The Drinfeld center of the category of Mackey functors (Q934081)

Let \(G\) be a finite group and \(k\) a commutative ring. Let \({\mathcal S}\) be the category of finite \(G\)-sets and \({\mathbb M}({\mathcal S})\) the category of Mackey functors for \(G\) over \(k\). Then \({\mathbb M}({\mathcal S})\) has a tensor category structure induced by the cartesian product in \({\mathcal S}\). The author proves that the Drinfeld center \({\mathbb Z}({\mathbb M}({\mathcal S}))\) of \({\mathbb M}({\mathcal S})\) is equivalent to the category \({\mathbb M}({\mathcal T}_{c*})\) of Mackey functors over the category \({\mathcal T}_{c*}\), whose objects are pairs \((X,a)\), where \(X\in {\mathcal S}\) and \(a:X\to X\) is an automorphism that leaves all \(G\)-orbits in \(X\) stable. Note that the category \({\mathcal T}_{c*}\) has finite sums and pullbacks, hence the category \({\mathbb M}({\mathcal T}_{c*})\) is defined. The tensor structure of \({\mathbb Z}({\mathbb M}({\mathcal S}))\) through the above equivalence will be described in another paper.