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On the 3-representations of groups and the 2-categorical traces

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Wei Wang

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.

Published 2015-12-31.

http://www.tac.mta.ca/tac/volumes/30/56/30-56.pdf

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