Corollary 18.5.6 (Multiplicative universal properties). Let \(D\) be a presentably symmetric monoidal \(\infty \)-category.
- (1)
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If \(D\) is pointed, then restriction along \(\An \to \An _*\) induces an equivalence \[ \Fun ^{\mathrm {L},\otimes }(\An _*,D)\iso \Fun ^{\mathrm {L},\otimes }(\An ,D). \]
- (2)
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If \(D\) is semiadditive, then restriction along \(\An \to \CMon (\An )\) induces an equivalence \[ \Fun ^{\mathrm {L},\otimes }(\CMon (\An ),D)\iso \Fun ^{\mathrm {L},\otimes }(\An ,D). \]
- (3)
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If \(D\) is additive, then restriction along \(\An \to \CGrp (\An )\) induces an equivalence \[ \Fun ^{\mathrm {L},\otimes }(\CGrp (\An ),D)\iso \Fun ^{\mathrm {L},\otimes }(\An ,D). \]
- (4)
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If \(D\) is stable, then restriction along \(\An \to \Sp \) induces an equivalence \[ \Fun ^{\mathrm {L},\otimes }(\Sp ,D)\iso \Fun ^{\mathrm {L},\otimes }(\An ,D). \]
Proof. Let \(M\) be any of the four modes in Theorem 18.5.3. If \(D\) is \(M\)-local, then the localization universal property of the idempotent commutative algebra \(M\) gives an equivalence \[ \Fun ^{\mathrm {L},\otimes }(M,D) \iso \Fun ^{\mathrm {L},\otimes }(\An ,D) \] by restriction along the unit \(\An \to M\). The four characterizations of \(M\)-local categories in Theorem 18.5.3 give the assertions. In the semiadditive case, this is also the specialization \(C=\An \) of Proposition 18.5.4. □
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