# What is the categorical operad whose algebras are the (based) symmetric monoidal categories?

What is the categorical operad whose algebras are the (based) symmetric monoidal categories?

Note that the definition of symmetric monoidal category can be found here. https://ncatlab.org/nlab/show/symmetric+monoidal+category#:~:text=A%20symmetric%20monoidal%20category%20is,categorical%20products%20may%20be%20commutative.

A symmetric monoidal category is based if the unit is strict.

I tried defining it myself but have had no luck. Can someone please tell me or point me in the right direction?

• Sorry what do you mean by based symmetric monoidal categories? – trujello Jun 30 at 1:30
• See this, ncatlab.org/nlab/show/…. – Johnathon Taylor Jun 30 at 3:04
• Also, based means the unit is strict. – Johnathon Taylor Jun 30 at 3:04
• I'm not quite sure what you mean by the unit being strict (i.e. perhaps $I \otimes A = A$) but one reference you might consult is Infinity Operads and Monoidal Categories with Group Equivariance. In that text Donald Yau outlines a general procedure for constructing all kinds of categorical operads that are in a coherent sense acted upon by a "group $G$ action operad". Then the algebras over those categorical operads "acted on" by a $G$-action operad give you braided, symmetric, ribbon monoidal categories by varying $G = B_n$ (braids), $G = S_n$, etc (I'm paraphrasing a lot here) – trujello Jun 30 at 3:32
• That is exactly what I mean by strict. – Johnathon Taylor Jun 30 at 3:54

• The algebras are actually symmetric monoidal $\infty$-categories. – Johnathon Taylor Jul 2 at 20:48