Axiomatic representation theory of finite groups by way of groupoids

TL;DR

This paper unifies various approaches to the axiomatic representation theory of finite groups using finite groupoids, clarifying their relationships and providing a comprehensive conceptual framework.

Contribution

It offers a systematic, unified perspective on Mackey and biset functors through the use of finite groupoids, including new comparisons and insights.

Findings

Comparison of different notions of Mackey and biset functors

Unified conceptual framework using finite groupoids

Clarification of relationships among various approaches

Abstract

We survey several notions of Mackey functors and biset functors found in the literature and prove some old and new theorems comparing them. While little here will surprise the experts, we draw a conceptual and unified picture by making systematic use of finite groupoids. This provides a road map for the various approaches to the axiomatic representation theory of finite groups, as well as some details which are hard to find in writing.