Cohomological Mackey 2-functors

TL;DR

This paper establishes a new categorical framework linking Mackey 2-motives with blocks of representation theory by introducing cohomological relations, thereby categorifying Yoshida's Theorem for cohomological Mackey functors.

Contribution

It introduces a quotient bicategory of Mackey 2-motives that categorifies Yoshida's Theorem and connects Mackey 2-motives to classical representation theory blocks.

Findings

Bicategory of finite groupoids and permutation bimodules is a quotient of Mackey 2-motives.

Categorification of Yoshida's Theorem for cohomological Mackey functors.

Provides a direct link between Mackey 2-motives and representation theory blocks.

Abstract

We show that the bicategory of finite groupoids and right-free permutation bimodules is a quotient of the bicategory of Mackey 2-motives introduced in arXiv:1808.04902, obtained by modding out the so-called cohomological relations. This categorifies Yoshida's Theorem for ordinary cohomological Mackey functors, and provides a direct connection between Mackey 2-motives and the usual blocks of representation theory.