Mackey 2-functors and Mackey 2-motives

TL;DR

This paper introduces Mackey 2-functors and Mackey 2-motives, categorifying classical Mackey functors, and explores their properties, examples, and connections to crossed Burnside rings using string diagram calculus.

Contribution

It defines Mackey 2-functors and Mackey 2-motives, establishing foundational properties and providing a new categorical framework with examples and computational tools.

Findings

Separable monadicity of restriction to subgroups

Identification of Mackey 2-motives with initial structure

Isomorphism between endomorphism ring and crossed Burnside ring

Abstract

We study collections of additive categories $\mathcal{M}(G)$, indexed by finite groups $G$ and related by induction and restriction in a way that categorifies usual Mackey functors. We call them `Mackey 2-functors'. We provide a large collection of examples in particular thanks to additive derivators. We prove the first properties of Mackey 2-functors, including separable monadicity of restriction to subgroups. We then isolate the initial such structure, leading to what we call `Mackey 2-motives'. We also exhibit a convenient calculus of morphisms in Mackey 2-motives, by means of string diagrams. Finally, we show that the 2-endomorphism ring of the identity of $G$ in this 2-category of Mackey 2-motives is isomorphic to the so-called crossed Burnside ring of $G$.