nLab
Mon(∞,1)Cat

Idea

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

Variants

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

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

Properties

Model category presentations

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

category: category