Pullback Crossed Modules in the Category of Racks

TL;DR

This paper introduces the concept of pullback crossed modules within the category of racks, extending the theory of crossed modules and demonstrating the preservation of these structures under the conjugation functor.

Contribution

It defines pullback crossed modules in racks and proves that the conjugation functor preserves these structures, bridging group and rack theories.

Findings

Pullback crossed modules are well-defined in the category of racks.

The conjugation functor preserves pullback crossed modules.

Establishes a connection between group and rack crossed modules.

Abstract

In this paper, we define the pullback crossed modules in the category of racks which mainly based on a pullback diagram of rack morphisms with extra crossed module data on some of its arrows. Furthermore we prove that the conjugation functor, which is defined between the category of crossed modules of groups and of racks, preserves the pullback crossed modules.