Actor of categories internal to groups

TL;DR

This paper explores the concept of actors within internal categories called group-groupoids, utilizing the Brown-Spencer theorem and Norrie's work to interpret actions, semi-direct products, and holomorphs of these structures.

Contribution

It provides a new interpretation of actors in group-groupoids and explicitly constructs their actions, semi-direct products, and holomorphs, extending existing theoretical frameworks.

Findings

Interpretation of actors in group-groupoids using Brown-Spencer theorem

Explicit construction of semi-direct products of group-groupoids

Development of holomorphs for group-groupoids

Abstract

In this study, using the Brown-Spencer theorem and in the ligth of the works of Norrie, in the category of internal categories within groups, also called group-groupoids, we interpret the notion of actor of a crossed module over groups. Further, we construct the action of a group-groupoid on a group-groupoid. Moreover, we give the explicit construction of semi-direct product of two group-groupoids and of holomorph of group-groupoids.