Biset functors as module Mackey functors, and its relation to derivators

TL;DR

This paper establishes a categorical relationship between biset functors, Mackey functors, and derivators, showing that biset functors form a reflective subcategory of Mackey functors and linking them to modules over the Burnside functor.

Contribution

It demonstrates that biset functors form a reflective monoidal subcategory of Mackey functors and connects them to derivators via module categories over the Burnside functor.

Findings

Biset functors form a reflective subcategory of Mackey functors.

The category of biset functors is equivalent to modules over the Burnside functor.

Any derivator on finite categories can be associated with a biset functor.

Abstract

In this article, we will show that the category of biset functors can be regarded as a reflective monoidal subcategory of the category of Mackey functors on the 2-category of finite groupoids. This reflective subcategory is equivalent to the category of modules over the Burnside functor. As a consequence of the reflectivity, we can associate a biset functor to any derivator on the 2-category of finite categories.