Presentably symmetric monoidal categories as commutative algebras

Remark 22.5.2. The suboperad description in Section 22.4 identifies presentably symmetric monoidal \(\infty \)-categories with commutative algebra objects of \((\PrL ,\otimes )\). Indeed, a commutative algebra object is an operad map \[ \Comm \longrightarrow \Mm _{(\PrL ,\otimes )} \subseteq \OpCart _{\widehat {\Cat }_{\infty }}. \] After forgetting the displayed factorization, this is a symmetric monoidal \(\infty \)-category \((C,\otimes _C)\). Its underlying \(\infty \)-category is presentable, and each of its multimorphism functors \(C^n\to C\) preserves small colimits separately in every variable because the operad map lands in \(\Mm _{(\PrL ,\otimes )}\). In particular, its binary tensor product has the property required in Definition 22.5.1.

Conversely, let \((C,\otimes _C)\) be presentably symmetric monoidal. Its symmetric monoidal structure gives an operad map \(\Comm \to \OpCart _{\widehat {\Cat }_{\infty }}\). The binary tensor product preserves small colimits separately by assumption, and hence so do all iterated tensor products. The operad map therefore factors through \(\Mm _{(\PrL ,\otimes )}\). The same factorization criterion for morphisms identifies maps in \(\CAlg (\PrL )\) with colimit-preserving symmetric monoidal functors. This gives the claimed identification of \(\infty \)-categories.