Centers, Commutators, and Holomorphs of 2-Groups

Abstract

This paper explores actor structures in group-groupoids, using the Brown-Spencer theorem to establish actors as universal objects and split extension classifiers. We construct the center and commutator subgroup of a group-groupoid, revealing their roles in internal symmetries, and introduce the holomorph as a categorical generalization of the classical holomorph of a group. These results extend group-theoretic concepts to 2-groups, bridging algebra and topology. By connecting actor theory with split extensions and intrinsic algebraic properties, we provide new tools for analyzing symmetries and automorphisms in higher-dimensional algebraic structures.

Keywords

• 2-group
• action
• actor
• center
• holomorph

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