# Mackey 2-Functors and Mackey 2-Motives

### Paul Balmer

University of California, Los Angeles, USA### Ivo Dell'Ambrogio

Université de Lille, France

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This book is dedicated to equivariant mathematics, specifically the study of additive categories of objects with actions of finite groups. The framework of Mackey 2-functors axiomatizes the variance of such categories as a function of the group. In other words, it provides a categorification of the widely used notion of Mackey functor, familiar to representation theorists and topologists.

The book contains an extended catalogue of examples of such Mackey 2-functors that are already in use in many mathematical fields from algebra to topology, from geometry to KK-theory. Among the first results of the theory, the ambidexterity theorem gives a way to construct further examples and the separable monadicity theorem explains how the value of a Mackey 2-functor at a subgroup can be carved out of the value at a larger group, by a construction that generalizes ordinary localization in the same way that the étale topology generalizes the Zariski topology. The second part of the book provides a motivic approach to Mackey 2-functors, 2-categorifying the well-known span construction of Dress and Lindner. This motivic theory culminates with the following application: The idempotents of Yoshida’s crossed Burnside ring are the universal source of block decompositions.

The book is self-contained, with appendices providing extensive background and terminology. It is written for graduate students and more advanced researchers interested in category theory, representation theory and topology.