I would like to ask for possible references for the following very general situation, a categorified version of Mackey functors.


The question is if there are other known constructions to associate to any subgroup $H$ of $G$ a category $C(H)$ and for any $H\leq K \leq G$ pairs of adjoint functors $Ind\_H^K:C(H)\rightarrow C(K)$, $Res\_H^K:C(K)\rightarrow C(H)$ satisfying analogues axioms to those from group theory ( i.e., when $C(H)=Rep(H)$)?


There are some very nice papers by Ocha & al considering Clifford theory for Mackey functors. Next natural question is can one define vertices and sources for these categorified Mackey functors? What about Green' s theorem in this categorified version from the previous question, are there other known examples?


Thanks!



Categorified versions of Mackey's functor
Have a look at page 4 of <http://www.math.umn.edu/~webb/Publications/GuideToMF.ps> for a few examples of Mackey functors in different categories.


