A symmetric $1$-coloured operad can be defined in any symmetric monoidal category. In particular, it can be defined for the cartesian monoidal category $\mathsf{Cat}$ of small categories and functors.
- Have such $\mathsf{Cat}$-valued operads and their algebras been studied (systematically) somewhere in the literature? Some texts I could find are Donald Yau's Infinity Operads and Monoidal Categories with Group Equivariance and Elmendorf's Operads for Symmetric Monoidal Categories.
- Is there a characterization of their algebras (like the one for non-symmetric $\mathsf{Set}$-valued operads given in Leinster's Higher Operads, Higher Categories: A monad on $\mathsf{Set}$ is operadic iff it is strongly regular)? Or more genrally, what categories-with-structure can be realized as algebras over an operad?
- Does such an operadic perspective on categories-with-structure yield interesting insights?