On the $3$-representations of groups and the $2$-categorical Traces

Abstract

To $2$-categorify the theory of group representations, we introduce the notions of the $3$-representation of a group in a strict $3$-category and the strict $2$-categorical action of a group on a strict $2$-category. We also $2$-categorify the concept of the trace by introducing the $2$-categorical trace of a $1$-endomorphism in a strict $3$-category. For a $3$-representation $\rho$ of a group $G$ and an element $f$ of $G$, the $2$-categorical trace $\mathbb{T}{r}_2{\rho }_f$ is a category. Moreover, the centralizer of $f$ in $G$ acts categorically on this $2$-categorical trace. We construct the induced strict $2$-categorical action of a finite group, and show that the $2$-categorical trace $\mathbb{T}{r}_2$ takes an induced strict $2$-categorical action into an induced categorical action of the initia groupoid. As a corollary, we get the $3$-character formula of the induced strict $2$-categorical action.

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References in corpus (2)

• Higher-Dimensional Algebra V: 2-Groups
• On unitary 2-representations of finite groups and topological quantum field theory

Cited by in corpus (1)

• On the global $2$-holonomy for a $2$-connection on a $2$-bundle