Mackey functors and bisets

TL;DR

This paper introduces a new bivariant functor linking finite G-sets to conjugation bisets, characterizes Mackey functors derived from this construction, and explores their properties.

Contribution

It defines a novel bivariant functor from G-sets to bisets and characterizes the Mackey functors obtainable through this framework.

Findings

Defines a bivariant functor from G-sets to conjugation bisets.

Characterizes Mackey functors arising from additive functors on the biset category.

Provides a new perspective on the structure of Mackey functors in relation to bisets.

Abstract

For any finite group G, we define a bivariant functor from the Dress category of finite G-sets to the conjugation biset category, whose objects are subgroups of G, and whose morphisms are generated by certain bifree bisets. Any additive functor from the conjugation biset category to abelian groups yields a Mackey functor by composition. We characterize the Mackey functors which arise in this way.