Mackey 2-functors and Mackey 2-motives

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 $G$.

arXiv comment213 pages, many beautiful figures

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Cited by in corpus (8)

• Cohomological Mackey 2-functors
• Green equivalences in equivariant mathematics
• A categorification of the Cartan-Eilenberg formula
• Bi-incomplete Tambara functors as $\mathcal{O}$-commutative monoids
• Universality of Barwick's unfurling construction
• Parametrized stability and the universal property of global spectra
• An introduction to Mackey and Green 2-functors
• Partial parametrized presentability and the universal property of equivariant spectra