Lemma 18.1.8 (Properties inherited by full suboperads). Let \(\Oo \subseteq \Pp \) be a full suboperad.
- (1)
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If \(\Pp \) is pointed and \(\Oo _{\lra {1}}\) contains its zero object, then \(\Oo \) is pointed.
- (2)
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If \(\Pp \) is semiadditive or additive and \(\Oo _{\lra {1}}\) is closed under finite biproducts in \(\Pp _{\lra {1}}\), then \(\Oo \) is semiadditive or additive, respectively.
- (3)
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If \(\Pp \) is stable and \(\Oo _{\lra {1}}\) is closed under finite limits and finite colimits in \(\Pp _{\lra {1}}\), then \(\Oo \) is stable.
Proof. In each case the relevant (co)limits in \(\Oo _{\lra {1}}\) are inherited from \(\Pp _{\lra {1}}\). This gives the asserted property of the underlying \(\infty \)-category. Since \(\Pp \) has the corresponding operadic (co)limits and \(\Oo \subseteq \Pp \) is full, the same multimorphism animae compute the operadic (co)limit conditions in \(\Oo \). □
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