A Monadic Interpretation of Categorical Mackey Functors

We study a monadic version of categorical Mackey functors proposed by Bonventre which we call $\hat{\Sigma_G} \wr (-)$ algebras or $\Sigma$GAs. These are algebras over the monad $\hat{\Sigma_G} \wr (-)$ in categories fibered over $Fin^G$ satisfying an additivity condition. The monad operation encode genuine commutative operations, which we can also interpret as transfers. These have several conditions we can strengthen or weaken, offering substantial flexibility. By varying the conditions on these algebras, we show we can recover the permutative Mackey functors of Bohmann-Osorno and the symmetric monoidal Mackey functors of Hill-Hopkins. In the process we construct a convenient strict (2,1)-category of spans. We also define G-commutative monoids in a $\Sigma$GA.