Categorified versions of Mackey's functor 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!
 A: 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.
A: To answer the question regarding vertices & co: Yes, vertices, sources and Green correspondence all work for categorified Mackey functors provided that the categories $M(H)$ are $\mathbb{Z}_{(p)}$-linear Krull-Schmidt categories (i.e. every object decomposes into indecomposables, all indecomposables have local endomorphism ring, and all idempotents split; $\mathbb{Z}_{(p)}$ is necessary in order for all vertices being $p$-subgroups).
One way to see this is to stare intensely at the usual proofs for Green correspondence until one realises that they show that restriction and induction induce a pair of equivalences between certain subquotients of the module categories. And then you can write the statement for your "Mackey 2-functors" and use the same proof with minor modifications.
More precisely: If $P\leq N_G(P)\leq H\leq G$, set $\mathfrak{X} := \{A\leq P\cap{^g P} | g\in G\setminus H\}$ and $\mathfrak{Y} := \{A\leq H\cap{^g P} | g\in G\setminus P\}$. Let $M^{\leq P}(U)$ be the full subcategory of $M(U)$ consisting of relatively $P$-projective objects (i.e. all indecomposable summands have a vertex $\leq P$) and $I_\mathfrak{X}$ the ideal generated by all relatively $\mathfrak{X}$-projective modules. Then the quotients $M^{\leq P}(H) / I_\mathfrak{X}$ and $M^{\leq P}(G) / I_\mathfrak{Y}$ are equivalent via restriction and induction.
