Operadic limits and colimits

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.