Operadic (co)limits

Definition 18.1.1. Let \(\Oo \) be an \(\infty \)-operad and let \(F\colon I \to \Oo _{\lra {1}}\) be a functor.

(1)

A cone \(\eta \colon \const _y \to F\) in \(\Oo _{\lra {1}}\) is called an operadic limit if for every \(n \geq 0\) and all colors \(x_1, \dots , x_n \in \Oo ^{\simeq }\) the induced map \[ \Oo ((x_1, \dots , x_n); y) \to \lim _{i \in I} \Oo ((x_1, \dots , x_n); F(i)) \] is an equivalence of animae.

(2)

A cocone \(\epsilon \colon F \to \const _x\) in \(\Oo _{\lra {1}}\) is called an operadic colimit if for every \(n \geq 1\) and all colors \(x_2, \dots , x_n, y \in \Oo ^{\simeq }\) the induced map \[ \Oo ((x, x_2, \dots , x_n); y) \to \lim _{i \in I} \Oo ((F(i), x_2, \dots , x_n); y) \] is an equivalence of animae.

Remark 18.1.2. These conditions are direct operadic analogues of the mapping-anima characterizations of ordinary limits and colimits, with unary mapping animae replaced by multimorphism animae. Taking \(n=1\) shows that an operadic limit is a limit in \(\Oo _{\lra {1}}\), and an operadic colimit is a colimit in \(\Oo _{\lra {1}}\). We will therefore usually assume that the relevant (co)limits already exist in \(\Oo _{\lra {1}}\) and only ask whether they are operadic.

For limits, there is also a useful reformulation through the envelope. Every object of \(\Env (\Oo )\) is a tensor \(x_1 \otimes \dots \otimes x_n\) of colors, and by Lemma 17.3.17 the multimorphism animae compute the mapping animae out of such tensors, \(\Oo ((x_1, \dots , x_n); y) \simeq \Hom _{\Env (\Oo )}(x_1 \otimes \dots \otimes x_n, y)\). Testing against all colors \(x_1, \dots , x_n\) therefore shows that a cone in \(\Oo _{\lra {1}}\) is an operadic limit if and only if it becomes a limit cone in \(\Env (\Oo )\) under the inclusion \(\Oo _{\lra {1}} \hookrightarrow \Env (\Oo )\).

For colimits, the corresponding statement is only relative: after tensoring with any fixed colors \(x_2,\dots ,x_n\), the cocone is colimiting when tested by mapping into colors of \(\Oo \). It need not be a colimit cocone in the whole envelope; in particular, an operadic initial object need not be initial there. Compare Example 18.1.7.

Example 18.1.3. If \(\Oo = \Mm _C\) for a symmetric monoidal \(\infty \)-category \(C\), then any limit in \(C\) is automatically operadic. This is because the tensor product provides a natural equivalence \[ \Mm _C((x_1, \dots , x_n); -) \simeq \Hom _C(x_1 \otimes \dots \otimes x_n, -), \] and the functor \(\Hom _C(X,-)\) preserves limits for any object \(X\).

Example 18.1.4. The claim from the previous example does not hold for general operads. For instance, consider the trivial operad \(\Triv \). Its underlying \(\infty \)-category is the terminal category \(*\), which has all limits, and in particular a terminal object. However, for \(n \neq 1\), we have \(\Triv ((x_1, \dots , x_n); -) = \emptyset \), so this terminal object is not an operadic terminal object.

Example 18.1.5. If \(\Oo = \Mm _C\) for a symmetric monoidal \(\infty \)-category \(C\), then a colimit in \(C\) is operadic if and only if it is preserved by the functor \(x \otimes - \colon C \to C\) for every object \(x \in C\); the argument is similar to Example 18.1.3.

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.

Example 18.1.7 (Pointed objects). Let \(C\) satisfy the hypotheses of Lemma 16.3.4, and equip \(C_*\) with its smash product monoidal structure. Its zero object is both operadic terminal and operadic initial, so \(\Mm _{C_*}\) is a pointed \(\infty \)-operad. Indeed, operadic terminality is automatic by Example 18.1.3, while operadic initiality says precisely that tensoring with the zero object gives the zero object.

Lemma 18.1.8 (Properties inherited by full suboperads). Let \(\Oo \subseteq \Pp \) be a full suboperad.

(1)

If \(\Pp \) is pointed and \(\Oo _{\lra {1}}\) contains its zero object, then \(\Oo \) is pointed.

(2)

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)

If \(\Pp \) is stable and \(\Oo _{\lra {1}}\) is closed under finite limits and finite colimits in \(\Pp _{\lra {1}}\), then \(\Oo \) is stable.