Pointed semiadditive additive stable operads

Definition 18.1.6. An \(\infty \)-operad \(\Oo \) is called:

(1)

Pointed if \(\Oo _{\lra {1}}\) has a zero object which is both operadic initial and operadic terminal.

(2)

Semiadditive if \(\Oo _{\lra {1}}\) is semiadditive and all finite products and coproducts are operadic.

(3)

Additive if \(\Oo _{\lra {1}}\) is additive and all finite products and coproducts are operadic.

(4)

Stable if \(\Oo _{\lra {1}}\) is stable and all finite limits and colimits are operadic.

This allows us to define various subcategories of the \(\infty \)-category \(\Op _{\infty }\) of \(\infty \)-operads:

• We define \(\Op _{\infty }^* \subseteq \Op _{\infty }\) as the (non-full) subcategory spanned by \(\infty \)-operads with an operadic terminal object and maps preserving them. We denote by \(\Op _{\infty }^{\pt } \subseteq \Op _{\infty }^*\) the full subcategory spanned by pointed \(\infty \)-operads.
• We define \(\Op _{\infty }^{\mathrm {prod}} \subseteq \Op _{\infty }\) as the (non-full) subcategory spanned by \(\infty \)-operads with finite operadic products and maps preserving them. We denote by \(\Op _{\infty }^{\sadd }, \Op _{\infty }^{\add } \subseteq \Op _{\infty }^{\mathrm {prod}}\) the full subcategories spanned by semiadditive and additive \(\infty \)-operads, respectively.
• We define \(\Op _{\infty }^{\lex } \subseteq \Op _{\infty }\) as the (non-full) subcategory spanned by \(\infty \)-operads with finite operadic limits and maps preserving them. We denote by \(\Op _{\infty }^{\st } \subseteq \Op _{\infty }^{\lex }\) the full subcategory spanned by stable \(\infty \)-operads.