Commutative algebras

Definition 8.1.3 (Commutative algebra). Let \((C,\otimes ,\unit )\) be a symmetric monoidal \(\infty \)-category. We define a commutative algebra in \(C\) to be a symmetric monoidal functor \[ A \colon \Fin \to C, \] where we equip the category \(\Fin \) of finite sets with the cocartesian monoidal structure; this functor packages the coherent \(n\)-fold multiplications described below. We refer to the object \(A(*)\) in \(C\) as the underlying object of \(A\). We will frequently abuse notation and simply write \(A\) for \(A(*)\).

We define the \(\infty \)-category of commutative algebras in \(C\) as \[ \CAlg (C) \quad := \quad \Fun ^{\otimes }(\Fin ,C). \]