On the regular representation of an (essentially) finite 2-group

TL;DR

This paper explores the regular representation of finite 2-groups within 2-vector spaces, computes cohomology invariants, and establishes a key equivalence with functor categories, advancing the understanding of 2-group representations.

Contribution

It defines the regular representation of an essentially finite 2-group in 2-vector spaces and proves its properties, including cohomology invariants and a fundamental equivalence with functor categories.

Findings

Hom-categories in the representation are 2-vector spaces.

A formula for intertwining numbers is derived, showing symmetry.

The regular representation is a representing object for the forgetful 2-functor.

Abstract

The regular representation of an essentially finite 2-group $\mathbb{G}$ in the 2-category $\mathbf{2Vect}_k$ of (Kapranov and Voevodsky) 2-vector spaces is defined and cohomology invariants classifying it computed. It is next shown that all hom-categories in $\mathbf{Rep}_{\mathbf{2Vect}_k}(\mathbb{G})$ are 2-vector spaces under quite standard assumptions on the field $k$, and a formula giving the corresponding "intertwining numbers" is obtained which proves they are symmetric. Finally, it is shown that the forgetful 2-functor ${\boldmath$\omega$}:\mathbf{Rep}_{\mathbf{2Vect}_k}(\mathbb{G})\To\mathbf{2Vect}_k$ is representable with the regular representation as representing object. As a consequence we obtain a $k$-linear equivalence between the 2-vector space $\mathbf{Vect}_k^{\mathcal{G}}$ of functors from the underlying groupoid of $\mathbb{G}$ to $\mathbf{Vect}_k$, on the one hand, and the $k$-linear category $\mathcal{E} nd({\boldmath$\omega$})$ of pseudonatural endomorphisms of ${\boldmath$\omega$}$, on the other hand. We conclude that $\mathcal{E} nd({\boldmath$\omega$})$ is a 2-vector space, and we (partially) describe a basis of it.