Fused Mackey functors

Abstract

Let \(G\) be a finite group. In [6], the question of comparing Mackey functors for \(G\) and biset functors defined on subgroups of \(G\) and bifree bisets as morphisms has been considered. This paper proposes a different approach to this problem, from the point of view of various categories of \(G\)-sets. In particular, the category \(G\hbox {-}{\underline{\mathsf{set}}}\) of fused\(G\)-sets is introduced, as well as the category \(\underline{\mathbf {S}}(G)\) of spans in \(G\hbox {-}{\underline{\mathsf{set}}}\). The fused Mackey functors for \(G\) over a commutative ring \(R\) are defined as \(R\)-linear functors from \(R\,\underline{\mathbf {S}}(G)\) to \(R\)-modules. They form an abelian subcategory \(\mathsf {Mack}_R^f(G)\) of the category of Mackey functors for \(G\) over \(R\). The category \(\mathsf {Mack}_\mathbb {Z}^f(G)\) is equivalent to the category of conjugation Mackey functors of [6]. The category \(\mathsf {Mack}_R^f(G)\) is also equivalent to the category of modules over the fused Mackey algebra\(\mu _R^f(G)\), which is a quotient of the usual Mackey algebra \(\mu _R(G)\) of \(G\) over \(R\).