Symmetric monoidal ∞-categories

Definition 8.1.1. A symmetric monoidal \(\infty \)-category is defined to be a commutative monoid in \(\Cat _{\infty }\). In particular, it is an \(\infty \)-category \(C\) which comes equipped with a tensor product functor \[ - \otimes - \colon C \times C \to C \] and a monoidal unit \(\unit \in C\), and the tensor product is coherently unital, associative and commutative. We will often denote symmetric monoidal \(\infty \)-categories as \((C,\otimes ,\unit )\), or often also just as \(C\). Given symmetric monoidal \(\infty \)-categories \(C\) and \(D\), a symmetric monoidal functor \(F\colon C \to D\) is a morphism in \(\CMon (\Cat _{\infty })\). We write \(\Map ^{\otimes }(C,D)\) for the hom anima in \(\CMon (\Cat _{\infty })\).