ScalingStacks

A.8.3 \(\mathcal O\)-monoidal \(\infty\)-categories[00IS]

Given an \(\infty\)-operad \(\mathcal O\), an \(\mathcal O\)-monoidal \(\infty\)-category \(\mathcal C\) is a coCartesian fibration \(\mathcal C^{\otimes} \rightarrow\mathcal O^{\otimes}\) satisfying analogous Segal conditions, which are equivalent to the condition that the composite \(\mathcal C^{\otimes} \rightarrow\mathcal O^{\otimes}\rightarrow\mathrm{Fin}_*\) is also an \(\infty\)-operad. In particular, an \(\mathcal O\)-monoidal \(\infty\)-category can be equivalently specified by the straightening \(\mathcal O^{\otimes} \rightarrow\mathrm{Cat}_\infty\) of this coCartesian fibration. ⁵⁵ We often abuse the notation by denoting an \(\mathcal O\)-monoidal \(\infty\)-category by its source operad \(\mathcal C\). Altogether, \(\mathcal O\)-monoidal \(\infty\)-categories define a full subcategory \(\mathrm{Alg}_\mathcal O(\mathrm{Cat}_\infty) \subseteq {\textup{coCart}}_{\mathcal O^{\otimes}} \simeq \mathrm{Fun}(\mathcal O^{\otimes},\mathrm{Cat}_\infty)\). As special cases, we write \(\mathrm{Alg}(\mathrm{Cat}_\infty) \coloneqq \mathrm{Alg}_{\textup{Assoc}}(\mathrm{Cat}_\infty)\) for the \(\infty\)-category of monoidal \(\infty\)-categories and \(\mathrm{CAlg}(\mathrm{Cat}_\infty) \coloneqq \mathrm{Alg}_\text{Comm}(\mathrm{Cat}_\infty)\) for the \(\infty\)-category of symmetric monoidal \(\infty\)-categories. The restricted coCartesian fibration \(\underline{\mathcal C} \rightarrow\underline{\mathcal O}\) (or simply its source) may be thought of as the “underlying \(\infty\)-category” of \(\mathcal C\), although this is most immediately meaningful when \(\mathcal O\) is single-colored.

An \(\infty\)-category that admits finite products canonically upgrades to a Cartesian symmetric monoidal \(\infty\)-category. We note that it is merely a condition for a symmetric monoidal \(\infty\)-category to be Cartesian symmetric monoidal. Dual remarks apply in the case of finite coproducts.