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 $Tr_2\rho_f$ 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 $Tr_2$ 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.
Keywords: the 3-representation of a group in a 3-category; the 2-categorical trace; the 3-cocycle condition; the induced strict 2-categorical actions; the 3-character; 2-categorification
2010 MSC: 18D05; 18D99; 20J99; 20C99
Theory and Applications of Categories, Vol. 30, 2015, No. 56, pp 1999-2047.