Fused Mackey functors

TL;DR

This paper introduces fused Mackey functors for finite groups, providing a new categorical framework that relates to existing Mackey functors and algebraic structures, offering an alternative perspective and tools.

Contribution

It defines fused Mackey functors via categories of $G$-sets and spans, establishing their equivalence to modules over a quotient of the Mackey algebra and connecting to conjugation Mackey functors.

Findings

Fused Mackey functors form an abelian subcategory of Mackey functors.

They are equivalent to modules over the fused Mackey algebra.

The approach offers a new categorical perspective on 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.