nLab Mon(∞,1)Cat

Idea

The (∞,2)-category of monoidal (∞,1)-categories.

Variants

$\mathcal{O}$-monoidal $(\infty,1)$-category

A monoidal (∞,1)-category $(C, \otimes)$ is equivalently a coCartesian fibration of (∞,1)-operads over Assoc.

A symmetric monoidal (∞,1)-category $(C, \otimes)$ is equivalently a coCartesian fibration of (∞,1)-operads over Comm.

Accordingly, for $\mathcal{O}$ any (∞,1)-operad, a coCartesian fibration of $(\infty,1)$-operads over $\mathcal{O}$ may be called an $\mathcal{O}$-monoidal $(\infty,1)$-category.

$$O Mon(\infty,1)Cat := coCart_{\mathcal{O}} \,.$$

Properties

Model category presentations

see table - models for (infinity,1)-operads

category: category