Higher-dimensional self-acting groups: From crossed modules to internal categories

In many algebraic settings, the categories of crossed modules, <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>cat</m:mi> <m:mn>1</m:mn> </m:msup> </m:math> {cat¹} -objects, and internal categories are known to be categorically equivalent. In this paper, we develop this correspondence in the category in self-acting groups. We first construct crossed modules, <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>cat</m:mi> <m:mn>1</m:mn> </m:msup> </m:math> {cat¹} -objects, and internal categories in this context and establish their fundamental structural properties. We then prove that the categories of these three types of objects in the category of self-acting groups are equivalent. This provides a unified categorical framework for interpreting higher-dimensional algebraic structures arising from self-actions.