Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Cohomological Mackey 2-functors

Paul Balmer, Ivo Dell'Ambrogio

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.

Cohomological Mackey 2-functors

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.

Paper Structure

This paper contains 7 sections, 22 theorems, 57 equations.

Table of Contents

  1. Introduction
  2. Recollections
  3. Hom-decategorification
  4. Mackey 2-motives via bisets
  5. Cohomological Mackey 2-functors
  6. Cohomological Mackey 2-motives
  7. Motivic decompositions

Key Result

Theorem 1.2

Let $\mathcal{M}\colon \mathsf{gpd}^{\mathrm{op}}\to \mathsf{ADD}$ be a Mackey 2-functor. Let $G$ be a finite group and let $X,Y\in \mathcal{M}(G)$ be two objects. Then there is an ordinary Mackey functor $M=M_{\mathcal{M},G,X,Y}$ for $G$ whose value on every subgroup $H\le G$ is given by the abelia

Theorems & Definitions (83)

  • Definition 1.1
  • Theorem 1.2: Hom-Decategorification
  • Theorem 1.4: Mackey 2-motives via bisets
  • Theorem 1.5: Cohomological Mackey 2-motives
  • Definition 2.3
  • Remark 2.10
  • Proposition 2.12: Linearization
  • proof
  • Definition 2.13
  • Definition 3.2: Mackey 1-functors
  • ...and 73 more