Mackey 2-Functors and Mackey 2-Motives

Paul Balmer, Ivo Dell’Ambrogio
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引用次数: 18

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 $G$.
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麦基二函子和麦基二动机
我们研究了加性范畴$\mathcal{M}(G)$的集合,它们由有限群$G$索引,并通过归纳和限制联系起来,以一种对通常的麦基函子进行分类的方式。我们称它们为麦基二函子。我们提供了大量的例子,特别是由于加性衍生。证明了Mackey 2泛函子的第一性质,包括子群的可分离单性。然后,我们分离出最初的这种结构,从而得出我们所谓的“麦基二动机”。我们还利用弦图展示了麦基二动机中态射的一个方便的演算。最后,我们证明了$G$的2-范畴的单位的2-自同态环与$G$的所谓交叉Burnside环是同构的。
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