∞ operad

Definition 14.1.1. An \(\infty \)-operad (or: \(\Span (\Fin )\)-\(\infty \)-operad) is a pair \(\Oo = (\Oo ^{\otimes },p_{\Oo })\) consisting of an \(\infty \)-category \(\Oo ^{\otimes }\) equipped with a functor \(p_{\Oo }\colon \Oo ^{\otimes } \to \Span (\Fin )\) satisfying the following conditions:

(1)

The \(\infty \)-category \(\Oo ^{\otimes }\) admits finite products, and the functor \(p_{\Oo }\) preserves finite products;

(2)

For every finite set \(I\), the product in \(\Oo ^{\otimes }\) defines an equivalence \[ \prod _{i \in I} \Oo ^{\otimes }_{\{i\}} \iso \Oo ^{\otimes }_I, \] where \(\Oo ^{\otimes }_I\) denotes the fiber of \(p_{\Oo }\) over the set \(I\);

(3)

For a morphism \(f\colon J \to I\) in \(\Fin \) and objects \(X_i \in \Oo ^{\otimes }\), the map \(\widetilde {f}\colon \prod _{i\in I} X_i \to \prod _{j \in J} X_{f(j)}\) whose \(j\)-th component is the projection to \(X_{f(j)}\) is \(p_{\Oo }\)-cocartesian.

The objects \(X_i\) in condition (3) are not required to be colors; this more general form will be used in Proposition 14.1.9.

If \(\Pp = (\Pp ^{\otimes }, p_{\Pp })\) is another \(\infty \)-operad, then a morphism of \(\infty \)-operads \(f\colon \Oo \to \Pp \) is a commutative diagram

such that the functor \(f^{\otimes }\) preserves finite products.

If we write \(\Cat ^{\mathrm {prod}}_{\infty }\) for the (very large) \(\infty \)-category of \(\infty \)-categories that admit finite products and functors that preserve finite products, we obtain a full subcategory \[ \Op _{\infty } \quad \subseteq \quad (\Cat ^{\mathrm {prod}}_{\infty })_{/\Span (\Fin )} \] spanned by the \(\infty \)-operads. Given two \(\infty \)-operads \(\Oo \) and \(\Pp \), we will denote by \(\Fun _{\Op _{\infty }}(\Oo ,\Pp )\) the \(\infty \)-category of morphisms of \(\infty \)-operads from \(\Oo \) to \(\Pp \), defined as the following fiber:

In certain contexts, morphisms of \(\infty \)-operads \(\Oo \to \Pp \) are also known as \(\Oo \)-algebras in \(\Pp \), and a common alternative notation for the \(\infty \)-category \(\Fun _{\Op _{\infty }}(\Oo ,\Pp )\) is \(\Alg _{\Oo }(\Pp )\).