Green 2-functors

TL;DR

This paper introduces Green 2-functors, extending Mackey 2-functors, and demonstrates how classical Green functors are decategorifications of these higher structures, enriching the theoretical framework.

Contribution

It defines Green 2-functors, provides foundational results and examples, and connects them to classical Green functors through decategorification.

Findings

Green 2-functors generalize Mackey 2-functors.

Classical Green functors arise from Green 2-functors via decategorification.

The paper offers numerous examples illustrating the theory.

Abstract

We extend the theory of Mackey 2-functors introduced in arXiv:1808.04902 by defining the appropriate notion of rings, namely Green 2-functors. After providing the first results of our theory and abundant examples, we show how all classical Green functors familiar from representation theory and topology arise by decategorification, in various ways, of some Green 2-functor occurring in Nature.