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

Abstract

The regular representation of an essentially finite 2-group $\mathbb{G}$ in the 2-category ${2Vect}_k$ of (Kapranov and Voevodsky) 2-vector spaces is defined and cohomology invariants classifying it computed. It is next shown that all hom-categories in ${\mathbf{Rep}}_{{2Vect}_k}(\mathbb{G})$ 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 ${\boldmath$ω$}:\mathbf{Rep}_{\mathbf{2Vect}_k}(\mathbb{G})\To\mathbf{2Vect}_k$ 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 $\mathcal{E} nd({\boldmath$ω$})$ of pseudonatural endomorphisms of ${\boldmath$ω$}$, on the other hand. We conclude that $\mathcal{E} nd({\boldmath$ω$})$ is a 2-vector space, and we (partially) describe a basis of it.

29 pages

References in corpus (3)

• Higher-Dimensional Algebra V: 2-Groups
• Categorical representations of categorical groups
• Infinite-Dimensional Representations of 2-Groups