oo )\) are pointed, semiadditive, and additive, respectively. Moreover, the resulting functors \[ (-)_*\colon \Op _{\infty }^* \to \Op _{\infty }^{\pt }, \qquad \oCMon \colon \Op _{\infty }^{\mathrm {prod}} \to \Op _{\infty }^{\sadd } \qquadtext { and } \oCGrp \colon \Op _{\infty }^{\mathrm {prod}} \to \Op _{\infty }^{\add } \] are right adjoint to the respective inclusion functors.
Proof. We first prove the pointed case. Choose a universe in which \(\Env (\Oo )\) is small and let \(P\) be the corresponding presheaf category. The multiplicative Yoneda embedding and Lemma 18.3.2 give full embeddings \[ \Oo _* \hookrightarrow (\Mm _{\Env (\Oo )})_* \hookrightarrow (\Mm _P)_* \] where the last target is understood in the chosen universe. By Lemma 18.4.10, this last operad is the multimorphism operad of the pointed presheaf category and is therefore pointed by Example 18.1.7. The suboperad \(\Oo _*\) contains its zero object, since the multiplicative Yoneda embedding preserves the operadic terminal object of \(\Oo \). It follows from Lemma 18.1.8(1) that \(\Oo _*\) is pointed.
Evaluation at the target gives a natural operad map \(\Oo _*\to \Oo \). If \(\Oo \) is pointed, this map is an equivalence by Lemma 18.4.11. Applying the pointed-object construction a second time gives the same conclusion because \(\Oo _*\) is pointed. The dual of Proposition 21.8.9, applied in \((\Op _{\infty }^*)\catop \), therefore identifies \((-)_*\) as right adjoint to the inclusion of pointed operads.
For commutative monoids and groups, choose a universe in which \(\Oo \) is small and apply the multiplicative Yoneda embedding to regard \(\Oo \) as a full suboperad of the presheaf operad on \(\Env (\Oo )\). In that presheaf category, Proposition 16.4.1 supplies the Day convolution operads of commutative monoids and commutative groups. Their underlying categories are semiadditive and additive by Proposition 5.3.20, and their tensor products preserve colimits separately in both variables. Hence their finite biproducts are operadic. These biproducts are computed pointwise. Since the multiplicative Yoneda image is closed under finite products, the full suboperads consisting of objects valued in that image are closed under these biproducts. They are precisely \(\oCMon (\Oo )\) and \(\oCGrp (\Oo )\), so Lemma 18.1.8(2) shows that the former is semiadditive and the latter additive.
If \(\Pp \) is semiadditive, every color of \(\Pp \) has a commutative-monoid structure supplied by its biproducts, compatibly with all multimorphisms because the biproducts are operadic. If \(\Pp \) is additive, these structures are grouplike. The cited theorem supplies the coherent lifts through \(\oCMon (\Oo )\to \Oo \) and \(\oCGrp (\Oo )\to \Oo \) and identifies them with the right adjoints to the respective inclusions. □
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