Categorical Green functors arising from group actions on categories

TL;DR

This paper introduces categorical Mackey functors derived from group actions on categories, enabling the construction of new Mackey and Green functors via K-theory and monoidal autoequivalences.

Contribution

It defines categorical Mackey functors and demonstrates how they produce new Mackey and Green functors from group actions on categories.

Findings

Categorical Mackey functors generalize classical Mackey functors.

K-theory of abelian categories yields new Mackey functors.

Green functor structures emerge from monoidal autoequivalence actions.

Abstract

In this paper we introduce the notion of a categorical Mackey functor. This categorical notion allows us to obtain new Mackey functors by passing to Quillen's $K$-theory of the corresponding abelian categories. In the case of an action by monoidal autoequivalences on a monoidal category the Mackey functor obtained at the level of Grothendieck rings has in fact a Green functor structure.