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Mackey 2-functors and Mackey 2-motives. (English) Zbl 1464.18021

EMS Monographs in Mathematics. Zürich: European Mathematical Society (EMS) (ISBN 978-3-03719-209-2/hbk; 978-3-03719-709-7/ebook). viii, 227 p. (2020).
Mackey functors for finite groups have been studied for a long time, by many people and in various contexts. The book under review takes this notion to a complete new level, the categorification of Mackey functors. This means to adapt the basic properties of Mackey functors to suitable families of additive categories parametrized by a finite group, so that such a Mackey 2-functor would assign an additive category (instead of an abelian group) to each finite group. In this setup, one also has additive functors and natural transformations between them, formalizing induction, restriction and the Mackey formula. Formally, a Mackey 2 -functor is a strict 2-functor \(\mathcal{M}\colon gpd^{op}\rightarrow ADD\) satisfying certain axioms, where \(gpd\) is the 2-category of finite groupoids and \(ADD\) is the 2-category of additive categories. For example, an axiom called Ambidexterity says that induction and coinduction are essentially the same. The authors also develop a motivic approach to their Mackey 2-functors, using a 2-category of so-called Mackey 2-motives which satisfies a universal property. The book has plenty of examples where Mackey 2-functors appear, including the theory of ordinary Mackey functors.

MSC:

18N40 Homotopical algebra, Quillen model categories, derivators
19A22 Frobenius induction, Burnside and representation rings
18B40 Groupoids, semigroupoids, semigroups, groups (viewed as categories)
18M30 String diagrams and graphical calculi
20J05 Homological methods in group theory
55P91 Equivariant homotopy theory in algebraic topology
18N10 2-categories, bicategories, double categories
18N25 Categorification