Operads with general groups of equivariance, and some 2-categorical aspects of operads in Cat

Understand

We give a definition of an operad with general groups of equivariance suitable for use in any symmetric monoidal category with appropriate colimits.

• We then apply this notion to study the 2-category of algebras over an operad in Cat.
• We show that any operad is finitary, that an operad is cartesian if and only if the group actions are nearly free (in a precise fashion), and that the existence of a pseudo-commutative structure largely depends on the groups of equivariance.
• We conclude by showing that the operad for strict braided monoidal categories has two canonical pseudo-commutative structures.