Fused Mackey functors

Abstract

Let $G$ be a finite group. In [HTW], Hambleton, Taylor and Williams have considered the question of comparing Mackey functors for $G$ and biset functors defined on subgroups of $G$ and bifree bisets as morphisms. This paper proposes a different approach to this problem, from the point of view of various categories of $G$-sets. In particular, the category of fused $G$-sets is introduced, as well its category of spans. The fused Mackey functors for $G$ over a commutative ring $R$ are defined as $R$-linear functors from this ($R$-linearized) category of spans to $R$-modules. They form an abelian subcategory of the category of Mackey functors for $G$ over $R$, equivalent (for $R=Z$) to the category to the category of conjugation Mackey functors of [HTW]. The category of fused Mackey functors is also equivalent to the category of modules over the fused Mackey algebra, which is a quotient of the usual Mackey algebra of $G$ over $R$. Reference: [HTW] I. Hambleton, L. R. Taylor, and E. B. Williams. Mackey functors and bisets. Geom. Dedicata, 148:157--174, 2010.