Fused Mackey functors

Let G be a finite group. In [5], 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 G-set of fused G-sets is introduced, as well as the category S(G) of spans in G-set. The fused Mackey functors for G over a commutative ring R are defined as R-linear functors from RS(G) to R-modules. They form an abelian subcategory MackfR(G) of the category of Mackey functors for G over R. The category MackfZ(G) is equivalent to the category of conjugation Mackey functors of [5]. The categoryMackfR(G) is also equivalent to the category of modules over the fused Mackey algebra μfR(G), which is a quotient of the usual Mackey algebra μR(G) of G over R. AMS Subject classification : 18A25, 19A22, 20J15.