Normality and quotient in crossed modules, cat$^1$-groups and internal groupoids within groups with operations

TL;DR

This paper explores the relationships between normality and quotient structures in crossed modules, cat$^1$-groups, and internal groupoids within groups with operations, establishing equivalences and characterizations.

Contribution

It introduces the notions of normal and quotient crossed modules in groups with operations and links these concepts through categorical equivalences to internal groupoids and cat$^1$-groups.

Findings

Normal and quotient concepts are related in categories of crossed modules and internal groupoids.

An equivalence between crossed modules and cat$^1$-groups with operations is established.

Normal and quotient objects are characterized within the category of cat$^1$-groups with operations.

Abstract

In this paper we define the notions of normal subcrossed module and quotient crossed module within groups with operations; and using the equivalence of crossed modules over groups with operations and internal groupoids we prove how normality and quotient concepts are related in these two categories. Further we prove an equivalence of crossed modules over groups with operations and cat$^1$-groups with operations for a certain algebraic category; and then by this equivalence we determine normal and quotient objects in the category of cat$^{1}$-groups with operations. Finally we characterize the coverings of cat$^{1}$-groups with operations.